Beispiel #1
0
def auw(T):
	""" 
	Area under water
	Triangle: T
	Waterlevel assumed to be at 0
	o = Point(-1, -1)
	p = Point(1, -1)
	q = Point(0, 1)
	T = Triangle(o, p, q)
	vuw(T)
	r = Point(-1, 1)
	s = Point(1, 1)
	t = Point(0, 2)
	V = Triangle(r, s, t)
	vuw(V)
	auw(V)
	from wpr import _pp
	figure()
	_pp(T)
	_pp(auw(T), color='g')
	"""
	v = vuw(T)
	if len(v) > 0:
		return Polygon(*(v + intersection(T, Line(Point(-10, 0), Point(10, 0)))))
	return None 
Beispiel #2
0
    def get_windows(self, building):
        p_obs = symg.Point(self.x, self.y)

        # collect the segments of the building
        sides = []
        for i in range(0, len(building.nodes)-1):
            node1 = building.nodes[i]
            node2 = building.nodes[i+1]
            p1 = symg.Point(node1.x, node1.y)
            p2 = symg.Point(node2.x, node2.y)
            sides.append(symg.Segment(p1, p2))

        # collect windows that sit on the segments
        for s in sides:
            perp_line = s.perpendicular_line(p_obs)
            p_list = symg.intersection(perp_line, s)
            if p_list:
                p_window = p_list[0]
                phi = get_angle_from_south(p_obs, p_window)
                distance = float(p_obs.distance(p_window))
                w = Window(
                    x=float(p_window.x), 
                    y=float(p_window.y), 
                    phi=phi, 
                    distance=distance)
                self.windows.append(w)

        # find the closest window
        self.closest_window = min(self.windows, key=lambda w: w.distance)
def telemetry(T,boxelist):
    a = radians(T.heading())
    P1,P2 = Point(T.xcor(),T.ycor()) , Point(T.xcor()+cos(a),T.ycor()+sin(a))
    P12 = P2 - P1
    intr = [N(P12.dot(p-P1)) for r in boxelist for p in intersection(Line(P1,P2),r) ]
    intr = [d for d in intr if d >= 0]
    #print intr
    return None if intr==[] else (min(intr)+np.random.normal(0,10))
Beispiel #4
0
def test_intersection():
    assert intersection(Point(0, 0)) == []
    raises(TypeError, lambda: intersection(Point(0, 0), 3))
    assert intersection(
            Segment((0, 0), (2, 0)),
            Segment((-1, 0), (1, 0)),
            Line((0, 0), (0, 1)), pairwise=True) == [
        Point(0, 0), Segment((0, 0), (1, 0))]
    assert intersection(
            Line((0, 0), (0, 1)),
            Segment((0, 0), (2, 0)),
            Segment((-1, 0), (1, 0)), pairwise=True) == [
        Point(0, 0), Segment((0, 0), (1, 0))]
    assert intersection(
            Line((0, 0), (0, 1)),
            Segment((0, 0), (2, 0)),
            Segment((-1, 0), (1, 0)),
            Line((0, 0), slope=1), pairwise=True) == [
        Point(0, 0), Segment((0, 0), (1, 0))]
Beispiel #5
0
def incircle( p,a,b,c):
    p = points[p]
    a = points[a]
    b = points[b]
    c = points[c]
    t = Triangle(a,b,c)

    if hasattr(t, "circumcircle") and t.circumcircle.encloses_point( p ):
        return 1
    if  hasattr(t, "circumcircle") and intersection(t.circumcenter,p): #and t.circumcenter.distance(p) == t.circumradius:
        return 0
    return -1
Beispiel #6
0
    def split(self, edge, rotate_up):
        """
        :param edge:
        :param rotate_up: 線より上の部分が回転するかどうか
        :return:
        """
        line = sg.Line(edge[0], edge[1])
        points = ConvexHull(self.vertices).vertices
        polygon_points = []
        for i in points:
            polygon_points.append(self.vertices[i])
        polygon = sg.Polygon(*polygon_points)
        new_vertices = sg.intersection(polygon, line)

        up = []
        down = []
        on_line = 0
        for v in self.vertices:
            check = is_up(v, edge)
            if check > 0:
                up.append(v)
            else:
                if check == 0:
                    on_line += 1
                down.append(v)
        if on_line == len(down):
            return None, None
        nv_cnt = 0
        for nv in new_vertices:
            if type(nv) == Segment:
                continue
            up.append(nv)
            down.append(nv)
            nv_cnt += 1
        if nv_cnt != 2:
            return None, None

        if rotate_up:
            up = [get_symmetric_point(p, edge) for p in up]
        else:
            down = [get_symmetric_point(p, edge) for p in down]

        up_polygon = PolygonNode(edge, rotate_up, up, self.node_id)
        down_polygon = PolygonNode(edge, not rotate_up, down, self.node_id)

        # GC
        self.vertices = []

        return up_polygon, down_polygon
    def check_intersect(self, ellipses):
        """
        Check if one ellipse either intersects with another ellipse, or is contained within it.
        Returns true if they intersect or one is within the other.
        Returns false if the ellipses do not touch
        """
        
        ellipse1 = Ellipse.__GenerateSymPyEllipse(self)

        for ellipse in ellipses:
            ellipse2 = Ellipse.__GenerateSymPyEllipse(ellipse)

            if len(intersection(ellipse1, ellipse2)) != 0 or ellipse1.encloses(ellipse2) or ellipse2.encloses(ellipse1): 
                return True

        return False
def findcross(one_line, second_line):
    """Линии состоят из [[x1,y1],[x2,y2]] для поиска пересечений,
    исключая крайние точки. Если пересечения нет, на выходе """
    x1, y1, x2, y2 = one_line[0][0], one_line[0][1], one_line[1][0], one_line[1][1]
    x3, y3, x4, y4 = second_line[0][0], second_line[0][1], second_line[1][0], second_line[1][1]
    p1, p2, p3, p4 = symg.Point(x1, y1), symg.Point(x2, y2), symg.Point(x3, y3), symg.Point(x4, y4)
    l1, l2 = symg.Segment(p1, p2), symg.Segment(p3, p4)
    pointxy = symg.intersection(l1, l2)

    if pointxy:
        pointxy = [pointxy[0].x, pointxy[0].y]
    """ кажется, что if работает быстрее
    try:
        pointxy = [pointxy[0].x, pointxy[0].y]
    except IndexError:
        return pointxy
    """
    return pointxy
Beispiel #9
0
def test_ellipse():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x2)
    p4 = Point(0, 1)
    p5 = Point(-1, 0)

    e1 = Ellipse(p1, 1, 1)
    e2 = Ellipse(p2, half, 1)
    e3 = Ellipse(p1, y1, y1)
    c1 = Circle(p1, 1)
    c2 = Circle(p2,1)
    c3 = Circle(Point(sqrt(2),sqrt(2)),1)

    # Test creation with three points
    cen, rad = Point(3*half, 2), 5*half
    assert Circle(Point(0,0), Point(3,0), Point(0,4)) == Circle(cen, rad)
    raises(GeometryError, "Circle(Point(0,0), Point(1,1), Point(2,2))")

    # Basic Stuff
    assert e1 == c1
    assert e1 != e2
    assert p4 in e1
    assert p2 not in e2
    assert e1.area == pi
    assert e2.area == pi/2
    assert e3.area == pi*(y1**2)
    assert c1.area == e1.area
    assert c1.circumference == e1.circumference
    assert e3.circumference == 2*pi*y1

    # with generic symbols, the hradius is assumed to contain the major radius
    M = Symbol('M')
    m = Symbol('m')
    c = Ellipse(p1, M, m).circumference
    _x = c.atoms(Dummy).pop()
    assert c == \
        4*M*C.Integral(sqrt((1 - _x**2*(M**2 - m**2)/M**2)/(1 - _x**2)), (_x, 0, 1))

    assert e2.arbitrary_point() in e2

    # Foci
    f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
    ef = Ellipse(Point(0, 0), 4, 2)
    assert ef.foci in [(f1, f2), (f2, f1)]

    # Tangents
    v = sqrt(2) / 2
    p1_1 = Point(v, v)
    p1_2 = p2 + Point(half, 0)
    p1_3 = p2 + Point(0, 1)
    assert e1.tangent_lines(p4) == c1.tangent_lines(p4)
    assert e2.tangent_lines(p1_2) == [Line(p1_2, p2 + Point(half, 1))]
    assert e2.tangent_lines(p1_3) == [Line(p1_3, p2 + Point(half, 1))]
    assert c1.tangent_lines(p1_1) == [Line(p1_1, Point(0, sqrt(2)))]
    assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
    assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
    assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
    assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) == False

    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(0, 0)) == \
    [Line(Point(0, 0), Point(S(77)/25, S(132)/25)),
     Line(Point(0, 0), Point(S(33)/5, S(22)/5))]
    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(3, 4)) == \
    [Line(Point(3, 4), Point(3, 5)), Line(Point(3, 4), Point(5, 4))]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(3, 3)) == \
    [Line(Point(3, 3), Point(3, 5)), Line(Point(3, 3), Point(5, 3))]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(5 - 2*sqrt(2), 5)) == \
    [Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 - sqrt(2))),
     Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 + sqrt(2))),]

    # Properties
    major = 3
    minor = 1
    e4 = Ellipse(p2, minor, major)
    assert e4.focus_distance == sqrt(major**2 - minor**2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major*(1 - ecc)
    assert e4.apoapsis == major*(1 + ecc)
    # independent of orientation
    e4 = Ellipse(p2, major, minor)
    assert e4.focus_distance == sqrt(major**2 - minor**2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major*(1 - ecc)
    assert e4.apoapsis == major*(1 + ecc)

    # Intersection
    l1 = Line(Point(1, -5), Point(1, 5))
    l2 = Line(Point(-5, -1), Point(5, -1))
    l3 = Line(Point(-1, -1), Point(1, 1))
    l4 = Line(Point(-10, 0), Point(0, 10))
    pts_c1_l3 = [Point(sqrt(2)/2, sqrt(2)/2), Point(-sqrt(2)/2, -sqrt(2)/2)]

    assert intersection(e2, l4) == []
    assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
    assert intersection(c1, l1) == [Point(1, 0)]
    assert intersection(c1, l2) == [Point(0, -1)]
    assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
    assert intersection(c1, c2) in [[(1,0), (0,1)],[(0,1),(1,0)]]
    assert intersection(c1, c3) == [(sqrt(2)/2, sqrt(2)/2)]

    # some special case intersections
    csmall = Circle(p1, 3)
    cbig = Circle(p1, 5)
    cout = Circle(Point(5, 5), 1)
    # one circle inside of another
    assert csmall.intersection(cbig) == []
    # separate circles
    assert csmall.intersection(cout) == []
    # coincident circles
    assert csmall.intersection(csmall) == csmall

    v = sqrt(2)
    t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
    points = intersection(t1, c1)
    assert len(points) == 4
    assert Point(0, 1) in points
    assert Point(0, -1) in points
    assert Point(v/2, v/2) in points
    assert Point(v/2, -v/2) in points

    circ = Circle(Point(0, 0), 5)
    elip = Ellipse(Point(0, 0), 5, 20)
    assert intersection(circ, elip) in \
        [[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
    assert elip.tangent_lines(Point(0, 0)) == []
    elip = Ellipse(Point(0, 0), 3, 2)
    assert elip.tangent_lines(Point(3, 0)) == [Line(Point(3, 0), Point(3, -12))]

    e1 = Ellipse(Point(0, 0), 5, 10)
    e2 = Ellipse(Point(2, 1), 4, 8)
    a = S(53)/17
    c = 2*sqrt(3991)/17
    assert e1.intersection(e2) == [Point(a - c/8, a/2 + c), Point(a + c/8, a/2 - c)]

    # Combinations of above
    assert e3.is_tangent(e3.tangent_lines(p1 + Point(y1, 0))[0])

    e = Ellipse((1, 2), 3, 2)
    assert e.tangent_lines(Point(10, 0)) == \
       [Line(Point(10, 0), Point(1, 0)),
        Line(Point(10, 0), Point(S(14)/5, S(18)/5))]

    # encloses_point
    e = Ellipse((0, 0), 1, 2)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(e.center + Point(e.hradius + Rational(1, 10), 0)) is False
    e = Ellipse((0, 0), 2, 1)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(e.center + Point(e.hradius + Rational(1, 10), 0)) is False
Beispiel #10
0
    def plot_map(self):
        if self.launch_location == 'izu':
            #for IZU URA-SABAKU!!
            # Set limit range in maps
            self.set_coordinate_izu()

            # for tamura version
            # Set map image
            img_map = Image.open("./map/Izu_map_mag.png")
            img_list = np.asarray(img_map)
            img_height = img_map.size[0]
            img_width = img_map.size[1]
            img_origin = np.array(
                [722, 749])  # TODO : compute by lat/long of launcher point

            #pixel2meter = (139.431463 - 139.41283)/1800.0 * lon2met
            pixel2meter = 0.946981208125

            # Define image range
            img_left = -1.0 * img_origin[0] * pixel2meter
            img_right = (img_width - img_origin[0]) * pixel2meter
            img_top = img_origin[1] * pixel2meter
            img_bottom = -1.0 * (img_height - img_origin[1]) * pixel2meter

            fig = plt.figure(figsize=(12, 10))

            # plot setting
            ax = fig.add_subplot(111)
            color_line = '#ffff33'  # Yellow
            color_circle = 'r'  # Red

            # Set circle object
            cir_rail = patches.Circle(xy=self.xy_rail,
                                      radius=self.lim_radius,
                                      ec=color_circle,
                                      fill=False)
            cir_switch = patches.Circle(xy=self.xy_switch,
                                        radius=self.lim_radius,
                                        ec=color_circle,
                                        fill=False)
            cir_tent = patches.Circle(xy=self.xy_tent,
                                      radius=self.lim_radius,
                                      ec=color_circle,
                                      fill=False)
            ax.add_patch(cir_rail)
            ax.add_patch(cir_switch)
            ax.add_patch(cir_tent)

            # plot map
            plt.imshow(img_list,
                       extent=(img_left, img_right, img_bottom, img_top))

            # Write landing permission range
            plt.plot(self.xy_rail[0],
                     self.xy_rail[1],
                     'r.',
                     color=color_circle,
                     markersize=12)
            plt.plot(self.xy_switch[0],
                     self.xy_switch[1],
                     '.',
                     color=color_circle)
            plt.plot(self.xy_tent[0], self.xy_tent[1], '.', color=color_circle)
            plt.plot(self.xy_range[:, 0],
                     self.xy_range[:, 1],
                     '--',
                     color=color_line)
            """
            # plot landing point for 2018/3/23
            plt.plot(self.xy_land[0], self.xy_land[1], 'r*', markersize = 12, label='actual langing point')
            """

        elif self.launch_location == 'noshiro_sea':
            #for NOSHIRO SEA!!
            # Set limit range in maps
            self.set_coordinate_noshiro()

            # Set map image
            img_map = Image.open("./map/noshiro_new_rotate.png")
            img_list = np.asarray(img_map)
            img_height = img_map.size[1]
            # print(img_map.size)
            img_width = img_map.size[0]
            img_origin = np.array(
                [894, 647])  # TODO : compute by lat/long of launcher point

            #pixel2meter
            pixel2meter = 8.96708

            # Define image range
            img_left = -1.0 * img_origin[0] * pixel2meter
            img_right = (img_width - img_origin[0]) * pixel2meter
            img_top = img_origin[1] * pixel2meter
            img_bottom = -1.0 * (img_height - img_origin[1]) * pixel2meter

            #calculate intersections of "inside_circle" and "over_line"
            center1 = sg.Point(self.xy_center[0], self.xy_center[1])
            radius1 = self.hachiya_radius
            circle1 = sg.Circle(center1, radius1)
            line = sg.Line(sg.Point(self.xy_point[0, 0], self.xy_point[0, 1]),
                           sg.Point(self.xy_point[1, 0], self.xy_point[1, 1]))
            result1 = sg.intersection(circle1, line)
            intersection1_1 = np.array(
                [float(result1[0].x), float(result1[0].y)])
            intersection1_2 = np.array(
                [float(result1[1].x), float(result1[1].y)])

            #caluculate equation of hachiya_line(="over_line")
            self.a = (self.xy_point[1, 1] - self.xy_point[0, 1]) / (
                self.xy_point[1, 0] - self.xy_point[0, 0])
            self.b = (self.xy_point[0, 1] * self.xy_point[1, 0] -
                      self.xy_point[1, 1] * self.xy_point[0, 0]) / (
                          self.xy_point[1, 0] - self.xy_point[0, 0])
            self.x = np.arange(intersection1_1[0], intersection1_2[0], 1)
            self.y = self.a * self.x + self.b
            self.hachiya_line = np.array([self.a, self.b])

            # plot setting
            plt.figure(figsize=(10, 10))
            ax = plt.axes()
            color_line = '#ffff33'  # Yellow
            color_circle = 'r'  # Red

            # Set circle object
            cir_rail = patches.Circle(xy=self.xy_rail,
                                      radius=self.lim_radius,
                                      ec=color_line,
                                      fill=False)
            #cir_switch = patches.Circle(xy=self.xy_switch, radius=self.lim_radius, ec=color_circle, fill=False)
            #cir_tent = patches.Circle(xy=self.xy_tent, radius=self.lim_radius, ec=color_circle, fill=False)
            cir_center = patches.Circle(xy=self.xy_center,
                                        radius=self.hachiya_radius,
                                        ec=color_circle,
                                        fill=False)

            ax.add_patch(cir_rail)
            #ax.add_patch(cir_switch)
            #ax.add_patch(cir_tent)
            ax.add_patch(cir_center)

            # plot map
            plt.imshow(img_list,
                       extent=(img_left, img_right, img_bottom, img_top))

            # Write landing permission range
            plt.plot(self.x, self.y, "r")
            plt.plot(self.xy_rail[0], self.xy_rail[1], '.', color=color_circle)
            #plt.plot(self.xy_switch[0], self.xy_switch[1], '.', color=color_circle)
            #plt.plot(self.xy_tent[0], self.xy_tent[1], '.', color=color_circle)
            #plt.plot(self.xy_range[:,0], self.xy_range[:,1], '--', color=color_line)
            plt.plot(self.xy_center[0],
                     self.xy_center[1],
                     '.',
                     color=color_circle)

        else:
            raise NotImplementedError(
                'Available location is: izu or noshiro_sea')

        return None
Beispiel #11
0
def test_ellipse_geom():
    x = Symbol("x", real=True)
    y = Symbol("y", real=True)
    t = Symbol("t", real=True)
    y1 = Symbol("y1", real=True)
    half = S.Half
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p4 = Point(0, 1)

    e1 = Ellipse(p1, 1, 1)
    e2 = Ellipse(p2, half, 1)
    e3 = Ellipse(p1, y1, y1)
    c1 = Circle(p1, 1)
    c2 = Circle(p2, 1)
    c3 = Circle(Point(sqrt(2), sqrt(2)), 1)
    l1 = Line(p1, p2)

    # Test creation with three points
    cen, rad = Point(3 * half, 2), 5 * half
    assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
    assert Circle(Point(0, 0), Point(1, 1),
                  Point(2, 2)) == Segment2D(Point2D(0, 0), Point2D(2, 2))

    raises(ValueError, lambda: Ellipse(None, None, None, 1))
    raises(GeometryError, lambda: Circle(Point(0, 0)))

    # Basic Stuff
    assert Ellipse(None, 1, 1).center == Point(0, 0)
    assert e1 == c1
    assert e1 != e2
    assert e1 != l1
    assert p4 in e1
    assert p2 not in e2
    assert e1.area == pi
    assert e2.area == pi / 2
    assert e3.area == pi * y1 * abs(y1)
    assert c1.area == e1.area
    assert c1.circumference == e1.circumference
    assert e3.circumference == 2 * pi * y1
    assert e1.plot_interval() == e2.plot_interval() == [t, -pi, pi]
    assert e1.plot_interval(x) == e2.plot_interval(x) == [x, -pi, pi]

    assert c1.minor == 1
    assert c1.major == 1
    assert c1.hradius == 1
    assert c1.vradius == 1

    assert Ellipse((1, 1), 0, 0) == Point(1, 1)
    assert Ellipse((1, 1), 1, 0) == Segment(Point(0, 1), Point(2, 1))
    assert Ellipse((1, 1), 0, 1) == Segment(Point(1, 0), Point(1, 2))

    # Private Functions
    assert hash(c1) == hash(Circle(Point(1, 0), Point(0, 1), Point(0, -1)))
    assert c1 in e1
    assert (Line(p1, p2) in e1) is False
    assert e1.__cmp__(e1) == 0
    assert e1.__cmp__(Point(0, 0)) > 0

    # Encloses
    assert e1.encloses(Segment(Point(-0.5, -0.5), Point(0.5, 0.5))) is True
    assert e1.encloses(Line(p1, p2)) is False
    assert e1.encloses(Ray(p1, p2)) is False
    assert e1.encloses(e1) is False
    assert (e1.encloses(
        Polygon(Point(-0.5, -0.5), Point(-0.5, 0.5), Point(0.5, 0.5))) is True)
    assert e1.encloses(RegularPolygon(p1, 0.5, 3)) is True
    assert e1.encloses(RegularPolygon(p1, 5, 3)) is False
    assert e1.encloses(RegularPolygon(p2, 5, 3)) is False

    assert e2.arbitrary_point() in e2

    # Foci
    f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
    ef = Ellipse(Point(0, 0), 4, 2)
    assert ef.foci in [(f1, f2), (f2, f1)]

    # Tangents
    v = sqrt(2) / 2
    p1_1 = Point(v, v)
    p1_2 = p2 + Point(half, 0)
    p1_3 = p2 + Point(0, 1)
    assert e1.tangent_lines(p4) == c1.tangent_lines(p4)
    assert e2.tangent_lines(p1_2) == [
        Line(Point(Rational(3, 2), 1), Point(Rational(3, 2), S.Half))
    ]
    assert e2.tangent_lines(p1_3) == [
        Line(Point(1, 2), Point(Rational(5, 4), 2))
    ]
    assert c1.tangent_lines(p1_1) != [Line(p1_1, Point(0, sqrt(2)))]
    assert c1.tangent_lines(p1) == []
    assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
    assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
    assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
    assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) is False
    assert c1.is_tangent(e1) is True
    assert c1.is_tangent(Ellipse(Point(2, 0), 1, 1)) is True
    assert c1.is_tangent(Polygon(Point(1, 1), Point(1, -1), Point(2,
                                                                  0))) is True
    assert c1.is_tangent(Polygon(Point(1, 1), Point(1, 0), Point(2,
                                                                 0))) is False
    assert Circle(Point(5, 5), 3).is_tangent(Circle(Point(0, 5), 1)) is False

    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(0, 0)) == [
        Line(Point(0, 0), Point(Rational(77, 25), Rational(132, 25))),
        Line(Point(0, 0), Point(Rational(33, 5), Rational(22, 5))),
    ]
    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(3, 4)) == [
        Line(Point(3, 4), Point(4, 4)),
        Line(Point(3, 4), Point(3, 5)),
    ]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(3, 3)) == [
        Line(Point(3, 3), Point(4, 3)),
        Line(Point(3, 3), Point(3, 4)),
    ]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(5 - 2 * sqrt(2), 5)) == [
        Line(Point(5 - 2 * sqrt(2), 5), Point(5 - sqrt(2), 5 - sqrt(2))),
        Line(Point(5 - 2 * sqrt(2), 5), Point(5 - sqrt(2), 5 + sqrt(2))),
    ]

    # for numerical calculations, we shouldn't demand exact equality,
    # so only test up to the desired precision
    def lines_close(l1, l2, prec):
        """ tests whether l1 and 12 are within 10**(-prec)
        of each other """
        return abs(l1.p1 - l2.p1) < 10**(-prec) and abs(l1.p2 -
                                                        l2.p2) < 10**(-prec)

    def line_list_close(ll1, ll2, prec):
        return all(lines_close(l1, l2, prec) for l1, l2 in zip(ll1, ll2))

    e = Ellipse(Point(0, 0), 2, 1)
    assert e.normal_lines(Point(0, 0)) == [
        Line(Point(0, 0), Point(0, 1)),
        Line(Point(0, 0), Point(1, 0)),
    ]
    assert e.normal_lines(Point(1, 0)) == [Line(Point(0, 0), Point(1, 0))]
    assert e.normal_lines((0, 1)) == [Line(Point(0, 0), Point(0, 1))]
    assert line_list_close(
        e.normal_lines(Point(1, 1), 2),
        [
            Line(
                Point(Rational(-51, 26), Rational(-1, 5)),
                Point(Rational(-25, 26), Rational(17, 83)),
            ),
            Line(
                Point(Rational(28, 29), Rational(-7, 8)),
                Point(Rational(57, 29), Rational(-9, 2)),
            ),
        ],
        2,
    )
    # test the failure of Poly.intervals and checks a point on the boundary
    p = Point(sqrt(3), S.Half)
    assert p in e
    assert line_list_close(
        e.normal_lines(p, 2),
        [
            Line(
                Point(Rational(-341, 171), Rational(-1, 13)),
                Point(Rational(-170, 171), Rational(5, 64)),
            ),
            Line(
                Point(Rational(26, 15), Rational(-1, 2)),
                Point(Rational(41, 15), Rational(-43, 26)),
            ),
        ],
        2,
    )
    # be sure to use the slope that isn't undefined on boundary
    e = Ellipse((0, 0), 2, 2 * sqrt(3) / 3)
    assert line_list_close(
        e.normal_lines((1, 1), 2),
        [
            Line(
                Point(Rational(-64, 33), Rational(-20, 71)),
                Point(Rational(-31, 33), Rational(2, 13)),
            ),
            Line(Point(1, -1), Point(2, -4)),
        ],
        2,
    )
    # general ellipse fails except under certain conditions
    e = Ellipse((0, 0), x, 1)
    assert e.normal_lines((x + 1, 0)) == [Line(Point(0, 0), Point(1, 0))]
    raises(NotImplementedError, lambda: e.normal_lines((x + 1, 1)))
    # Properties
    major = 3
    minor = 1
    e4 = Ellipse(p2, minor, major)
    assert e4.focus_distance == sqrt(major**2 - minor**2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major * (1 - ecc)
    assert e4.apoapsis == major * (1 + ecc)
    assert e4.semilatus_rectum == major * (1 - ecc**2)
    # independent of orientation
    e4 = Ellipse(p2, major, minor)
    assert e4.focus_distance == sqrt(major**2 - minor**2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major * (1 - ecc)
    assert e4.apoapsis == major * (1 + ecc)

    # Intersection
    l1 = Line(Point(1, -5), Point(1, 5))
    l2 = Line(Point(-5, -1), Point(5, -1))
    l3 = Line(Point(-1, -1), Point(1, 1))
    l4 = Line(Point(-10, 0), Point(0, 10))
    pts_c1_l3 = [
        Point(sqrt(2) / 2,
              sqrt(2) / 2),
        Point(-sqrt(2) / 2, -sqrt(2) / 2)
    ]

    assert intersection(e2, l4) == []
    assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
    assert intersection(c1, l1) == [Point(1, 0)]
    assert intersection(c1, l2) == [Point(0, -1)]
    assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
    assert intersection(c1, c2) == [Point(0, 1), Point(1, 0)]
    assert intersection(c1, c3) == [Point(sqrt(2) / 2, sqrt(2) / 2)]
    assert e1.intersection(l1) == [Point(1, 0)]
    assert e2.intersection(l4) == []
    assert e1.intersection(Circle(Point(0, 2), 1)) == [Point(0, 1)]
    assert e1.intersection(Circle(Point(5, 0), 1)) == []
    assert e1.intersection(Ellipse(Point(2, 0), 1, 1)) == [Point(1, 0)]
    assert e1.intersection(Ellipse(Point(5, 0), 1, 1)) == []
    assert e1.intersection(Point(2, 0)) == []
    assert e1.intersection(e1) == e1
    assert intersection(Ellipse(Point(0, 0), 2, 1),
                        Ellipse(Point(3, 0), 1, 2)) == [Point(2, 0)]
    assert intersection(Circle(Point(0, 0), 2), Circle(Point(3, 0),
                                                       1)) == [Point(2, 0)]
    assert intersection(Circle(Point(0, 0), 2), Circle(Point(7, 0), 1)) == []
    assert intersection(Ellipse(Point(0, 0), 5, 17),
                        Ellipse(Point(4, 0), 1, 0.2)) == [Point(5, 0)]
    assert (intersection(Ellipse(Point(0, 0), 5, 17),
                         Ellipse(Point(4, 0), 0.999, 0.2)) == [])
    assert Circle((0, 0),
                  S.Half).intersection(Triangle((-1, 0), (1, 0), (0, 1))) == [
                      Point(Rational(-1, 2), 0),
                      Point(S.Half, 0),
                  ]
    raises(TypeError, lambda: intersection(e2, Line((0, 0, 0), (0, 0, 1))))
    raises(TypeError, lambda: intersection(e2, Rational(12)))
    # some special case intersections
    csmall = Circle(p1, 3)
    cbig = Circle(p1, 5)
    cout = Circle(Point(5, 5), 1)
    # one circle inside of another
    assert csmall.intersection(cbig) == []
    # separate circles
    assert csmall.intersection(cout) == []
    # coincident circles
    assert csmall.intersection(csmall) == csmall

    v = sqrt(2)
    t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
    points = intersection(t1, c1)
    assert len(points) == 4
    assert Point(0, 1) in points
    assert Point(0, -1) in points
    assert Point(v / 2, v / 2) in points
    assert Point(v / 2, -v / 2) in points

    circ = Circle(Point(0, 0), 5)
    elip = Ellipse(Point(0, 0), 5, 20)
    assert intersection(circ, elip) in [
        [Point(5, 0), Point(-5, 0)],
        [Point(-5, 0), Point(5, 0)],
    ]
    assert elip.tangent_lines(Point(0, 0)) == []
    elip = Ellipse(Point(0, 0), 3, 2)
    assert elip.tangent_lines(Point(3,
                                    0)) == [Line(Point(3, 0), Point(3, -12))]

    e1 = Ellipse(Point(0, 0), 5, 10)
    e2 = Ellipse(Point(2, 1), 4, 8)
    a = Rational(53, 17)
    c = 2 * sqrt(3991) / 17
    ans = [Point(a - c / 8, a / 2 + c), Point(a + c / 8, a / 2 - c)]
    assert e1.intersection(e2) == ans
    e2 = Ellipse(Point(x, y), 4, 8)
    c = sqrt(3991)
    ans = [
        Point(-c / 68 + a,
              c * Rational(2, 17) + a / 2),
        Point(c / 68 + a,
              c * Rational(-2, 17) + a / 2),
    ]
    assert [p.subs({x: 2, y: 1}) for p in e1.intersection(e2)] == ans

    # Combinations of above
    assert e3.is_tangent(e3.tangent_lines(p1 + Point(y1, 0))[0])

    e = Ellipse((1, 2), 3, 2)
    assert e.tangent_lines(Point(10, 0)) == [
        Line(Point(10, 0), Point(1, 0)),
        Line(Point(10, 0), Point(Rational(14, 5), Rational(18, 5))),
    ]

    # encloses_point
    e = Ellipse((0, 0), 1, 2)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(e.center +
                            Point(e.hradius + Rational(1, 10), 0)) is False
    e = Ellipse((0, 0), 2, 1)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(e.center +
                            Point(e.hradius + Rational(1, 10), 0)) is False
    assert c1.encloses_point(Point(1, 0)) is False
    assert c1.encloses_point(Point(0.3, 0.4)) is True

    assert e.scale(2, 3) == Ellipse((0, 0), 4, 3)
    assert e.scale(3, 6) == Ellipse((0, 0), 6, 6)
    assert e.rotate(pi) == e
    assert e.rotate(pi, (1, 2)) == Ellipse(Point(2, 4), 2, 1)
    raises(NotImplementedError, lambda: e.rotate(pi / 3))

    # Circle rotation tests (Issue #11743)
    # Link - https://github.com/sympy/sympy/issues/11743
    cir = Circle(Point(1, 0), 1)
    assert cir.rotate(pi / 2) == Circle(Point(0, 1), 1)
    assert cir.rotate(pi / 3) == Circle(Point(S.Half, sqrt(3) / 2), 1)
    assert cir.rotate(pi / 3, Point(1, 0)) == Circle(Point(1, 0), 1)
    assert cir.rotate(pi / 3, Point(0, 1)) == Circle(
        Point(S.Half + sqrt(3) / 2, S.Half + sqrt(3) / 2), 1)
Beispiel #12
0
def test_intersection_2d():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x1)
    p4 = Point(y1, y1)

    l1 = Line(p1, p2)
    l3 = Line(Point(0, 0), Point(3, 4))

    r1 = Ray(Point(1, 1), Point(2, 2))
    r2 = Ray(Point(0, 0), Point(3, 4))
    r4 = Ray(p1, p2)
    r6 = Ray(Point(0, 1), Point(1, 2))
    r7 = Ray(Point(0.5, 0.5), Point(1, 1))

    s1 = Segment(p1, p2)
    s2 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
    s3 = Segment(Point(0, 0), Point(3, 4))

    assert intersection(l1, p1) == [p1]
    assert intersection(l1, Point(x1, 1 + x1)) == []
    assert intersection(l1, Line(p3, p4)) in [[l1], [Line(p3, p4)]]
    assert intersection(l1, l1.parallel_line(Point(x1, 1 + x1))) == []
    assert intersection(l3, l3) == [l3]
    assert intersection(l3, r2) == [r2]
    assert intersection(l3, s3) == [s3]
    assert intersection(s3, l3) == [s3]
    assert intersection(Segment(Point(-10, 10), Point(10, 10)), Segment(Point(-5, -5), Point(-5, 5))) == []
    assert intersection(r2, l3) == [r2]
    assert intersection(r1, Ray(Point(2, 2), Point(0, 0))) == [Segment(Point(1, 1), Point(2, 2))]
    assert intersection(r1, Ray(Point(1, 1), Point(-1, -1))) == [Point(1, 1)]
    assert intersection(r1, Segment(Point(0, 0), Point(2, 2))) == [Segment(Point(1, 1), Point(2, 2))]

    assert r4.intersection(s2) == [s2]
    assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
    assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert r4.intersection(Ray(p2, p1)) == [s1]
    assert Ray(p2, p1).intersection(r6) == []
    assert r4.intersection(r7) == r7.intersection(r4) == [r7]
    assert Ray3D((0, 0), (3, 0)).intersection(Ray3D((1, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
    assert Ray3D((1, 0), (3, 0)).intersection(Ray3D((0, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
    assert Ray(Point(0, 0), Point(0, 4)).intersection(Ray(Point(0, 1), Point(0, -1))) == \
           [Segment(Point(0, 0), Point(0, 1))]

    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((1, 0), (2, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((1, 0), (2, 0)).intersection(
        Segment3D((0, 0), (3, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((3, 0), (4, 0))) == [Point3D((3, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((2, 0), (5, 0))) == [Segment3D((2, 0), (3, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((-2, 0), (1, 0))) == [Segment3D((0, 0), (1, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((-2, 0), (0, 0))) == [Point3D(0, 0)]
    assert s1.intersection(Segment(Point(1, 1), Point(2, 2))) == [Point(1, 1)]
    assert s1.intersection(Segment(Point(0.5, 0.5), Point(1.5, 1.5))) == [Segment(Point(0.5, 0.5), p2)]
    assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
    assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
    assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert s1.intersection(Line(Point(1, 0), Point(2, 1))) == []
    assert s1.intersection(s2) == [s2]
    assert s2.intersection(s1) == [s2]

    assert asa(120, 8, 52) == \
           Triangle(
               Point(0, 0),
               Point(8, 0),
               Point(-4 * cos(19 * pi / 90) / sin(2 * pi / 45),
                     4 * sqrt(3) * cos(19 * pi / 90) / sin(2 * pi / 45)))
    assert Line((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)]
    assert Line((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (10, 10)).contains(Segment((1, 1), (2, 2))) is True
    assert Segment((1, 1), (2, 2)) in Line((0, 0), (10, 10))

    # 16628 - this should be fast
    p0 = Point2D(S(249)/5, S(497999)/10000)
    p1 = Point2D((-58977084786*sqrt(405639795226) + 2030690077184193 +
        20112207807*sqrt(630547164901) + 99600*sqrt(255775022850776494562626))
        /(2000*sqrt(255775022850776494562626) + 1991998000*sqrt(405639795226)
        + 1991998000*sqrt(630547164901) + 1622561172902000),
        (-498000*sqrt(255775022850776494562626) - 995999*sqrt(630547164901) +
        90004251917891999 +
        496005510002*sqrt(405639795226))/(10000*sqrt(255775022850776494562626)
        + 9959990000*sqrt(405639795226) + 9959990000*sqrt(630547164901) +
        8112805864510000))
    p2 = Point2D(S(497)/10, -S(497)/10)
    p3 = Point2D(-S(497)/10, -S(497)/10)
    l = Line(p0, p1)
    s = Segment(p2, p3)
    n = (-52673223862*sqrt(405639795226) - 15764156209307469 -
        9803028531*sqrt(630547164901) +
        33200*sqrt(255775022850776494562626))
    d = sqrt(405639795226) + 315274080450 + 498000*sqrt(
        630547164901) + sqrt(255775022850776494562626)
    assert intersection(l, s) == [
        Point2D(n/d*S(3)/2000, -S(497)/10)]
Beispiel #13
0
def test_polygon():
    p1 = Polygon(
        Point(0, 0), Point(3,-1),
        Point(6, 0), Point(4, 5),
        Point(2, 3), Point(0, 3))
    p2 = Polygon(
        Point(6, 0), Point(3,-1),
        Point(0, 0), Point(0, 3),
        Point(2, 3), Point(4, 5))
    p3 = Polygon(
        Point(0, 0), Point(3, 0),
        Point(5, 2), Point(4, 4))
    p4 = Polygon(
        Point(0, 0), Point(4, 4),
        Point(5, 2), Point(3, 0))

    #
    # General polygon
    #
    assert p1 == p2
    assert len(p1) == Rational(6)
    assert len(p1.sides) == 6
    assert p1.perimeter == 5+2*sqrt(10)+sqrt(29)+sqrt(8)
    assert p1.area == 22
    assert not p1.is_convex()
    assert p3.is_convex()
    assert p4.is_convex()  # ensure convex for both CW and CCW point specification

    #
    # Regular polygon
    #
    p1 = RegularPolygon(Point(0, 0), 10, 5)
    p2 = RegularPolygon(Point(0, 0), 5, 5)

    assert p1 != p2
    assert p1.interior_angle == 3*pi/5
    assert p1.exterior_angle == 2*pi/5
    assert p2.apothem == 5*cos(pi/5)
    assert p2.circumcircle == Circle(Point(0, 0), 5)
    assert p2.incircle == Circle(Point(0, 0), p2.apothem)
    assert p1.is_convex()

    #
    # Angles
    #
    angles = p4.angles
    assert feq(angles[Point(0, 0)].evalf(), Real("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Real("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Real("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Real("2.3561944901923449"))

    angles = p3.angles
    assert feq(angles[Point(0, 0)].evalf(), Real("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Real("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Real("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Real("2.3561944901923449"))

    #
    # Triangle
    #
    p1 = Point(0, 0)
    p2 = Point(5, 0)
    p3 = Point(0, 5)
    t1 = Triangle(p1, p2, p3)
    t2 = Triangle(p1, p2, Point(Rational(5,2), sqrt(Rational(75,4))))
    t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
    s1 = t1.sides
    s2 = t2.sides
    s3 = t3.sides

    # Basic stuff
    assert t1.area == Rational(25,2)
    assert t1.is_right()
    assert t2.is_right() == False
    assert t3.is_right()
    assert p1 in t1
    assert Point(5, 5) not in t2
    assert t1.is_convex()
    assert feq(t1.angles[p1].evalf(), pi.evalf()/2)

    assert t1.is_equilateral() == False
    assert t2.is_equilateral()
    assert t3.is_equilateral() == False
    assert are_similar(t1, t2) == False
    assert are_similar(t1, t3)
    assert are_similar(t2, t3) == False

    # Bisectors
    bisectors = t1.bisectors
    assert bisectors[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    ic = (250 - 125*sqrt(2)) / 50
    assert t1.incenter == Point(ic, ic)

    # Inradius
    assert t1.inradius == 5 - 5*2**(S(1)/2)/2
    assert t2.inradius == 5*3**(S(1)/2)/6
    assert t3.inradius == (2*x1**2*Abs(x1) - 2**(S(1)/2)*x1**2*Abs(x1))/(2*x1**2)

    # Medians + Centroid
    m = t1.medians
    assert t1.centroid == Point(Rational(5,3), Rational(5,3))
    assert m[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    assert t3.medians[p1] == Segment(p1, Point(x1/2, x1/2))
    assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]

    # Perpendicular
    altitudes = t1.altitudes
    assert altitudes[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    assert altitudes[p2] == s1[0]
    assert altitudes[p3] == s1[2]

    # Ensure
    assert len(intersection(*bisectors.values())) == 1
    assert len(intersection(*altitudes.values())) == 1
    assert len(intersection(*m.values())) == 1

    # Distance
    p1 = Polygon(
        Point(0, 0), Point(1, 0),
        Point(1, 1), Point(0, 1))
    p2 = Polygon(
        Point(0, Rational(5)/4), Point(1, Rational(5)/4),
        Point(1, Rational(9)/4), Point(0,  Rational(9)/4))
    p3 = Polygon(
        Point(1, 2), Point(2, 2),
        Point(2, 1))
    p4 = Polygon(
        Point(1, 1), Point(Rational(6)/5, 1),
        Point(1, Rational(6)/5))
    p5 = Polygon(
        Point(half, 3**(half)/2), Point(-half, 3**(half)/2),
        Point(-1, 0), Point(-half, -(3)**(half)/2),
        Point(half, -(3)**(half)/2), Point(1, 0))
    p6 = Polygon(Point(2, Rational(3)/10), Point(Rational(17)/10, 0),
                 Point(2, -Rational(3)/10), Point(Rational(23)/10, 0))
    pt1 = Point(half, half)
    pt2 = Point(1, 1)

    '''Polygon to Point'''
    assert p1.distance(pt1) == half
    assert p1.distance(pt2) == 0
    assert p2.distance(pt1) == Rational(3)/4
    assert p3.distance(pt2) == sqrt(2)/2

    '''Polygon to Polygon'''
    assert p1.distance(p2) == half/2
    assert p1.distance(p3) == sqrt(2)/2
    assert p3.distance(p4) == (sqrt(2)/2 - sqrt(Rational(2)/25)/2)
    assert p5.distance(p6) == Rational(7)/10
Beispiel #14
0
def test_polygon():
    t = Triangle(Point(0, 0), Point(2, 0), Point(3, 3))
    assert Polygon(Point(0, 0), Point(1, 0), Point(2, 0), Point(3, 3)) == t
    assert Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(0, 0)) == t
    assert Polygon(Point(2, 0), Point(3, 3), Point(0, 0), Point(1, 0)) == t

    p1 = Polygon(
        Point(0, 0), Point(3,-1),
        Point(6, 0), Point(4, 5),
        Point(2, 3), Point(0, 3))
    p2 = Polygon(
        Point(6, 0), Point(3,-1),
        Point(0, 0), Point(0, 3),
        Point(2, 3), Point(4, 5))
    p3 = Polygon(
        Point(0, 0), Point(3, 0),
        Point(5, 2), Point(4, 4))
    p4 = Polygon(
        Point(0, 0), Point(4, 4),
        Point(5, 2), Point(3, 0))
    p5 = Polygon(
        Point(0, 0), Point(4, 4),
        Point(0, 4))

    #
    # General polygon
    #
    assert p1 == p2
    assert len(p1.args) == 6
    assert len(p1.sides) == 6
    assert p1.perimeter == 5+2*sqrt(10)+sqrt(29)+sqrt(8)
    assert p1.area == 22
    assert not p1.is_convex()
    assert p3.is_convex()
    assert p4.is_convex()  # ensure convex for both CW and CCW point specification
    dict5 = p5.angles
    assert dict5[Point(0, 0)] == pi / 4
    assert dict5[Point(0, 4)] == pi / 2
    assert p5.encloses_point(Point(x, y)) == None
    assert p5.encloses_point(Point(1, 3))
    assert p5.encloses_point(Point(0, 0)) == False
    assert p5.encloses_point(Point(4, 0)) == False
    p5.plot_interval('x') == [x, 0, 1]
    assert p5.distance(Polygon(Point(10, 10), Point(14, 14), Point(10, 14))) == 6 * sqrt(2)
    assert p5.distance(Polygon(Point(1, 8), Point(5, 8), Point(8, 12), Point(1, 12))) == 4
    raises(UserWarning,
           'Polygon(Point(0, 0), Point(1, 0), Point(1,1)).distance(Polygon(Point(0, 0), Point(0, 1), Point(1, 1)))')
    assert hash(p5) == hash(Polygon(Point(0, 0), Point(4, 4), Point(0, 4)))
    assert p5 == Polygon(Point(4, 4), Point(0, 4), Point(0, 0))
    assert Polygon(Point(4, 4), Point(0, 4), Point(0, 0)) in p5
    assert p5 != Point(0, 4)
    assert Point(0, 1) in p5
    assert p5.arbitrary_point('t').subs(Symbol('t', real=True), 0) == Point(0, 0)
    raises(ValueError, "Polygon(Point(x, 0), Point(0, y), Point(x, y)).arbitrary_point('x')")

    #
    # Regular polygon
    #
    p1 = RegularPolygon(Point(0, 0), 10, 5)
    p2 = RegularPolygon(Point(0, 0), 5, 5)
    raises(GeometryError, 'RegularPolygon(Point(0, 0), Point(0, 1), Point(1, 1))')
    raises(GeometryError, 'RegularPolygon(Point(0, 0), 1, 2)')
    raises(ValueError, 'RegularPolygon(Point(0, 0), 1, 2.5)')

    assert p1 != p2
    assert p1.interior_angle == 3*pi/5
    assert p1.exterior_angle == 2*pi/5
    assert p2.apothem == 5*cos(pi/5)
    assert p2.circumcenter == p1.circumcenter == Point(0, 0)
    assert p1.circumradius == p1.radius == 10
    assert p2.circumcircle == Circle(Point(0, 0), 5)
    assert p2.incircle == Circle(Point(0, 0), p2.apothem)
    assert p2.inradius == p2.apothem == (5 * (1 + sqrt(5)) / 4)
    p2.spin(pi / 10)
    dict1 = p2.angles
    assert dict1[Point(0, 5)] == 3 * pi / 5
    assert p1.is_convex()
    assert p1.rotation == 0
    assert p1.encloses_point(Point(0, 0))
    assert p1.encloses_point(Point(11, 0)) == False
    assert p2.encloses_point(Point(0, 4.9))
    p1.spin(pi/3)
    assert p1.rotation == pi/3
    assert p1.vertices[0] == Point(5, 5*sqrt(3))
    for var in p1.args:
        if isinstance(var, Point):
            assert var == Point(0, 0)
        else:
            assert var == 5 or var == 10 or var == pi / 3
    assert p1 != Point(0, 0)
    assert p1 != p5

    # while spin works in place (notice that rotation is 2pi/3 below)
    # rotate returns a new object
    p1_old = p1
    assert p1.rotate(pi/3) == RegularPolygon(Point(0, 0), 10, 5, 2*pi/3)
    assert p1 == p1_old

    assert `p1` == str(p1)

    #
    # Angles
    #
    angles = p4.angles
    assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))

    angles = p3.angles
    assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))

    #
    # Triangle
    #
    p1 = Point(0, 0)
    p2 = Point(5, 0)
    p3 = Point(0, 5)
    t1 = Triangle(p1, p2, p3)
    t2 = Triangle(p1, p2, Point(Rational(5,2), sqrt(Rational(75,4))))
    t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
    s1 = t1.sides
    assert Triangle(p1, p2, p1) == Polygon(p1, p2, p1) == Segment(p1, p2)
    raises(GeometryError, 'Triangle(Point(0, 0))')

    # Basic stuff
    assert Triangle(p1, p1, p1) == p1
    assert Triangle(p2, p2*2, p2*3) == Segment(p2, p2*3)
    assert t1.area == Rational(25,2)
    assert t1.is_right()
    assert t2.is_right() == False
    assert t3.is_right()
    assert p1 in t1
    assert t1.sides[0] in t1
    assert Segment((0, 0), (1, 0)) in t1
    assert Point(5, 5) not in t2
    assert t1.is_convex()
    assert feq(t1.angles[p1].evalf(), pi.evalf()/2)

    assert t1.is_equilateral() == False
    assert t2.is_equilateral()
    assert t3.is_equilateral() == False
    assert are_similar(t1, t2) == False
    assert are_similar(t1, t3)
    assert are_similar(t2, t3) == False
    assert t1.is_similar(Point(0, 0)) == False

    # Bisectors
    bisectors = t1.bisectors()
    assert bisectors[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    ic = (250 - 125*sqrt(2)) / 50
    assert t1.incenter == Point(ic, ic)

    # Inradius
    assert t1.inradius == t1.incircle.radius == 5 - 5*sqrt(2)/2
    assert t2.inradius == t2.incircle.radius == 5*sqrt(3)/6
    assert t3.inradius == t3.incircle.radius == x1**2/((2 + sqrt(2))*Abs(x1))

    # Circumcircle
    assert t1.circumcircle.center == Point(2.5, 2.5)

    # Medians + Centroid
    m = t1.medians
    assert t1.centroid == Point(Rational(5,3), Rational(5,3))
    assert m[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    assert t3.medians[p1] == Segment(p1, Point(x1/2, x1/2))
    assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]
    assert t1.medial == Triangle(Point(2.5, 0), Point(0, 2.5), Point(2.5, 2.5))

    # Perpendicular
    altitudes = t1.altitudes
    assert altitudes[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    assert altitudes[p2] == s1[0]
    assert altitudes[p3] == s1[2]
    assert t1.orthocenter == p1

    # Ensure
    assert len(intersection(*bisectors.values())) == 1
    assert len(intersection(*altitudes.values())) == 1
    assert len(intersection(*m.values())) == 1

    # Distance
    p1 = Polygon(
        Point(0, 0), Point(1, 0),
        Point(1, 1), Point(0, 1))
    p2 = Polygon(
        Point(0, Rational(5)/4), Point(1, Rational(5)/4),
        Point(1, Rational(9)/4), Point(0,  Rational(9)/4))
    p3 = Polygon(
        Point(1, 2), Point(2, 2),
        Point(2, 1))
    p4 = Polygon(
        Point(1, 1), Point(Rational(6)/5, 1),
        Point(1, Rational(6)/5))
    pt1 = Point(half, half)
    pt2 = Point(1, 1)

    '''Polygon to Point'''
    assert p1.distance(pt1) == half
    assert p1.distance(pt2) == 0
    assert p2.distance(pt1) == Rational(3)/4
    assert p3.distance(pt2) == sqrt(2)/2
Beispiel #15
0
def test_line():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x1)
    p4 = Point(y1, y1)
    p5 = Point(x1, 1 + x1)
    p6 = Point(1, 0)
    p7 = Point(0, 1)
    p8 = Point(2, 0)
    p9 = Point(2, 1)

    l1 = Line(p1, p2)
    l2 = Line(p3, p4)
    l3 = Line(p3, p5)
    l4 = Line(p1, p6)
    l5 = Line(p1, p7)
    l6 = Line(p8, p9)
    l7 = Line(p2, p9)

    # Basic stuff
    assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
    assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
    assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
    raises(ValueError, "Line((1, 1), 1)")
    assert Line(p1, p2) == Line(p2, p1)
    assert l1 == l2
    assert l1 != l3
    assert l1.slope == 1
    assert l3.slope == oo
    assert l4.slope == 0
    assert l4.coefficients == (0, 1, 0)
    assert l4.equation(x=x, y=y) == y
    assert l5.slope == oo
    assert l5.coefficients == (1, 0, 0)
    assert l5.equation() == x
    assert l6.equation() == x - 2
    assert l7.equation() == y - 1
    assert p1 in l1  # is p1 on the line l1?
    assert p1 not in l3

    assert simplify(l1.equation()) in (x - y, y - x)
    assert simplify(l3.equation()) in (x - x1, x1 - x)

    assert l2.arbitrary_point() in l2
    for ind in xrange(0, 5):
        assert l3.random_point() in l3

    # Orthogonality
    p1_1 = Point(-x1, x1)
    l1_1 = Line(p1, p1_1)
    assert l1.perpendicular_line(p1) == l1_1
    assert Line.is_perpendicular(l1, l1_1)
    assert Line.is_perpendicular(l1, l2) == False

    # Parallelity
    p2_1 = Point(-2 * x1, 0)
    l2_1 = Line(p3, p5)
    assert l2.parallel_line(p1_1) == Line(p2_1, p1_1)
    assert l2_1.parallel_line(p1) == Line(p1, Point(0, 2))
    assert Line.is_parallel(l1, l2)
    assert Line.is_parallel(l2, l3) == False
    assert Line.is_parallel(l2, l2.parallel_line(p1_1))
    assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))

    # Intersection
    assert intersection(l1, p1) == [p1]
    assert intersection(l1, p5) == []
    assert intersection(l1, l2) in [[l1], [l2]]
    assert intersection(l1, l1.parallel_line(p5)) == []

    # Concurrency
    l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
    assert Line.is_concurrent(l1, l3)
    assert Line.is_concurrent(l1, l3, l3_1)
    assert Line.is_concurrent(l1, l1_1, l3) == False

    # Projection
    assert l2.projection(p4) == p4
    assert l1.projection(p1_1) == p1
    assert l3.projection(p2) == Point(x1, 1)

    # Finding angles
    l1_1 = Line(p1, Point(5, 0))
    assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf() / 4)

    # Testing Rays and Segments (very similar to Lines)
    assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2))
    assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0))
    assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
    assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1))
    assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1))
    assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
    # XXX don't know why this fails without str
    assert str(Ray((1, 1), angle=4.2 * pi)) == str(Ray(Point(1, 1), Point(2, 1 + C.tan(0.2 * pi))))
    assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + C.tan(5)))
    raises(ValueError, "Ray((1, 1), 1)")

    r1 = Ray(p1, Point(-1, 5))
    r2 = Ray(p1, Point(-1, 1))
    r3 = Ray(p3, p5)
    assert l1.projection(r1) == Ray(p1, p2)
    assert l1.projection(r2) == p1
    assert r3 != r1
    t = Symbol("t", real=True)
    assert Ray((1, 1), angle=pi / 4).arbitrary_point() == Point(1 / (1 - t), 1 / (1 - t))

    s1 = Segment(p1, p2)
    s2 = Segment(p1, p1_1)
    assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
    assert s2.length == sqrt(2 * (x1 ** 2))
    assert s1.perpendicular_bisector() == Line(Point(0, 1), Point(1, 0))
    assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t)

    # Segment contains
    a, b = symbols("a,b")
    s = Segment((0, a), (0, b))
    assert Point(0, (a + b) / 2) in s
    s = Segment((a, 0), (b, 0))
    assert Point((a + b) / 2, 0) in s
    assert (Point(2 * a, 0) in s) is False  # XXX should be None?

    # Testing distance from a Segment to an object
    s1 = Segment(Point(0, 0), Point(1, 1))
    s2 = Segment(Point(half, half), Point(1, 0))
    pt1 = Point(0, 0)
    pt2 = Point(Rational(3) / 2, Rational(3) / 2)
    assert s1.distance(pt1) == 0
    assert s2.distance(pt1) == 2 ** (half) / 2
    assert s2.distance(pt2) == 2 ** (half)

    # Special cases of projection and intersection
    r1 = Ray(Point(1, 1), Point(2, 2))
    r2 = Ray(Point(2, 2), Point(0, 0))
    r3 = Ray(Point(1, 1), Point(-1, -1))
    r4 = Ray(Point(0, 4), Point(-1, -5))
    assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
    assert intersection(r1, r3) == [Point(1, 1)]
    assert r1.projection(r3) == Point(1, 1)
    assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))

    r5 = Ray(Point(0, 0), Point(0, 1))
    r6 = Ray(Point(0, 0), Point(0, 2))
    assert r5 in r6
    assert r6 in r5

    s1 = Segment(Point(0, 0), Point(2, 2))
    s2 = Segment(Point(-1, 5), Point(-5, -10))
    s3 = Segment(Point(0, 4), Point(-2, 2))
    assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
    assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
    assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))

    l1 = Line(Point(0, 0), Point(3, 4))
    r1 = Ray(Point(0, 0), Point(3, 4))
    s1 = Segment(Point(0, 0), Point(3, 4))
    assert intersection(l1, l1) == [l1]
    assert intersection(l1, r1) == [r1]
    assert intersection(l1, s1) == [s1]
    assert intersection(r1, l1) == [r1]
    assert intersection(s1, l1) == [s1]

    entity1 = Segment(Point(-10, 10), Point(10, 10))
    entity2 = Segment(Point(-5, -5), Point(-5, 5))
    assert intersection(entity1, entity2) == []
Beispiel #16
0
def test_ellipse_geom():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p4 = Point(0, 1)

    e1 = Ellipse(p1, 1, 1)
    e2 = Ellipse(p2, half, 1)
    e3 = Ellipse(p1, y1, y1)
    c1 = Circle(p1, 1)
    c2 = Circle(p2, 1)
    c3 = Circle(Point(sqrt(2), sqrt(2)), 1)

    # Test creation with three points
    cen, rad = Point(3*half, 2), 5*half
    assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
    raises(
        GeometryError, lambda: Circle(Point(0, 0), Point(1, 1), Point(2, 2)))

    raises(ValueError, lambda: Ellipse(None, None, None, 1))
    raises(GeometryError, lambda: Circle(Point(0, 0)))

    # Basic Stuff
    assert Ellipse(None, 1, 1).center == Point(0, 0)
    assert e1 == c1
    assert e1 != e2
    assert p4 in e1
    assert p2 not in e2
    assert e1.area == pi
    assert e2.area == pi/2
    assert e3.area == pi*y1*abs(y1)
    assert c1.area == e1.area
    assert c1.circumference == e1.circumference
    assert e3.circumference == 2*pi*y1
    assert e1.plot_interval() == e2.plot_interval() == [t, -pi, pi]
    assert e1.plot_interval(x) == e2.plot_interval(x) == [x, -pi, pi]
    assert Ellipse(None, 1, None, 1).circumference == 2*pi
    assert c1.minor == 1
    assert c1.major == 1
    assert c1.hradius == 1
    assert c1.vradius == 1

    # Private Functions
    assert hash(c1) == hash(Circle(Point(1, 0), Point(0, 1), Point(0, -1)))
    assert c1 in e1
    assert (Line(p1, p2) in e1) is False
    assert e1.__cmp__(e1) == 0
    assert e1.__cmp__(Point(0, 0)) > 0

    # Encloses
    assert e1.encloses(Segment(Point(-0.5, -0.5), Point(0.5, 0.5))) is True
    assert e1.encloses(Line(p1, p2)) is False
    assert e1.encloses(Ray(p1, p2)) is False
    assert e1.encloses(e1) is False
    assert e1.encloses(
        Polygon(Point(-0.5, -0.5), Point(-0.5, 0.5), Point(0.5, 0.5))) is True
    assert e1.encloses(RegularPolygon(p1, 0.5, 3)) is True
    assert e1.encloses(RegularPolygon(p1, 5, 3)) is False
    assert e1.encloses(RegularPolygon(p2, 5, 3)) is False

    # with generic symbols, the hradius is assumed to contain the major radius
    M = Symbol('M')
    m = Symbol('m')
    c = Ellipse(p1, M, m).circumference
    _x = c.atoms(Dummy).pop()
    assert c == 4*M*Integral(
        sqrt((1 - _x**2*(M**2 - m**2)/M**2)/(1 - _x**2)), (_x, 0, 1))

    assert e2.arbitrary_point() in e2

    # Foci
    f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
    ef = Ellipse(Point(0, 0), 4, 2)
    assert ef.foci in [(f1, f2), (f2, f1)]

    # Tangents
    v = sqrt(2) / 2
    p1_1 = Point(v, v)
    p1_2 = p2 + Point(half, 0)
    p1_3 = p2 + Point(0, 1)
    assert e1.tangent_lines(p4) == c1.tangent_lines(p4)
    assert e2.tangent_lines(p1_2) == [Line(Point(3/2, 1), Point(3/2, 1/2))]
    assert e2.tangent_lines(p1_3) == [Line(Point(1, 2), Point(5/4, 2))]
    assert c1.tangent_lines(p1_1) != [Line(p1_1, Point(0, sqrt(2)))]
    assert c1.tangent_lines(p1) == []
    assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
    assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
    assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
    assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) is False
    assert c1.is_tangent(e1) is False
    assert c1.is_tangent(Ellipse(Point(2, 0), 1, 1)) is True
    assert c1.is_tangent(
        Polygon(Point(1, 1), Point(1, -1), Point(2, 0))) is True
    assert c1.is_tangent(
        Polygon(Point(1, 1), Point(1, 0), Point(2, 0))) is False
    assert Circle(Point(5, 5), 3).is_tangent(Circle(Point(0, 5), 1)) is False

    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(0, 0)) == \
        [Line(Point(0, 0), Point(77/25, 132/25)),
     Line(Point(0, 0), Point(33/5, 22/5))]
    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(3, 4)) == \
        [Line(Point(3, 4), Point(4, 4)), Line(Point(3, 4), Point(3, 5))]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(3, 3)) == \
        [Line(Point(3, 3), Point(4, 3)), Line(Point(3, 3), Point(3, 4))]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(5 - 2*sqrt(2), 5)) == \
        [Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 - sqrt(2))),
     Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 + sqrt(2))), ]

    e = Ellipse(Point(0, 0), 2, 1)
    assert e.normal_lines(Point(0, 0)) == \
        [Line(Point(0, 0), Point(0, 1)), Line(Point(0, 0), Point(1, 0))]
    assert e.normal_lines(Point(1, 0)) == \
        [Line(Point(0, 0), Point(1, 0))]
    assert e.normal_lines((0, 1)) == \
        [Line(Point(0, 0), Point(0, 1))]
    assert e.normal_lines(Point(1, 1), 2) == [
        Line(Point(-51/26, -1/5), Point(-25/26, 17/83)),
        Line(Point(28/29, -7/8), Point(57/29, -9/2))]
    # test the failure of Poly.intervals and checks a point on the boundary
    p = Point(sqrt(3), S.Half)
    assert p in e
    assert e.normal_lines(p, 2) == [
        Line(Point(-341/171, -1/13), Point(-170/171, 5/64)),
        Line(Point(26/15, -1/2), Point(41/15, -43/26))]
    # be sure to use the slope that isn't undefined on boundary
    e = Ellipse((0, 0), 2, 2*sqrt(3)/3)
    assert e.normal_lines((1, 1), 2) == [
        Line(Point(-64/33, -20/71), Point(-31/33, 2/13)),
        Line(Point(1, -1), Point(2, -4))]
    # general ellipse fails except under certain conditions
    e = Ellipse((0, 0), x, 1)
    assert e.normal_lines((x + 1, 0)) == [Line(Point(0, 0), Point(1, 0))]
    raises(NotImplementedError, lambda: e.normal_lines((x + 1, 1)))


    # Properties
    major = 3
    minor = 1
    e4 = Ellipse(p2, minor, major)
    assert e4.focus_distance == sqrt(major**2 - minor**2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major*(1 - ecc)
    assert e4.apoapsis == major*(1 + ecc)
    # independent of orientation
    e4 = Ellipse(p2, major, minor)
    assert e4.focus_distance == sqrt(major**2 - minor**2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major*(1 - ecc)
    assert e4.apoapsis == major*(1 + ecc)

    # Intersection
    l1 = Line(Point(1, -5), Point(1, 5))
    l2 = Line(Point(-5, -1), Point(5, -1))
    l3 = Line(Point(-1, -1), Point(1, 1))
    l4 = Line(Point(-10, 0), Point(0, 10))
    pts_c1_l3 = [Point(sqrt(2)/2, sqrt(2)/2), Point(-sqrt(2)/2, -sqrt(2)/2)]

    assert intersection(e2, l4) == []
    assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
    assert intersection(c1, l1) == [Point(1, 0)]
    assert intersection(c1, l2) == [Point(0, -1)]
    assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
    assert intersection(c1, c2) == [Point(0, 1), Point(1, 0)]
    assert intersection(c1, c3) == [Point(sqrt(2)/2, sqrt(2)/2)]
    assert e1.intersection(l1) == [Point(1, 0)]
    assert e2.intersection(l4) == []
    assert e1.intersection(Circle(Point(0, 2), 1)) == [Point(0, 1)]
    assert e1.intersection(Circle(Point(5, 0), 1)) == []
    assert e1.intersection(Ellipse(Point(2, 0), 1, 1)) == [Point(1, 0)]
    assert e1.intersection(Ellipse(Point(5, 0), 1, 1,)) == []
    assert e1.intersection(Point(2, 0)) == []
    assert e1.intersection(e1) == e1

    # some special case intersections
    csmall = Circle(p1, 3)
    cbig = Circle(p1, 5)
    cout = Circle(Point(5, 5), 1)
    # one circle inside of another
    assert csmall.intersection(cbig) == []
    # separate circles
    assert csmall.intersection(cout) == []
    # coincident circles
    assert csmall.intersection(csmall) == csmall

    v = sqrt(2)
    t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
    points = intersection(t1, c1)
    assert len(points) == 4
    assert Point(0, 1) in points
    assert Point(0, -1) in points
    assert Point(v/2, v/2) in points
    assert Point(v/2, -v/2) in points

    circ = Circle(Point(0, 0), 5)
    elip = Ellipse(Point(0, 0), 5, 20)
    assert intersection(circ, elip) in \
        [[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
    assert elip.tangent_lines(Point(0, 0)) == []
    elip = Ellipse(Point(0, 0), 3, 2)
    assert elip.tangent_lines(Point(3, 0)) == \
        [Line(Point(3, 0), Point(3, -12))]

    e1 = Ellipse(Point(0, 0), 5, 10)
    e2 = Ellipse(Point(2, 1), 4, 8)
    a = 53/17
    c = 2*sqrt(3991)/17
    ans = [Point(a - c/8, a/2 + c), Point(a + c/8, a/2 - c)]
    assert e1.intersection(e2) == ans
    e2 = Ellipse(Point(x, y), 4, 8)
    c = sqrt(3991)
    ans = [Point(-c/68 + a, 2*c/17 + a/2), Point(c/68 + a, -2*c/17 + a/2)]
    assert [p.subs({x: 2, y:1}) for p in e1.intersection(e2)] == ans

    # Combinations of above
    assert e3.is_tangent(e3.tangent_lines(p1 + Point(y1, 0))[0])

    e = Ellipse((1, 2), 3, 2)
    assert e.tangent_lines(Point(10, 0)) == \
        [Line(Point(10, 0), Point(1, 0)),
        Line(Point(10, 0), Point(14/5, 18/5))]

    # encloses_point
    e = Ellipse((0, 0), 1, 2)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(
        e.center + Point(e.hradius + Rational(1, 10), 0)) is False
    e = Ellipse((0, 0), 2, 1)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(
        e.center + Point(e.hradius + Rational(1, 10), 0)) is False
    assert c1.encloses_point(Point(1, 0)) is False
    assert c1.encloses_point(Point(0.3, 0.4)) is True

    assert e.scale(2, 3) == Ellipse((0, 0), 4, 3)
    assert e.scale(3, 6) == Ellipse((0, 0), 6, 6)
    assert e.rotate(pi) == e
    assert e.rotate(pi, (1, 2)) == Ellipse(Point(2, 4), 2, 1)
    raises(NotImplementedError, lambda: e.rotate(pi/3))

    # transformations
    c = Circle((1, 1), 2)
    assert c.scale(-1) == Circle((-1, 1), 2)
    assert c.scale(y=-1) == Circle((1, -1), 2)
    assert c.scale(2) == Ellipse((2, 1), 4, 2)
Beispiel #17
0
def test_intersection_3d():
    p1 = Point3D(0, 0, 0)
    p2 = Point3D(1, 1, 1)

    l1 = Line3D(p1, p2)
    l2 = Line3D(Point3D(0, 0, 0), Point3D(3, 4, 0))

    r1 = Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
    r2 = Ray3D(Point3D(0, 0, 0), Point3D(3, 4, 0))

    s1 = Segment3D(Point3D(0, 0, 0), Point3D(3, 4, 0))

    assert intersection(l1, p1) == [p1]
    assert intersection(l1, Point3D(x1, 1 + x1, 1)) == []
    assert intersection(l1, l1.parallel_line(p1)) == [
        Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
    ]
    assert intersection(l2, r2) == [r2]
    assert intersection(l2, s1) == [s1]
    assert intersection(r2, l2) == [r2]
    assert intersection(r1, Ray3D(Point3D(1, 1, 1),
                                  Point3D(-1, -1, -1))) == [Point3D(1, 1, 1)]
    assert intersection(r1, Segment3D(Point3D(0, 0, 0), Point3D(
        2, 2, 2))) == [Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
    assert intersection(Ray3D(Point3D(1, 0, 0), Point3D(-1, 0, 0)), Ray3D(Point3D(0, 1, 0), Point3D(0, -1, 0))) \
           == [Point3D(0, 0, 0)]
    assert intersection(r1, Ray3D(Point3D(2, 2, 2), Point3D(0, 0, 0))) == \
           [Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
    assert intersection(s1, r2) == [s1]

    assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).intersection(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) == \
           [Point3D(2, 2, 1)]
    assert Line3D((0, 1, 2),
                  (0, 2, 3)).intersection(Line3D(
                      (0, 1, 2), (0, 1, 1))) == [Point3D(0, 1, 2)]
    assert Line3D((0, 0), (t, t)).intersection(Line3D((0, 1), (t, t))) == \
           [Point3D(t, t)]

    assert Ray3D(Point3D(0, 0, 0), Point3D(0, 4, 0)).intersection(
        Ray3D(Point3D(0, 1, 1), Point3D(0, -1, 1))) == []
Beispiel #18
0
def test_line3d():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    z = Symbol('z', real=True)
    k = Symbol('k', real=True)
    x1 = Symbol('x1', real=True)
    y1 = Symbol('y1', real=True)
    p1 = Point3D(0, 0, 0)
    p2 = Point3D(1, 1, 1)
    p3 = Point3D(x1, x1, x1)
    p4 = Point3D(y1, y1, y1)
    p5 = Point3D(x1, 1 + x1, 1)
    p6 = Point3D(1, 0, 1)
    p7 = Point3D(0, 1, 1)
    p8 = Point3D(2, 0, 3)
    p9 = Point3D(2, 1, 4)

    l1 = Line3D(p1, p2)
    l2 = Line3D(p3, p4)
    l3 = Line3D(p3, p5)
    l4 = Line3D(p1, p6)
    l5 = Line3D(p1, p7)
    l6 = Line3D(p8, p9)
    l7 = Line3D(p2, p9)
    raises(ValueError, lambda: Line3D(Point3D(0, 0, 0), Point3D(0, 0, 0)))

    assert Line3D((1, 1, 1), direction_ratio=[2, 3, 4]) == \
        Line3D(Point3D(1, 1, 1), Point3D(3, 4, 5))
    assert Line3D((1, 1, 1), direction_ratio=[1, 5, 7 ]) == \
        Line3D(Point3D(1, 1, 1), Point3D(2, 6, 8))
    assert Line3D((1, 1, 1), direction_ratio=[1, 2, 3]) == \
        Line3D(Point3D(1, 1, 1), Point3D(2, 3, 4))
    raises(TypeError, lambda: Line3D((1, 1), 1))
    assert Line3D(p1, p2) != Line3D(p2, p1)
    assert l1 != l3
    assert l1.is_parallel(l1)  # same as in 2D
    assert l1 != l2
    assert l1.direction_ratio == [1, 1, 1]
    assert l1.length == oo
    assert l1.equation() == (x, y, z, k)
    assert l2.equation() == \
        ((x - x1)/(-x1 + y1), (-x1 + y)/(-x1 + y1), (-x1 + z)/(-x1 + y1), k)
    assert p1 in l1
    assert p1 not in l3

    # Orthogonality
    p1_1 = Point3D(x1, x1, x1)
    l1_1 = Line3D(p1, p1_1)
    assert Line3D.is_perpendicular(l1, l2) is False
    p = l1.arbitrary_point()
    raises(NotImplementedError , lambda: l1.perpendicular_segment(p))

    # Parallelity
    assert l1.parallel_line(p1_1) == Line3D(Point3D(x1, x1, x1),
        Point3D(x1 + 1, x1 + 1, x1 + 1))
    assert l1.parallel_line(p1_1.args) == \
        Line3D(Point3D(x1, x1, x1), Point3D(x1 + 1, x1 + 1, x1 + 1))

    # Intersection
    assert intersection(l1, p1) == [p1]
    assert intersection(l1, p5) == []
    assert intersection(l1, l1.parallel_line(p1)) == [
        Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1))]
    # issue 8517
    line3 = Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1))
    line4 = Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))
    assert line3.intersection(line4) == [Point3D(2, 2, 1)]
    assert line3.is_parallel(line4) is False
    assert Line3D((0, 1, 2), (0, 2, 3)).intersection(
        Line3D((0, 1, 2), (0, 1, 1))) == []
    ray0 = Ray3D((0, 0), (3, 0))
    ray1 = Ray3D((1, 0), (3, 0))
    assert ray0.intersection(ray1) == [ray1]
    assert ray1.intersection(ray0) == [ray1]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((1, 0), (2, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((1, 0), (2, 0)).intersection(
        Segment3D((0, 0), (3, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((3, 0), (4, 0))) == [Point3D((3, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((2, 0), (5, 0))) == [Segment3D((3, 0), (2, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((-2, 0), (1, 0))) == [Segment3D((0, 0), (1, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((-2, 0), (0, 0))) == [Point3D(0, 0, 0)]
    # issue 7757
    p = Ray3D(Point3D(1, 0, 0), Point3D(-1, 0, 0))
    q = Ray3D(Point3D(0, 1, 0), Point3D(0, -1, 0))
    assert intersection(p, q) == [Point3D(0, 0, 0)]

    # Concurrency
    assert Line3D.are_concurrent(l1) is False
    assert Line3D.are_concurrent(l1, l2)
    assert Line3D.are_concurrent(l1, l1_1, l3) is False
    parallel_1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
    parallel_2 = Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))
    assert Line3D.are_concurrent(parallel_1, parallel_2) == False

    # Finding angles
    l1_1 = Line3D(p1, Point3D(5, 0, 0))
    assert Line3D.angle_between(l1, l1_1), acos(sqrt(3)/3)

    # Testing Rays and Segments (very similar to Lines)
    assert Ray3D((1, 1, 1), direction_ratio=[4, 4, 4]) == \
        Ray3D(Point3D(1, 1, 1), Point3D(5, 5, 5))
    assert Ray3D((1, 1, 1), direction_ratio=[1, 2, 3]) == \
        Ray3D(Point3D(1, 1, 1), Point3D(2, 3, 4))
    assert Ray3D((1, 1, 1), direction_ratio=[1, 1, 1]) == \
        Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2))

    r1 = Ray3D(p1, Point3D(-1, 5, 0))
    r2 = Ray3D(p1, Point3D(-1, 1, 1))
    r3 = Ray3D(p1, p2)
    r4 = Ray3D(p2, p1)
    r5 = Ray3D(Point3D(0, 1, 1), Point3D(1, 2, 0))
    assert l1.projection(r1) == [
        Ray3D(Point3D(0, 0, 0), Point3D(4/3, 4/3, 4/3))]
    assert l1.projection(r2) == [
        Ray3D(Point3D(0, 0, 0), Point3D(1/3, 1/3, 1/3))]
    assert r3 != r1
    t = Symbol('t', real=True)
    assert Ray3D((1, 1, 1), direction_ratio=[1, 2, 3]).arbitrary_point() == \
        Point3D(t + 1, 2*t + 1, 3*t + 1)
    r6 = Ray3D(Point3D(0, 0, 0), Point3D(0, 4, 0))
    r7 = Ray3D(Point3D(0, 1, 1), Point3D(0, -1, 1))
    assert r6.intersection(r7) == []

    s1 = Segment3D(p1, p2)
    s2 = Segment3D(p3, p4)
    assert s1.midpoint == \
        Point3D(Rational(1, 2), Rational(1, 2), Rational(1, 2))
    assert s2.length == sqrt(3)*sqrt((x1 - y1)**2)
    assert Segment3D((1, 1, 1), (2, 3, 4)).arbitrary_point() == \
        Point3D(t + 1, 2*t + 1, 3*t + 1)

    # Segment contains
    s = Segment3D((0, 1, 0), (0, 1, 0))
    assert Point3D(0, 1, 0) in s
    s = Segment3D((1, 0, 0), (1, 0, 0))
    assert Point3D(1, 0, 0) in s

    # Testing distance from a Segment to an object
    s1 = Segment3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
    s2 = Segment3D(Point3D(1/2, 1/2, 1/2), Point3D(1, 0, 1))
    pt1 = Point3D(0, 0, 0)
    pt2 = Point3D(Rational(3)/2, Rational(3)/2, Rational(3)/2)
    assert s1.distance(pt1) == 0
    assert s2.distance(pt1) == sqrt(3)/2
    assert s2.distance(pt2) == 2
    assert s1.distance((0,0,0)) == 0
    assert s2.distance((0,0,0)) == sqrt(3)/2
    # Line to point
    p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1)
    s = Line3D(p1, p2)
    assert s.distance(Point3D(-1, 1, 1)) == 2*sqrt(6)/3
    assert s.distance(Point3D(1, -1, 1)) == 2*sqrt(6)/3
    assert s.distance(Point3D(2, 2, 2)) == 0
    assert s.distance((2, 2, 2)) == 0
    assert s.distance((1, -1, 1)) == 2*sqrt(6)/3
    assert Line3D((0, 0, 0), (0, 1, 0)).distance(p1) == 0
    assert Line3D((0, 0, 0), (0, 1, 0)).distance(p2) == sqrt(2)
    assert Line3D((0, 0, 0), (1, 0, 0)).distance(p1) == 0
    assert Line3D((0, 0, 0), (1, 0, 0)).distance(p2) == sqrt(2)
    # Ray to point
    r = Ray3D(p1, p2)
    assert r.distance(Point3D(-1, -1, -1)) == sqrt(3)
    assert r.distance(Point3D(1, 1, 1)) == 0
    assert r.distance((-1, -1, -1)) == sqrt(3)
    assert r.distance((1, 1, 1)) == 0
    assert Ray3D((1, 1, 1), (2, 2, 2)).distance(Point3D(1.5, 3, 1)) == \
        sqrt(17)/2


    # Special cases of projection and intersection
    r1 = Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
    r2 = Ray3D(Point3D(2, 2, 2), Point3D(0, 0, 0))
    r3 = Ray3D(Point3D(1, 1, 1), Point3D(-1, -1, -1))
    r4 = Ray3D(Point3D(0, 4, 2), Point3D(-1, -5, -1))
    r5 = Ray3D(Point3D(2, 2, 2), Point3D(3, 3, 3))
    assert intersection(r1, r2) == \
        [Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
    assert intersection(r1, r3) == [Point3D(1, 1, 1)]

    r5 = Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
    r6 = Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2))
    assert r5 in r6
    assert r6 in r5

    s1 = Segment3D(Point3D(0, 0, 0), Point3D(2, 2, 2))
    s2 = Segment3D(Point3D(-1, 5, 2), Point3D(-5, -10, 0))
    assert intersection(r1, s1) == [
        Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]

    l1 = Line3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
    r1 = Ray3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
    s1 = Segment3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
    assert intersection(l1, r1) == [r1]
    assert intersection(l1, s1) == [s1]
    assert intersection(r1, l1) == [r1]
    assert intersection(s1, r1) == [s1]

    # check that temporary symbol is Dummy
    assert Line3D((0, 0), (t, t)).perpendicular_line((0, 1)) == \
        Line3D(Point3D(0, 1, 0), Point3D(1/2, 1/2, 0))
    assert Line3D((0, 0), (t, t)).perpendicular_segment((0, 1)) == \
        Segment3D(Point3D(0, 1, 0), Point3D(1/2, 1/2, 0))
    assert Line3D((0, 0), (t, t)).intersection(Line3D((0, 1), (t, t))) == \
        [Point3D(t, t, 0)]
    assert Line3D((0, 0, 0), (x, y, z)).contains((2*x, 2*y, 2*z))

    # Test is_perpendicular
    perp_1 = Line3D(p1, Point3D(0, 1, 0))
    assert Line3D.is_perpendicular(parallel_1, perp_1) is True
    assert Line3D.is_perpendicular(parallel_1, parallel_2) is False

    # Test projection
    assert parallel_1.projection(Point3D(5, 5, 0)) == Point3D(5, 0, 0)
    assert parallel_1.projection(parallel_2) == [parallel_1]
    raises(GeometryError, lambda: parallel_1.projection(Plane(p1, p2, p6)))

    # Test __new__
    assert Line3D(perp_1) == perp_1
    raises(ValueError, lambda: Line3D(p1))

    # Test contains
    pt2d = Point(1.0, 1.0)
    assert perp_1.contains(pt2d) is False

    # Test equals
    assert perp_1.equals(pt2d) is False
    col1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
    col2 = Line3D(Point3D(-5, 0, 0), Point3D(-1, 0, 0))
    assert col1.equals(col2) is True
    assert col1.equals(perp_1) is False

    # Begin ray
    # Test __new__
    assert Ray3D(col1) == Ray3D(p1, Point3D(1, 0, 0))
    raises(ValueError, lambda: Ray3D(pt2d))

    # Test zdirection
    negz = Ray3D(p1, Point3D(0, 0, -1))
    assert negz.zdirection == S.NegativeInfinity

    # Test contains
    assert negz.contains(Segment3D(p1, Point3D(0, 0, -10))) is True
    assert negz.contains(Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))) is False
    posy = Ray3D(p1, Point3D(0, 1, 0))
    posz = Ray3D(p1, Point3D(0, 0, 1))
    assert posy.contains(p1) is True
    assert posz.contains(p1) is True
    assert posz.contains(pt2d) is False
    ray1 = Ray3D(Point3D(1, 1, 1), Point3D(1, 0, 0))
    raises(TypeError, lambda: ray1.contains([]))

    # Test equals
    assert negz.equals(pt2d) is False
    assert negz.equals(negz) is True

    assert ray1.is_similar(Line3D(Point3D(1, 1, 1), Point3D(1, 0, 0))) is True
    assert ray1.is_similar(perp_1) is False
    raises(NotImplementedError, lambda: ray1.is_similar(ray1))

    # Begin Segment
    seg1 = Segment3D(p1, Point3D(1, 0, 0))
    raises(TypeError, lambda: seg1.contains([]))
    seg2= Segment3D(Point3D(2, 2, 2), Point3D(3, 2, 2))
    assert seg1.contains(seg2) is False
Beispiel #19
0
def test_line_geom():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    x1 = Symbol('x1', real=True)
    y1 = Symbol('y1', real=True)
    half = Rational(1, 2)
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x1)
    p4 = Point(y1, y1)
    p5 = Point(x1, 1 + x1)
    p6 = Point(1, 0)
    p7 = Point(0, 1)
    p8 = Point(2, 0)
    p9 = Point(2, 1)

    l1 = Line(p1, p2)
    l2 = Line(p3, p4)
    l3 = Line(p3, p5)
    l4 = Line(p1, p6)
    l5 = Line(p1, p7)
    l6 = Line(p8, p9)
    l7 = Line(p2, p9)
    raises(ValueError, lambda: Line(Point(0, 0), Point(0, 0)))

    # Basic stuff
    assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
    assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
    assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
    raises(TypeError, lambda: Line((1, 1), 1))
    assert Line(p1, p2) == Line(p1, p2)
    assert Line(p1, p2) != Line(p2, p1)
    assert l1 != l2
    assert l1 != l3
    assert l1.slope == 1
    assert l1.length == oo
    assert l3.slope == oo
    assert l4.slope == 0
    assert l4.coefficients == (0, 1, 0)
    assert l4.equation(x=x, y=y) == y
    assert l5.slope == oo
    assert l5.coefficients == (1, 0, 0)
    assert l5.equation() == x
    assert l6.equation() == x - 2
    assert l7.equation() == y - 1
    assert p1 in l1  # is p1 on the line l1?
    assert p1 not in l3
    assert Line((-x, x), (-x + 1, x - 1)).coefficients == (1, 1, 0)

    assert simplify(l1.equation()) in (x - y, y - x)
    assert simplify(l3.equation()) in (x - x1, x1 - x)

    assert Line(p1, p2).scale(2, 1) == Line(p1, p9)

    assert l2.arbitrary_point() in l2
    for ind in range(0, 5):
        assert l3.random_point() in l3

    # Orthogonality
    p1_1 = Point(-x1, x1)
    l1_1 = Line(p1, p1_1)
    assert l1.perpendicular_line(p1.args).equals( Line(Point(0, 0), Point(1, -1)) )
    assert l1.perpendicular_line(p1).equals( Line(Point(0, 0), Point(1, -1)) )
    assert Line.is_perpendicular(l1, l1_1)
    assert Line.is_perpendicular(l1, l2) is False
    p = l1.random_point()
    assert l1.perpendicular_segment(p) == p

    # Parallelity
    l2_1 = Line(p3, p5)
    assert l2.parallel_line(p1_1).equals( Line(Point(-x1, x1), Point(-y1, 2*x1 - y1)) )
    assert l2_1.parallel_line(p1.args).equals( Line(Point(0, 0), Point(0, -1)) )
    assert l2_1.parallel_line(p1).equals( Line(Point(0, 0), Point(0, -1)) )
    assert Line.is_parallel(l1, l2)
    assert Line.is_parallel(l2, l3) is False
    assert Line.is_parallel(l2, l2.parallel_line(p1_1))
    assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))

    # Intersection
    assert intersection(l1, p1) == [p1]
    assert intersection(l1, p5) == []
    assert intersection(l1, l2) in [[l1], [l2]]
    assert intersection(l1, l1.parallel_line(p5)) == []

    # Concurrency
    l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
    assert Line.are_concurrent(l1) is False
    assert Line.are_concurrent(l1, l3)
    assert Line.are_concurrent(l1, l1, l1, l3)
    assert Line.are_concurrent(l1, l3, l3_1)
    assert Line.are_concurrent(l1, l1_1, l3) is False

    # Projection
    assert l2.projection(p4) == p4
    assert l1.projection(p1_1) == p1
    assert l3.projection(p2) == Point(x1, 1)
    raises(GeometryError, lambda: Line(Point(0, 0), Point(1, 0))
           .projection(Circle(Point(0, 0), 1)))

    # Finding angles
    l1_1 = Line(p1, Point(5, 0))
    assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf()/4)
    a = Point(1, 2, 3, 4)
    b = a.orthogonal_direction
    o = a.origin
    assert Line(a, o).angle_between(Line(b, o)) == pi/2

    # Testing Rays and Segments (very similar to Lines)
    assert Ray((1, 1), angle=pi/4) == Ray((1, 1), (2, 2))
    assert Ray((1, 1), angle=pi/2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=-pi/2) == Ray((1, 1), (1, 0))
    assert Ray((1, 1), angle=-3*pi/2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=5*pi/2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=5.0*pi/2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
    assert Ray((1, 1), angle=3.0*pi) == Ray((1, 1), (0, 1))
    assert Ray((1, 1), angle=4.0*pi) == Ray((1, 1), (2, 1))
    assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
    assert Ray((1, 1), angle=4.05*pi) == Ray(Point(1, 1),
               Point(2, -sqrt(5)*sqrt(2*sqrt(5) + 10)/4 - sqrt(2*sqrt(5) + 10)/4 + 2 + sqrt(5)))
    assert Ray((1, 1), angle=4.02*pi) == Ray(Point(1, 1),
               Point(2, 1 + tan(4.02*pi)))
    assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + tan(5)))
    raises(TypeError, lambda: Ray((1, 1), 1))

    # issue 7963
    r = Ray((0, 0), angle=x)
    assert r.subs(x, 3*pi/4) == Ray((0, 0), (-1, 1))
    assert r.subs(x, 5*pi/4) == Ray((0, 0), (-1, -1))
    assert r.subs(x, -pi/4) == Ray((0, 0), (1, -1))
    assert r.subs(x, pi/2) == Ray((0, 0), (0, 1))
    assert r.subs(x, -pi/2) == Ray((0, 0), (0, -1))

    r1 = Ray(p1, Point(-1, 5))
    r2 = Ray(p1, Point(-1, 1))
    r3 = Ray(p3, p5)
    r4 = Ray(p1, p2)
    r5 = Ray(p2, p1)
    r6 = Ray(Point(0, 1), Point(1, 2))
    r7 = Ray(Point(0.5, 0.5), Point(1, 1))
    assert l1.projection(r1) == Ray(Point(0, 0), Point(2, 2))
    assert l1.projection(r2) == p1
    assert r3 != r1
    t = Symbol('t', real=True)
    assert Ray((1, 1), angle=pi/4).arbitrary_point() == \
        Point(t + 1, t + 1)
    r8 = Ray(Point(0, 0), Point(0, 4))
    r9 = Ray(Point(0, 1), Point(0, -1))
    assert r8.intersection(r9) == [Segment(Point(0, 0), Point(0, 1))]

    s1 = Segment(p1, p2)
    s2 = Segment(p1, p1_1)
    assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
    assert s2.length == sqrt( 2*(x1**2) )
    assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2*t)
    aline = Line(Point(1/2, 1/2), Point(3/2, -1/2))
    assert s1.perpendicular_bisector().equals(aline)
    on_line = Segment(Point(1/2, 1/2), Point(3/2, -1/2)).midpoint
    assert s1.perpendicular_bisector(on_line) == Segment(s1.midpoint, on_line)
    assert s1.perpendicular_bisector(on_line + (1, 0)).equals(aline)
    # intersections
    assert s1.intersection(Line(p6, p9)) == []
    s3 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
    assert s1.intersection(s3) == [s3]
    assert s3.intersection(s1) == [s3]
    assert r4.intersection(s3) == [s3]
    assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
    assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == \
        [Segment(p1, Point(0.5, 0.5))]
    s3 = Segment(Point(1, 1), Point(2, 2))
    assert s1.intersection(s3) == [Point(1, 1)]
    s3 = Segment(Point(0.5, 0.5), Point(1.5, 1.5))
    assert s1.intersection(s3) == [Segment(Point(0.5, 0.5), p2)]
    assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
    assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
    assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == \
        [Segment(p1, Point(0.5, 0.5))]
    assert r4.intersection(r5) == [s1]
    assert r5.intersection(r6) == []
    assert r4.intersection(r7) == r7.intersection(r4) == [r7]

    # Segment contains
    a, b = symbols('a,b', real=True)
    s = Segment((0, a), (0, b))
    assert Point(0, (a + b)/2) in s
    s = Segment((a, 0), (b, 0))
    assert Point((a + b)/2, 0) in s

    raises(Undecidable, lambda: Point(2*a, 0) in s)

    # Testing distance from a Segment to an object
    s1 = Segment(Point(0, 0), Point(1, 1))
    s2 = Segment(Point(half, half), Point(1, 0))
    pt1 = Point(0, 0)
    pt2 = Point(Rational(3)/2, Rational(3)/2)
    assert s1.distance(pt1) == 0
    assert s1.distance((0, 0)) == 0
    assert s2.distance(pt1) == 2**(half)/2
    assert s2.distance(pt2) == 2**(half)
    # Line to point
    p1, p2 = Point(0, 0), Point(1, 1)
    s = Line(p1, p2)
    assert s.distance(Point(-1, 1)) == sqrt(2)
    assert s.distance(Point(1, -1)) == sqrt(2)
    assert s.distance(Point(2, 2)) == 0
    assert s.distance((-1, 1)) == sqrt(2)
    assert Line((0, 0), (0, 1)).distance(p1) == 0
    assert Line((0, 0), (0, 1)).distance(p2) == 1
    assert Line((0, 0), (1, 0)).distance(p1) == 0
    assert Line((0, 0), (1, 0)).distance(p2) == 1
    m = symbols('m', real=True)
    l = Line((0, 5), slope=m)
    p = Point(2, 3)
    assert (l.distance(p) - 2*abs(m + 1)/sqrt(m**2 + 1)).equals(0)
    # Ray to point
    r = Ray(p1, p2)
    assert r.distance(Point(-1, -1)) == sqrt(2)
    assert r.distance(Point(1, 1)) == 0
    assert r.distance(Point(-1, 1)) == sqrt(2)
    assert Ray((1, 1), (2, 2)).distance(Point(1.5, 3)) == 3*sqrt(2)/4
    assert r.distance((1, 1)) == 0

    #Line contains
    p1, p2 = Point(0, 1), Point(3, 4)
    l = Line(p1, p2)
    assert l.contains(p1) is True
    assert l.contains((0, 1)) is True
    assert l.contains((0, 0)) is False

    #Ray contains
    p1, p2 = Point(0, 0), Point(4, 4)
    r = Ray(p1, p2)
    assert r.contains(p1) is True
    assert r.contains((1, 1)) is True
    assert r.contains((1, 3)) is False
    s = Segment((1, 1), (2, 2))
    assert r.contains(s) is True
    s = Segment((1, 2), (2, 5))
    assert r.contains(s) is False
    r1 = Ray((2, 2), (3, 3))
    assert r.contains(r1) is True
    r1 = Ray((2, 2), (3, 5))
    assert r.contains(r1) is False


    # Special cases of projection and intersection
    r1 = Ray(Point(1, 1), Point(2, 2))
    r2 = Ray(Point(2, 2), Point(0, 0))
    r3 = Ray(Point(1, 1), Point(-1, -1))
    r4 = Ray(Point(0, 4), Point(-1, -5))
    r5 = Ray(Point(2, 2), Point(3, 3))
    assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
    assert intersection(r1, r3) == [Point(1, 1)]
    assert r1.projection(r3) == Point(1, 1)
    assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))

    r5 = Ray(Point(0, 0), Point(0, 1))
    r6 = Ray(Point(0, 0), Point(0, 2))
    assert r5 in r6
    assert r6 in r5

    s1 = Segment(Point(0, 0), Point(2, 2))
    s2 = Segment(Point(-1, 5), Point(-5, -10))
    s3 = Segment(Point(0, 4), Point(-2, 2))
    assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
    assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
    assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))

    l1 = Line(Point(0, 0), Point(3, 4))
    r1 = Ray(Point(0, 0), Point(3, 4))
    s1 = Segment(Point(0, 0), Point(3, 4))
    assert intersection(l1, l1) == [l1]
    assert intersection(l1, r1) == [r1]
    assert intersection(l1, s1) == [s1]
    assert intersection(r1, l1) == [r1]
    assert intersection(s1, l1) == [s1]

    entity1 = Segment(Point(-10, 10), Point(10, 10))
    entity2 = Segment(Point(-5, -5), Point(-5, 5))
    assert intersection(entity1, entity2) == []

    r1 = Ray(p1, Point(0, 1))
    r2 = Ray(Point(0, 1), p1)
    r3 = Ray(p1, p2)
    r4 = Ray(p2, p1)
    s1 = Segment(p1, Point(0, 1))
    assert Line(r1.source, r1.random_point()).slope == r1.slope
    assert Line(r2.source, r2.random_point()).slope == r2.slope
    assert Segment(Point(0, -1), s1.random_point()).slope == s1.slope
    p_r3 = r3.random_point()
    p_r4 = r4.random_point()
    assert p_r3.x >= p1.x and p_r3.y >= p1.y
    assert p_r4.x <= p2.x and p_r4.y <= p2.y
    p10 = Point(2000, 2000)
    s1 = Segment(p1, p10)
    p_s1 = s1.random_point()
    assert p1.x <= p_s1.x and p_s1.x <= p10.x and \
        p1.y <= p_s1.y and p_s1.y <= p10.y
    s2 = Segment(p10, p1)
    assert hash(s1) == hash(s2)
    p11 = p10.scale(2, 2)
    assert s1.is_similar(Segment(p10, p11))
    assert s1.is_similar(r1) is False
    assert (r1 in s1) is False
    assert Segment(p1, p2) in s1
    assert s1.plot_interval() == [t, 0, 1]
    assert s1 in Line(p1, p10)
    assert Line(p1, p10) != Line(p10, p1)
    assert Line(p1, p10) != p1
    assert Line(p1, p10).plot_interval() == [t, -5, 5]
    assert Ray((0, 0), angle=pi/4).plot_interval() == \
        [t, 0, 10]
Beispiel #20
0
def test_intersection_2d():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x1)
    p4 = Point(y1, y1)

    l1 = Line(p1, p2)
    l3 = Line(Point(0, 0), Point(3, 4))

    r1 = Ray(Point(1, 1), Point(2, 2))
    r2 = Ray(Point(0, 0), Point(3, 4))
    r4 = Ray(p1, p2)
    r6 = Ray(Point(0, 1), Point(1, 2))
    r7 = Ray(Point(0.5, 0.5), Point(1, 1))

    s1 = Segment(p1, p2)
    s2 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
    s3 = Segment(Point(0, 0), Point(3, 4))

    assert intersection(l1, p1) == [p1]
    assert intersection(l1, Point(x1, 1 + x1)) == []
    assert intersection(l1, Line(p3, p4)) in [[l1], [Line(p3, p4)]]
    assert intersection(l1, l1.parallel_line(Point(x1, 1 + x1))) == []
    assert intersection(l3, l3) == [l3]
    assert intersection(l3, r2) == [r2]
    assert intersection(l3, s3) == [s3]
    assert intersection(s3, l3) == [s3]
    assert intersection(Segment(Point(-10, 10), Point(10, 10)), Segment(Point(-5, -5), Point(-5, 5))) == []
    assert intersection(r2, l3) == [r2]
    assert intersection(r1, Ray(Point(2, 2), Point(0, 0))) == [Segment(Point(1, 1), Point(2, 2))]
    assert intersection(r1, Ray(Point(1, 1), Point(-1, -1))) == [Point(1, 1)]
    assert intersection(r1, Segment(Point(0, 0), Point(2, 2))) == [Segment(Point(1, 1), Point(2, 2))]

    assert r4.intersection(s2) == [s2]
    assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
    assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert r4.intersection(Ray(p2, p1)) == [s1]
    assert Ray(p2, p1).intersection(r6) == []
    assert r4.intersection(r7) == r7.intersection(r4) == [r7]
    assert Ray3D((0, 0), (3, 0)).intersection(Ray3D((1, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
    assert Ray3D((1, 0), (3, 0)).intersection(Ray3D((0, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
    assert Ray(Point(0, 0), Point(0, 4)).intersection(Ray(Point(0, 1), Point(0, -1))) == \
           [Segment(Point(0, 0), Point(0, 1))]

    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((1, 0), (2, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((1, 0), (2, 0)).intersection(
        Segment3D((0, 0), (3, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((3, 0), (4, 0))) == [Point3D((3, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((2, 0), (5, 0))) == [Segment3D((3, 0), (2, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((-2, 0), (1, 0))) == [Segment3D((0, 0), (1, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(
        Segment3D((-2, 0), (0, 0))) == [Point3D(0, 0)]
    assert s1.intersection(Segment(Point(1, 1), Point(2, 2))) == [Point(1, 1)]
    assert s1.intersection(Segment(Point(0.5, 0.5), Point(1.5, 1.5))) == [Segment(Point(0.5, 0.5), p2)]
    assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
    assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
    assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert s1.intersection(Line(Point(1, 0), Point(2, 1))) == []
    assert s1.intersection(s2) == [s2]
    assert s2.intersection(s1) == [s2]
Beispiel #21
0
def test_intersection_2d():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x1)
    p4 = Point(y1, y1)

    l1 = Line(p1, p2)
    l3 = Line(Point(0, 0), Point(3, 4))

    r1 = Ray(Point(1, 1), Point(2, 2))
    r2 = Ray(Point(0, 0), Point(3, 4))
    r4 = Ray(p1, p2)
    r6 = Ray(Point(0, 1), Point(1, 2))
    r7 = Ray(Point(0.5, 0.5), Point(1, 1))

    s1 = Segment(p1, p2)
    s2 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
    s3 = Segment(Point(0, 0), Point(3, 4))

    assert intersection(l1, p1) == [p1]
    assert intersection(l1, Point(x1, 1 + x1)) == []
    assert intersection(l1, Line(p3, p4)) in [[l1], [Line(p3, p4)]]
    assert intersection(l1, l1.parallel_line(Point(x1, 1 + x1))) == []
    assert intersection(l3, l3) == [l3]
    assert intersection(l3, r2) == [r2]
    assert intersection(l3, s3) == [s3]
    assert intersection(s3, l3) == [s3]
    assert intersection(Segment(Point(-10, 10), Point(10, 10)),
                        Segment(Point(-5, -5), Point(-5, 5))) == []
    assert intersection(r2, l3) == [r2]
    assert intersection(r1,
                        Ray(Point(2, 2),
                            Point(0,
                                  0))) == [Segment(Point(1, 1), Point(2, 2))]
    assert intersection(r1, Ray(Point(1, 1), Point(-1, -1))) == [Point(1, 1)]
    assert intersection(r1, Segment(Point(0, 0), Point(
        2, 2))) == [Segment(Point(1, 1), Point(2, 2))]

    assert r4.intersection(s2) == [s2]
    assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
    assert r4.intersection(Segment(Point(-1, -1), Point(
        0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert r4.intersection(Ray(p2, p1)) == [s1]
    assert Ray(p2, p1).intersection(r6) == []
    assert r4.intersection(r7) == r7.intersection(r4) == [r7]
    assert Ray3D((0, 0), (3, 0)).intersection(Ray3D(
        (1, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
    assert Ray3D((1, 0), (3, 0)).intersection(Ray3D(
        (0, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
    assert Ray(Point(0, 0), Point(0, 4)).intersection(Ray(Point(0, 1), Point(0, -1))) == \
           [Segment(Point(0, 0), Point(0, 1))]

    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (1, 0), (2, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((1, 0), (2, 0)).intersection(Segment3D(
        (0, 0), (3, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (3, 0), (4, 0))) == [Point3D((3, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (2, 0), (5, 0))) == [Segment3D((2, 0), (3, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (-2, 0), (1, 0))) == [Segment3D((0, 0), (1, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (-2, 0), (0, 0))) == [Point3D(0, 0)]
    assert s1.intersection(Segment(Point(1, 1), Point(2, 2))) == [Point(1, 1)]
    assert s1.intersection(Segment(Point(0.5, 0.5), Point(
        1.5, 1.5))) == [Segment(Point(0.5, 0.5), p2)]
    assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
    assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
    assert s1.intersection(Segment(Point(-1, -1), Point(
        0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert s1.intersection(Line(Point(1, 0), Point(2, 1))) == []
    assert s1.intersection(s2) == [s2]
    assert s2.intersection(s1) == [s2]

    assert asa(120, 8, 52) == \
           Triangle(
               Point(0, 0),
               Point(8, 0),
               Point(-4 * cos(19 * pi / 90) / sin(2 * pi / 45),
                     4 * sqrt(3) * cos(19 * pi / 90) / sin(2 * pi / 45)))
    assert Line((0, 0), (1, 1)).intersection(Ray((1, 0),
                                                 (1, 2))) == [Point(1, 1)]
    assert Line((0, 0), (1, 1)).intersection(Segment((1, 0),
                                                     (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (1, 1)).intersection(Ray((1, 0),
                                                (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (1, 1)).intersection(Segment((1, 0),
                                                    (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (10, 10)).contains(Segment((1, 1), (2, 2))) is True
    assert Segment((1, 1), (2, 2)) in Line((0, 0), (10, 10))
    assert s1.intersection(Ray((1, 1), (4, 4))) == [Point(1, 1)]
Beispiel #22
0
def test_intersection_3d():
    p1 = Point3D(0, 0, 0)
    p2 = Point3D(1, 1, 1)

    l1 = Line3D(p1, p2)
    l2 = Line3D(Point3D(0, 0, 0), Point3D(3, 4, 0))

    r1 = Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
    r2 = Ray3D(Point3D(0, 0, 0), Point3D(3, 4, 0))

    s1 = Segment3D(Point3D(0, 0, 0), Point3D(3, 4, 0))

    assert intersection(l1, p1) == [p1]
    assert intersection(l1, Point3D(x1, 1 + x1, 1)) == []
    assert intersection(l1, l1.parallel_line(p1)) == [Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1))]
    assert intersection(l2, r2) == [r2]
    assert intersection(l2, s1) == [s1]
    assert intersection(r2, l2) == [r2]
    assert intersection(r1, Ray3D(Point3D(1, 1, 1), Point3D(-1, -1, -1))) == [Point3D(1, 1, 1)]
    assert intersection(r1, Segment3D(Point3D(0, 0, 0), Point3D(2, 2, 2))) == [
        Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
    assert intersection(Ray3D(Point3D(1, 0, 0), Point3D(-1, 0, 0)), Ray3D(Point3D(0, 1, 0), Point3D(0, -1, 0))) \
           == [Point3D(0, 0, 0)]
    assert intersection(r1, Ray3D(Point3D(2, 2, 2), Point3D(0, 0, 0))) == \
           [Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
    assert intersection(s1, r2) == [s1]

    assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).intersection(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) == \
           [Point3D(2, 2, 1)]
    assert Line3D((0, 1, 2), (0, 2, 3)).intersection(Line3D((0, 1, 2), (0, 1, 1))) == [Point3D(0, 1, 2)]
    assert Line3D((0, 0), (t, t)).intersection(Line3D((0, 1), (t, t))) == \
           [Point3D(t, t)]

    assert Ray3D(Point3D(0, 0, 0), Point3D(0, 4, 0)).intersection(Ray3D(Point3D(0, 1, 1), Point3D(0, -1, 1))) == []
Beispiel #23
0
def test_line():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x1)
    p4 = Point(y1, y1)
    p5 = Point(x1, 1 + x1)
    p6 = Point(1, 0)
    p7 = Point(0, 1)
    p8 = Point(2, 0)
    p9 = Point(2, 1)

    l1 = Line(p1, p2)
    l2 = Line(p3, p4)
    l3 = Line(p3, p5)
    l4 = Line(p1, p6)
    l5 = Line(p1, p7)
    l6 = Line(p8, p9)
    l7 = Line(p2, p9)
    raises(ValueError, lambda: Line(Point(0, 0), Point(0, 0)))

    # Basic stuff
    assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
    assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
    assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
    raises(ValueError, lambda: Line((1, 1), 1))
    assert Line(p1, p2) == Line(p2, p1)
    assert l1 == l2
    assert l1 != l3
    assert l1.slope == 1
    assert l1.length == oo
    assert l3.slope == oo
    assert l4.slope == 0
    assert l4.coefficients == (0, 1, 0)
    assert l4.equation(x=x, y=y) == y
    assert l5.slope == oo
    assert l5.coefficients == (1, 0, 0)
    assert l5.equation() == x
    assert l6.equation() == x - 2
    assert l7.equation() == y - 1
    assert p1 in l1  # is p1 on the line l1?
    assert p1 not in l3
    assert Line((-x, x), (-x + 1, x - 1)).coefficients == (1, 1, 0)

    assert simplify(l1.equation()) in (x - y, y - x)
    assert simplify(l3.equation()) in (x - x1, x1 - x)

    assert Line(p1, p2).scale(2, 1) == Line(p1, p9)

    assert l2.arbitrary_point() in l2
    for ind in xrange(0, 5):
        assert l3.random_point() in l3

    # Orthogonality
    p1_1 = Point(-x1, x1)
    l1_1 = Line(p1, p1_1)
    assert l1.perpendicular_line(p1) == l1_1
    assert Line.is_perpendicular(l1, l1_1)
    assert Line.is_perpendicular(l1, l2) == False
    p = l1.random_point()
    assert l1.perpendicular_segment(p) == p

    # Parallelity
    p2_1 = Point(-2 * x1, 0)
    l2_1 = Line(p3, p5)
    assert l2.parallel_line(p1_1) == Line(p2_1, p1_1)
    assert l2_1.parallel_line(p1) == Line(p1, Point(0, 2))
    assert Line.is_parallel(l1, l2)
    assert Line.is_parallel(l2, l3) == False
    assert Line.is_parallel(l2, l2.parallel_line(p1_1))
    assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))

    # Intersection
    assert intersection(l1, p1) == [p1]
    assert intersection(l1, p5) == []
    assert intersection(l1, l2) in [[l1], [l2]]
    assert intersection(l1, l1.parallel_line(p5)) == []

    # Concurrency
    l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
    assert Line.is_concurrent(l1) == False
    assert Line.is_concurrent(l1, l3)
    assert Line.is_concurrent(l1, l3, l3_1)
    assert Line.is_concurrent(l1, l1_1, l3) == False

    # Projection
    assert l2.projection(p4) == p4
    assert l1.projection(p1_1) == p1
    assert l3.projection(p2) == Point(x1, 1)
    raises(GeometryError, lambda: Line(Point(0, 0), Point(1, 0)).projection(Circle(Point(0, 0), 1)))

    # Finding angles
    l1_1 = Line(p1, Point(5, 0))
    assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf() / 4)

    # Testing Rays and Segments (very similar to Lines)
    assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2))
    assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0))
    assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2))
    assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
    assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1))
    assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1))
    assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
    # XXX don't know why this fails without str
    assert str(Ray((1, 1), angle=4.2 * pi)) == str(Ray(Point(1, 1), Point(2, 1 + C.tan(0.2 * pi))))
    assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + C.tan(5)))
    raises(ValueError, lambda: Ray((1, 1), 1))

    r1 = Ray(p1, Point(-1, 5))
    r2 = Ray(p1, Point(-1, 1))
    r3 = Ray(p3, p5)
    r4 = Ray(p1, p2)
    r5 = Ray(p2, p1)
    r6 = Ray(Point(0, 1), Point(1, 2))
    r7 = Ray(Point(0.5, 0.5), Point(1, 1))
    assert l1.projection(r1) == Ray(p1, p2)
    assert l1.projection(r2) == p1
    assert r3 != r1
    t = Symbol("t", real=True)
    assert Ray((1, 1), angle=pi / 4).arbitrary_point() == Point(1 / (1 - t), 1 / (1 - t))

    s1 = Segment(p1, p2)
    s2 = Segment(p1, p1_1)
    assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
    assert s2.length == sqrt(2 * (x1 ** 2))
    assert s1.perpendicular_bisector() == Line(Point(0, 1), Point(1, 0))
    assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t)

    # intersections
    assert s1.intersection(Line(p6, p9)) == []
    s3 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
    assert s1.intersection(s3) == [s1]
    assert s3.intersection(s1) == [s3]
    assert r4.intersection(s3) == [s3]
    assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
    assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    s3 = Segment(Point(1, 1), Point(2, 2))
    assert s1.intersection(s3) == [Point(1, 1)]
    s3 = Segment(Point(0.5, 0.5), Point(1.5, 1.5))
    assert s1.intersection(s3) == [Segment(Point(0.5, 0.5), p2)]
    assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
    assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
    assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert r4.intersection(r5) == [s1]
    assert r5.intersection(r6) == []
    assert r4.intersection(r7) == r7.intersection(r4) == [r7]

    # Segment contains
    a, b = symbols("a,b")
    s = Segment((0, a), (0, b))
    assert Point(0, (a + b) / 2) in s
    s = Segment((a, 0), (b, 0))
    assert Point((a + b) / 2, 0) in s

    raises(Undecidable, lambda: Point(2 * a, 0) in s)

    # Testing distance from a Segment to an object
    s1 = Segment(Point(0, 0), Point(1, 1))
    s2 = Segment(Point(half, half), Point(1, 0))
    pt1 = Point(0, 0)
    pt2 = Point(Rational(3) / 2, Rational(3) / 2)
    assert s1.distance(pt1) == 0
    assert s2.distance(pt1) == 2 ** (half) / 2
    assert s2.distance(pt2) == 2 ** (half)

    # Special cases of projection and intersection
    r1 = Ray(Point(1, 1), Point(2, 2))
    r2 = Ray(Point(2, 2), Point(0, 0))
    r3 = Ray(Point(1, 1), Point(-1, -1))
    r4 = Ray(Point(0, 4), Point(-1, -5))
    r5 = Ray(Point(2, 2), Point(3, 3))
    assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
    assert intersection(r1, r3) == [Point(1, 1)]
    assert r1.projection(r3) == Point(1, 1)
    assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))

    r5 = Ray(Point(0, 0), Point(0, 1))
    r6 = Ray(Point(0, 0), Point(0, 2))
    assert r5 in r6
    assert r6 in r5

    s1 = Segment(Point(0, 0), Point(2, 2))
    s2 = Segment(Point(-1, 5), Point(-5, -10))
    s3 = Segment(Point(0, 4), Point(-2, 2))
    assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
    assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
    assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))

    l1 = Line(Point(0, 0), Point(3, 4))
    r1 = Ray(Point(0, 0), Point(3, 4))
    s1 = Segment(Point(0, 0), Point(3, 4))
    assert intersection(l1, l1) == [l1]
    assert intersection(l1, r1) == [r1]
    assert intersection(l1, s1) == [s1]
    assert intersection(r1, l1) == [r1]
    assert intersection(s1, l1) == [s1]

    entity1 = Segment(Point(-10, 10), Point(10, 10))
    entity2 = Segment(Point(-5, -5), Point(-5, 5))
    assert intersection(entity1, entity2) == []

    r1 = Ray(p1, Point(0, 1))
    r2 = Ray(Point(0, 1), p1)
    r3 = Ray(p1, p2)
    r4 = Ray(p2, p1)
    s1 = Segment(p1, Point(0, 1))
    assert Line(r1.source, r1.random_point()).slope == r1.slope
    assert Line(r2.source, r2.random_point()).slope == r2.slope
    assert Segment(Point(0, -1), s1.random_point()).slope == s1.slope
    p_r3 = r3.random_point()
    p_r4 = r4.random_point()
    assert p_r3.x >= p1.x and p_r3.y >= p1.y
    assert p_r4.x <= p2.x and p_r4.y <= p2.y
    p10 = Point(2000, 2000)
    s1 = Segment(p1, p10)
    p_s1 = s1.random_point()
    assert p1.x <= p_s1.x and p_s1.x <= p10.x and p1.y <= p_s1.y and p_s1.y <= p10.y
    s2 = Segment(p10, p1)

    assert hash(s1) == hash(s2)
    p11 = p10.scale(2, 2)
    assert s1.is_similar(Segment(p10, p11))
    assert s1.is_similar(r1) == False
    assert (r1 in s1) == False
    assert Segment(p1, p2) in s1
    assert s1.plot_interval() == [t, 0, 1]
    assert s1 in Line(p1, p10)
    assert Line(p1, p10) == Line(p10, p1)
    assert Line(p1, p10) != p1
    assert Line(p1, p10).plot_interval() == [t, -5, 5]
    assert Ray((0, 0), angle=pi / 4).plot_interval() == [t, 0, 5 * sqrt(2) / (1 + 5 * sqrt(2))]
Beispiel #24
0
def test_polygon():
    t = Triangle(Point(0, 0), Point(2, 0), Point(3, 3))
    assert Polygon(Point(0, 0), Point(1, 0), Point(2, 0), Point(3, 3)) == t
    assert Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(0, 0)) == t
    assert Polygon(Point(2, 0), Point(3, 3), Point(0, 0), Point(1, 0)) == t

    p1 = Polygon(Point(0, 0), Point(3, -1), Point(6, 0), Point(4, 5), Point(2, 3), Point(0, 3))
    p2 = Polygon(Point(6, 0), Point(3, -1), Point(0, 0), Point(0, 3), Point(2, 3), Point(4, 5))
    p3 = Polygon(Point(0, 0), Point(3, 0), Point(5, 2), Point(4, 4))
    p4 = Polygon(Point(0, 0), Point(4, 4), Point(5, 2), Point(3, 0))

    #
    # General polygon
    #
    assert p1 == p2
    assert len(p1) == 6
    assert len(p1.sides) == 6
    assert p1.perimeter == 5 + 2 * sqrt(10) + sqrt(29) + sqrt(8)
    assert p1.area == 22
    assert not p1.is_convex()
    assert p3.is_convex()
    assert p4.is_convex()  # ensure convex for both CW and CCW point specification

    #
    # Regular polygon
    #
    p1 = RegularPolygon(Point(0, 0), 10, 5)
    p2 = RegularPolygon(Point(0, 0), 5, 5)

    assert p1 != p2
    assert p1.interior_angle == 3 * pi / 5
    assert p1.exterior_angle == 2 * pi / 5
    assert p2.apothem == 5 * cos(pi / 5)
    assert p2.circumcircle == Circle(Point(0, 0), 5)
    assert p2.incircle == Circle(Point(0, 0), p2.apothem)
    assert p1.is_convex()

    #
    # Angles
    #
    angles = p4.angles
    assert feq(angles[Point(0, 0)].evalf(), Real("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Real("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Real("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Real("2.3561944901923449"))

    angles = p3.angles
    assert feq(angles[Point(0, 0)].evalf(), Real("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Real("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Real("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Real("2.3561944901923449"))

    #
    # Triangle
    #
    p1 = Point(0, 0)
    p2 = Point(5, 0)
    p3 = Point(0, 5)
    t1 = Triangle(p1, p2, p3)
    t2 = Triangle(p1, p2, Point(Rational(5, 2), sqrt(Rational(75, 4))))
    t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
    s1 = t1.sides
    s2 = t2.sides
    s3 = t3.sides

    # Basic stuff
    assert Triangle(p1, p1, p1) == p1
    assert Triangle(p2, p2 * 2, p2 * 3) == Segment(p2, p2 * 3)
    assert t1.area == Rational(25, 2)
    assert t1.is_right()
    assert t2.is_right() == False
    assert t3.is_right()
    assert p1 in t1
    assert Point(5, 5) not in t2
    assert t1.is_convex()
    assert feq(t1.angles[p1].evalf(), pi.evalf() / 2)

    assert t1.is_equilateral() == False
    assert t2.is_equilateral()
    assert t3.is_equilateral() == False
    assert are_similar(t1, t2) == False
    assert are_similar(t1, t3)
    assert are_similar(t2, t3) == False

    # Bisectors
    bisectors = t1.bisectors()
    assert bisectors[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    ic = (250 - 125 * sqrt(2)) / 50
    assert t1.incenter == Point(ic, ic)

    # Inradius
    assert t1.inradius == 5 - 5 * sqrt(2) / 2
    assert t2.inradius == 5 * sqrt(3) / 6
    assert t3.inradius == x1 ** 2 / ((2 + sqrt(2)) * Abs(x1))

    # Medians + Centroid
    m = t1.medians
    assert t1.centroid == Point(Rational(5, 3), Rational(5, 3))
    assert m[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    assert t3.medians[p1] == Segment(p1, Point(x1 / 2, x1 / 2))
    assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]

    # Perpendicular
    altitudes = t1.altitudes
    assert altitudes[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    assert altitudes[p2] == s1[0]
    assert altitudes[p3] == s1[2]

    # Ensure
    assert len(intersection(*bisectors.values())) == 1
    assert len(intersection(*altitudes.values())) == 1
    assert len(intersection(*m.values())) == 1

    # Distance
    p1 = Polygon(Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1))
    p2 = Polygon(
        Point(0, Rational(5) / 4), Point(1, Rational(5) / 4), Point(1, Rational(9) / 4), Point(0, Rational(9) / 4)
    )
    p3 = Polygon(Point(1, 2), Point(2, 2), Point(2, 1))
    p4 = Polygon(Point(1, 1), Point(Rational(6) / 5, 1), Point(1, Rational(6) / 5))
    p5 = Polygon(
        Point(half, 3 ** (half) / 2),
        Point(-half, 3 ** (half) / 2),
        Point(-1, 0),
        Point(-half, -(3) ** (half) / 2),
        Point(half, -(3) ** (half) / 2),
        Point(1, 0),
    )
    p6 = Polygon(
        Point(2, Rational(3) / 10),
        Point(Rational(17) / 10, 0),
        Point(2, -Rational(3) / 10),
        Point(Rational(23) / 10, 0),
    )
    pt1 = Point(half, half)
    pt2 = Point(1, 1)

    """Polygon to Point"""
    assert p1.distance(pt1) == half
    assert p1.distance(pt2) == 0
    assert p2.distance(pt1) == Rational(3) / 4
    assert p3.distance(pt2) == sqrt(2) / 2

    """Polygon to Polygon"""
    import warnings

    # p1.distance(p2) emits a warning
    # First, test the warning
    warnings.filterwarnings("error", "Polygons may intersect producing erroneous output")
    raises(UserWarning, "p1.distance(p2)")
    # now test the actual output
    warnings.filterwarnings("ignore", "Polygons may intersect producing erroneous output")
    assert p1.distance(p2) == half / 2
    # Keep testing reasonably thread safe, so reset the warning
    warnings.filterwarnings("default", "Polygons may intersect producing erroneous output")
    # Note, in Python 2.6+, this can be done more nicely using the
    # warnings.catch_warnings context manager.
    # See http://docs.python.org/library/warnings#testing-warnings.

    assert p1.distance(p3) == sqrt(2) / 2
    assert p3.distance(p4) == (sqrt(2) / 2 - sqrt(Rational(2) / 25) / 2)
    assert p5.distance(p6) == Rational(7) / 10
Beispiel #25
0
def test_polygon():
    t = Triangle(Point(0, 0), Point(2, 0), Point(3, 3))
    assert Polygon(Point(0, 0), Point(1, 0), Point(2, 0), Point(3, 3)) == t
    assert Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(0, 0)) == t
    assert Polygon(Point(2, 0), Point(3, 3), Point(0, 0), Point(1, 0)) == t

    p1 = Polygon(Point(0, 0), Point(3, -1), Point(6, 0), Point(4, 5), Point(2, 3), Point(0, 3))
    p2 = Polygon(Point(6, 0), Point(3, -1), Point(0, 0), Point(0, 3), Point(2, 3), Point(4, 5))
    p3 = Polygon(Point(0, 0), Point(3, 0), Point(5, 2), Point(4, 4))
    p4 = Polygon(Point(0, 0), Point(4, 4), Point(5, 2), Point(3, 0))
    p5 = Polygon(Point(0, 0), Point(4, 4), Point(0, 4))

    #
    # General polygon
    #
    assert p1 == p2
    assert len(p1.args) == 6
    assert len(p1.sides) == 6
    assert p1.perimeter == 5 + 2 * sqrt(10) + sqrt(29) + sqrt(8)
    assert p1.area == 22
    assert not p1.is_convex()
    assert p3.is_convex()
    assert p4.is_convex()  # ensure convex for both CW and CCW point specification
    dict5 = p5.angles
    assert dict5[Point(0, 0)] == pi / 4
    assert dict5[Point(0, 4)] == pi / 2
    assert p5.encloses_point(Point(x, y)) == None
    assert p5.encloses_point(Point(1, 3))
    assert p5.encloses_point(Point(0, 0)) == False
    assert p5.encloses_point(Point(4, 0)) == False
    p5.plot_interval("x") == [x, 0, 1]
    assert p5.distance(Polygon(Point(10, 10), Point(14, 14), Point(10, 14))) == 6 * sqrt(2)
    assert p5.distance(Polygon(Point(1, 8), Point(5, 8), Point(8, 12), Point(1, 12))) == 4
    raises(
        UserWarning,
        lambda: Polygon(Point(0, 0), Point(1, 0), Point(1, 1)).distance(Polygon(Point(0, 0), Point(0, 1), Point(1, 1))),
    )
    assert hash(p5) == hash(Polygon(Point(0, 0), Point(4, 4), Point(0, 4)))
    assert p5 == Polygon(Point(4, 4), Point(0, 4), Point(0, 0))
    assert Polygon(Point(4, 4), Point(0, 4), Point(0, 0)) in p5
    assert p5 != Point(0, 4)
    assert Point(0, 1) in p5
    assert p5.arbitrary_point("t").subs(Symbol("t", real=True), 0) == Point(0, 0)
    raises(ValueError, lambda: Polygon(Point(x, 0), Point(0, y), Point(x, y)).arbitrary_point("x"))

    #
    # Regular polygon
    #
    p1 = RegularPolygon(Point(0, 0), 10, 5)
    p2 = RegularPolygon(Point(0, 0), 5, 5)
    raises(GeometryError, lambda: RegularPolygon(Point(0, 0), Point(0, 1), Point(1, 1)))
    raises(GeometryError, lambda: RegularPolygon(Point(0, 0), 1, 2))
    raises(ValueError, lambda: RegularPolygon(Point(0, 0), 1, 2.5))

    assert p1 != p2
    assert p1.interior_angle == 3 * pi / 5
    assert p1.exterior_angle == 2 * pi / 5
    assert p2.apothem == 5 * cos(pi / 5)
    assert p2.circumcenter == p1.circumcenter == Point(0, 0)
    assert p1.circumradius == p1.radius == 10
    assert p2.circumcircle == Circle(Point(0, 0), 5)
    assert p2.incircle == Circle(Point(0, 0), p2.apothem)
    assert p2.inradius == p2.apothem == (5 * (1 + sqrt(5)) / 4)
    p2.spin(pi / 10)
    dict1 = p2.angles
    assert dict1[Point(0, 5)] == 3 * pi / 5
    assert p1.is_convex()
    assert p1.rotation == 0
    assert p1.encloses_point(Point(0, 0))
    assert p1.encloses_point(Point(11, 0)) == False
    assert p2.encloses_point(Point(0, 4.9))
    p1.spin(pi / 3)
    assert p1.rotation == pi / 3
    assert p1.vertices[0] == Point(5, 5 * sqrt(3))
    for var in p1.args:
        if isinstance(var, Point):
            assert var == Point(0, 0)
        else:
            assert var == 5 or var == 10 or var == pi / 3
    assert p1 != Point(0, 0)
    assert p1 != p5

    # while spin works in place (notice that rotation is 2pi/3 below)
    # rotate returns a new object
    p1_old = p1
    assert p1.rotate(pi / 3) == RegularPolygon(Point(0, 0), 10, 5, 2 * pi / 3)
    assert p1 == p1_old

    assert p1.area == (-250 * sqrt(5) + 1250) / (4 * tan(pi / 5))
    assert p1.length == 20 * sqrt(-sqrt(5) / 8 + S(5) / 8)
    assert p1.scale(2, 2) == RegularPolygon(p1.center, p1.radius * 2, p1._n, p1.rotation)
    assert RegularPolygon((0, 0), 1, 4).scale(2, 3) == Polygon(Point(2, 0), Point(0, 3), Point(-2, 0), Point(0, -3))

    assert ` p1 ` == str(p1)

    #
    # Angles
    #
    angles = p4.angles
    assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))

    angles = p3.angles
    assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))

    #
    # Triangle
    #
    p1 = Point(0, 0)
    p2 = Point(5, 0)
    p3 = Point(0, 5)
    t1 = Triangle(p1, p2, p3)
    t2 = Triangle(p1, p2, Point(Rational(5, 2), sqrt(Rational(75, 4))))
    t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
    s1 = t1.sides
    assert Triangle(p1, p2, p1) == Polygon(p1, p2, p1) == Segment(p1, p2)
    raises(GeometryError, lambda: Triangle(Point(0, 0)))

    # Basic stuff
    assert Triangle(p1, p1, p1) == p1
    assert Triangle(p2, p2 * 2, p2 * 3) == Segment(p2, p2 * 3)
    assert t1.area == Rational(25, 2)
    assert t1.is_right()
    assert t2.is_right() == False
    assert t3.is_right()
    assert p1 in t1
    assert t1.sides[0] in t1
    assert Segment((0, 0), (1, 0)) in t1
    assert Point(5, 5) not in t2
    assert t1.is_convex()
    assert feq(t1.angles[p1].evalf(), pi.evalf() / 2)

    assert t1.is_equilateral() == False
    assert t2.is_equilateral()
    assert t3.is_equilateral() == False
    assert are_similar(t1, t2) == False
    assert are_similar(t1, t3)
    assert are_similar(t2, t3) == False
    assert t1.is_similar(Point(0, 0)) == False

    # Bisectors
    bisectors = t1.bisectors()
    assert bisectors[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    ic = (250 - 125 * sqrt(2)) / 50
    assert t1.incenter == Point(ic, ic)

    # Inradius
    assert t1.inradius == t1.incircle.radius == 5 - 5 * sqrt(2) / 2
    assert t2.inradius == t2.incircle.radius == 5 * sqrt(3) / 6
    assert t3.inradius == t3.incircle.radius == x1 ** 2 / ((2 + sqrt(2)) * Abs(x1))

    # Circumcircle
    assert t1.circumcircle.center == Point(2.5, 2.5)

    # Medians + Centroid
    m = t1.medians
    assert t1.centroid == Point(Rational(5, 3), Rational(5, 3))
    assert m[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    assert t3.medians[p1] == Segment(p1, Point(x1 / 2, x1 / 2))
    assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]
    assert t1.medial == Triangle(Point(2.5, 0), Point(0, 2.5), Point(2.5, 2.5))

    # Perpendicular
    altitudes = t1.altitudes
    assert altitudes[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    assert altitudes[p2] == s1[0]
    assert altitudes[p3] == s1[2]
    assert t1.orthocenter == p1
    t = S(
        """Triangle(
    Point(100080156402737/5000000000000, 79782624633431/500000000000),
    Point(39223884078253/2000000000000, 156345163124289/1000000000000),
    Point(31241359188437/1250000000000, 338338270939941/1000000000000000))"""
    )
    assert t.orthocenter == S(
        """Point(-780660869050599840216997"""
        """79471538701955848721853/80368430960602242240789074233100000000000000,"""
        """20151573611150265741278060334545897615974257/16073686192120448448157"""
        """8148466200000000000)"""
    )

    # Ensure
    assert len(intersection(*bisectors.values())) == 1
    assert len(intersection(*altitudes.values())) == 1
    assert len(intersection(*m.values())) == 1

    # Distance
    p1 = Polygon(Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1))
    p2 = Polygon(
        Point(0, Rational(5) / 4), Point(1, Rational(5) / 4), Point(1, Rational(9) / 4), Point(0, Rational(9) / 4)
    )
    p3 = Polygon(Point(1, 2), Point(2, 2), Point(2, 1))
    p4 = Polygon(Point(1, 1), Point(Rational(6) / 5, 1), Point(1, Rational(6) / 5))
    pt1 = Point(half, half)
    pt2 = Point(1, 1)

    """Polygon to Point"""
    assert p1.distance(pt1) == half
    assert p1.distance(pt2) == 0
    assert p2.distance(pt1) == Rational(3) / 4
    assert p3.distance(pt2) == sqrt(2) / 2
Beispiel #26
0
def test_ellipse():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x2)
    p4 = Point(0, 1)
    p5 = Point(-1, 0)

    e1 = Ellipse(p1, 1, 1)
    e2 = Ellipse(p2, half, 1)
    e3 = Ellipse(p1, y1, y1)
    c1 = Circle(p1, 1)
    c2 = Circle(p2,1)
    c3 = Circle(Point(sqrt(2),sqrt(2)),1)

    # Test creation with three points
    cen,rad = Point(3*half, 2), 5*half
    assert Circle(Point(0,0), Point(3,0), Point(0,4)) == Circle(cen, rad)
    raises(GeometryError, "Circle(Point(0,0), Point(1,1), Point(2,2))")

    # Basic Stuff
    assert e1 == c1
    assert e1 != e2
    assert p4 in e1
    assert p2 not in e2
    assert e1.area == pi
    assert e2.area == pi/2
    assert e3.area == pi*(y1**2)
    assert c1.area == e1.area
    assert c1.circumference == e1.circumference
    assert e3.circumference == 2*pi*y1

    a = Symbol('a')
    b = Symbol('b')
    e5 = Ellipse(p1, a, b)
    assert e5.circumference == 4*a*C.Integral(((1 - x**2*Abs(b**2 - a**2)/a**2)/(1 - x**2))**(S(1)/2),\
                                            (x, 0, 1))

    assert e2.arbitrary_point() in e2

    # Foci
    f1,f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
    ef = Ellipse(Point(0, 0), 4, 2)
    assert ef.foci in [(f1, f2), (f2, f1)]

    # Tangents
    v = sqrt(2) / 2
    p1_1 = Point(v, v)
    p1_2 = p2 + Point(half, 0)
    p1_3 = p2 + Point(0, 1)
    assert e1.tangent_line(p4) == c1.tangent_line(p4)
    assert e2.tangent_line(p1_2) == Line(p1_2, p2 + Point(half, 1))
    assert e2.tangent_line(p1_3) == Line(p1_3, p2 + Point(half, 1))
    assert c1.tangent_line(p1_1) == Line(p1_1, Point(0, sqrt(2)))
    assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
    assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
    assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
    assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) == False

    # Intersection
    l1 = Line(Point(1, -5), Point(1, 5))
    l2 = Line(Point(-5, -1), Point(5, -1))
    l3 = Line(Point(-1, -1), Point(1, 1))
    l4 = Line(Point(-10, 0), Point(0, 10))
    pts_c1_l3 = [Point(sqrt(2)/2, sqrt(2)/2), Point(-sqrt(2)/2, -sqrt(2)/2)]

    assert intersection(e2, l4) == []
    assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
    assert intersection(c1, l1) == [Point(1, 0)]
    assert intersection(c1, l2) == [Point(0, -1)]
    assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
    assert intersection(c1, c2) in [[(1,0), (0,1)],[(0,1),(1,0)]]
    assert intersection(c1, c3) == [(sqrt(2)/2, sqrt(2)/2)]

    # some special case intersections
    csmall = Circle(p1, 3)
    cbig = Circle(p1, 5)
    cout = Circle(Point(5, 5), 1)
    # one circle inside of another
    assert csmall.intersection(cbig) == []
    # separate circles
    assert csmall.intersection(cout) == []
    # coincident circles
    assert csmall.intersection(csmall) == csmall

    v = sqrt(2)
    t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
    points = intersection(t1, c1)
    assert len(points) == 4
    assert Point(0, 1) in points
    assert Point(0, -1) in points
    assert Point(v/2, v/2) in points
    assert Point(v/2, -v/2) in points

    e1 = Circle(Point(0, 0), 5)
    e2 = Ellipse(Point(0, 0), 5, 20)
    assert intersection(e1, e2) in \
        [[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]

    # FAILING ELLIPSE INTERSECTION GOES HERE

    # Combinations of above
    assert e3.is_tangent(e3.tangent_line(p1 + Point(y1, 0)))

    major = 3
    minor = 1
    e4 = Ellipse(p2, major, minor)
    assert e4.focus_distance == sqrt(abs(major**2 - minor**2))
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major*(1 - ecc)
    assert e4.apoapsis == major*(1 + ecc)
    def json_to_rdkit(self, json_data, map={}, mol_type="reactant"):

        m0 = Chem.MolFromSmarts("")
        mw = Chem.RWMol(m0)

        atoms_cd = {}
        query_atoms = []

        for atom in json_data["a"]:
            atom_id = atom["i"]
            # Query atoms
            if "q" in atom:
                q = atom["q"]
                v = q["as"]["v"]
                if "a" in v:
                    smarts = "[*]"
                    a_defaut = "C"
                else:
                    smarts = v
                    # Manage aromaticity
                    if "A" in q:
                        if not q["A"]["n"]:
                            smarts = []
                        if q["A"]["v"]:
                            smarts = smarts + [a_.lower() for a_ in v]
                    smarts = "[{0}]".format(",".join([a_ for a_ in smarts]))
                    a_defaut = v[0]
                query_atoms.append((atom_id, smarts))
                a = Chem.Atom(a_defaut)
            # Non-query atoms
            else:
                symbol = "C" if "l" not in atom else atom["l"]
                a = Chem.Atom(symbol)
                if "c" in atom:
                    a.SetFormalCharge(atom["c"])
            atoms_cd[atom_id] = mw.AddAtom(a)

        if not "b" in json_data:
            json_data["b"] = []

        for bond in json_data["b"]:
            bond_id = bond["i"]
            if "o" in bond:
                bond_type = self.bond_type(bond["o"])
            else:
                bond_type = self.bond_type(1)
            mw.AddBond(bond["b"], bond["e"], bond_type) - 1

        bonds_cd1 = {
            i: mw.GetBondBetweenAtoms(b_cd["b"], b_cd["e"]).GetIdx()
            for i, b_cd in enumerate(json_data["b"])
        }

        bonds_cd = {
            "{0}-{1}".format(b_cd["b"], b_cd["e"]): i
            for i, b_cd in enumerate(json_data["b"])
        }

        atoms_rd = {v: k for k, v in atoms_cd.items()}
        bonds_rd = {v: k for k, v in bonds_cd.items()}

        def a_point(atom_rd):
            x, y = (json_data["a"][atom_rd][d] for d in ["x", "y"])
            return Point(x, y)

        def mol_atom(idx):
            return mw.GetAtoms()[idx]

        ### Manage chirality

        for atom in mw.GetAtoms():
            if atom.GetSymbol() == "C":
                s = []
                for b in atom.GetBonds():
                    id_1 = "{0}-{1}".format(b.GetBeginAtomIdx(),
                                            b.GetEndAtomIdx())
                    if id_1 in bonds_cd:
                        b_cd = json_data["b"][bonds_cd[id_1]]
                    else:
                        b_cd = json_data["b"][bonds_cd["{1}-{0}".format(
                            b.GetBeginAtomIdx(), b.GetEndAtomIdx())]]
                    if "s" in b_cd:
                        s.append(b_cd["s"])
                    else:
                        s.append("")
                if len(s) == 4 and s.count("protruding") * s.count(
                        "recessed") == 1:
                    points = []
                    for i, b in enumerate(atom.GetBonds()):
                        if b.GetBeginAtomIdx() != atom.GetIdx():
                            a = b.GetBeginAtomIdx()
                        else:
                            a = b.GetEndAtomIdx()

                        if s[i] == "protruding":
                            z = 1
                        elif s[i] == "recessed":
                            z = -1
                        else:
                            z = 0
                        points.append(Point([i for i in a_point(a)] + [z]))
                    int = Line(points[2].midpoint(points[3]),
                               points[1]).intersection(
                                   Line(points[1].midpoint(points[2]),
                                        points[3]))[0]
                    N = CoordSys3D("N")
                    coords = [N.i, N.j, N.k]
                    vectors = []
                    for p in points:
                        v = Vector.zero
                        for v_ in [
                                n * coords[v] for v, n in enumerate(p - int)
                        ]:
                            v += v_
                        vectors.append(v)
                    if vectors[1].cross(vectors[3]).dot(vectors[0]) > 0:
                        atom.SetChiralTag(
                            Chem.rdchem.ChiralType.CHI_TETRAHEDRAL_CCW)
                    else:
                        atom.SetChiralTag(
                            Chem.rdchem.ChiralType.CHI_TETRAHEDRAL_CW)

        ### Manage stereo

        for bond in mw.GetBonds():
            if (bond.GetBondType() == Chem.rdchem.BondType.DOUBLE
                    and bond.GetBeginAtom().GetSymbol() == "C"
                    and bond.GetEndAtom().GetSymbol() == "C"):
                carbons = [bond.GetBeginAtomIdx(), bond.GetEndAtomIdx()]
                # Check if stero can be applied
                if True in [len(mol_atom(c).GetBonds()) > 1 for c in carbons]:
                    sa = [
                        [
                            b.GetBeginAtomIdx() if b.GetBeginAtomIdx() != c
                            else b.GetEndAtomIdx()  # ,
                            for b in mol_atom(c).GetBonds()
                            if b.GetBondType() == Chem.rdchem.BondType.SINGLE
                        ] for c in carbons
                    ]
                    if not [] in sa:
                        sa = [sa[i][0] for i in range(2)]
                        bond.SetStereoAtoms(sa[0], sa[1])

                        if (len(
                                intersection(
                                    Segment(a_point(sa[0]), a_point(sa[1])),
                                    Line(a_point(carbons[0]),
                                         a_point(carbons[1])),
                                )) > 0):
                            bond.SetStereo(Chem.rdchem.BondStereo.STEREOTRANS)
                        else:
                            bond.SetStereo(Chem.rdchem.BondStereo.STEREOCIS)

        # replace query_atoms
        if len(query_atoms) > 0:
            for id, smarts in query_atoms:
                m_a = Chem.MolFromSmarts(smarts)
                Chem.SanitizeMol(m_a)
                mw.ReplaceAtom(atoms_cd[id], m_a.GetAtoms()[0])

        # Map atoms
        i = 1
        for m in map:
            if m["t"] == "AtomMapping":
                a_rd_id = None
                for a in ["a1", "a2"]:
                    if m[a] in atoms_cd:
                        a_rd_id = atoms_cd[m[a]]
                if a_rd_id is not None:
                    mw.GetAtoms()[a_rd_id].SetAtomMapNum(i)
                i += 1

        mol = mw.GetMol()
        RDKit.apply_aromaticity(mol)
        Chem.rdDepictor.Compute2DCoords(mol)

        return mol
Beispiel #28
0
def test_line():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x1)
    p4 = Point(y1, y1)
    p5 = Point(x1, 1 + x1)

    l1 = Line(p1, p2)
    l2 = Line(p3, p4)
    l3 = Line(p3, p5)

    # Basic stuff
    assert Line(p1, p2) == Line(p2, p1)
    assert l1 == l2
    assert l1 != l3
    assert l1.slope == 1
    assert l3.slope == oo
    assert p1 in l1 # is p1 on the line l1?
    assert p1 not in l3

    assert simplify(l1.equation()) in (x-y, y-x)
    assert simplify(l3.equation()) in (x-x1, x1-x)

    assert l2.arbitrary_point() in l2
    for ind in xrange(0, 5):
        assert l3.random_point() in l3

    # Orthogonality
    p1_1 = Point(-x1, x1)
    l1_1 = Line(p1, p1_1)
    assert l1.perpendicular_line(p1) == l1_1
    assert Line.is_perpendicular(l1, l1_1)
    assert Line.is_perpendicular(l1 , l2) == False

    # Parallelity
    p2_1 = Point(-2*x1, 0)
    l2_1 = Line(p3, p5)
    assert l2.parallel_line(p1_1) == Line(p2_1, p1_1)
    assert l2_1.parallel_line(p1) == Line(p1, Point(0, 2))
    assert Line.is_parallel(l1, l2)
    assert Line.is_parallel(l2, l3) == False
    assert Line.is_parallel(l2, l2.parallel_line(p1_1))
    assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))

    # Intersection
    assert intersection(l1, p1) == [p1]
    assert intersection(l1, p5) == []
    assert intersection(l1, l2) in [[l1], [l2]]
    assert intersection(l1, l1.parallel_line(p5)) == []

    # Concurrency
    l3_1 = Line(Point(5, x1), Point(-Rational(3,5), x1))
    assert Line.is_concurrent(l1, l3)
    assert Line.is_concurrent(l1, l3, l3_1)
    assert Line.is_concurrent(l1, l1_1, l3) == False

    # Projection
    assert l2.projection(p4) == p4
    assert l1.projection(p1_1) == p1
    assert l3.projection(p2) == Point(x1, 1)

    # Finding angles
    l1_1 = Line(p1, Point(5, 0))
    assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf()/4)

    # Testing Rays and Segments (very similar to Lines)
    r1 = Ray(p1, Point(-1, 5))
    r2 = Ray(p1, Point(-1, 1))
    r3 = Ray(p3, p5)
    assert l1.projection(r1) == Ray(p1, p2)
    assert l1.projection(r2) == p1
    assert r3 != r1

    s1 = Segment(p1, p2)
    s2 = Segment(p1, p1_1)
    assert s1.midpoint == Point(Rational(1,2), Rational(1,2))
    assert s2.length == sqrt( 2*(x1**2) )
    assert s1.perpendicular_bisector() == Line(Point(0, 1), Point(1, 0))

    # Testing distance from a Segment to an object
    s1 = Segment(Point(0, 0), Point(1, 1))
    s2 = Segment(Point(half, half), Point(1, 0))
    pt1 = Point(0, 0)
    pt2 = Point(Rational(3)/2, Rational(3)/2)
    assert s1.distance(pt1) == 0
    assert s2.distance(pt1) == 2**(half)/2
    assert s2.distance(pt2) == 2**(half)

    # Special cases of projection and intersection
    r1 = Ray(Point(1, 1), Point(2, 2))
    r2 = Ray(Point(2, 2), Point(0, 0))
    r3 = Ray(Point(1, 1), Point(-1, -1))
    r4 = Ray(Point(0, 4), Point(-1, -5))
    assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
    assert intersection(r1, r3) == [Point(1, 1)]
    assert r1.projection(r3) == Point(1, 1)
    assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))

    r5 = Ray(Point(0, 0), Point(0, 1))
    r6 = Ray(Point(0, 0), Point(0, 2))
    assert r5 in r6
    assert r6 in r5

    s1 = Segment(Point(0, 0), Point(2, 2))
    s2 = Segment(Point(-1, 5), Point(-5, -10))
    s3 = Segment(Point(0, 4), Point(-2, 2))
    assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
    assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
    assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))

    l1 = Line(Point(0, 0), Point(3, 4))
    r1 = Ray(Point(0, 0), Point(3, 4))
    s1 = Segment(Point(0, 0), Point(3, 4))
    assert intersection(l1, l1) == [l1]
    assert intersection(l1, r1) == [r1]
    assert intersection(l1, s1) == [s1]
    assert intersection(r1, l1) == [r1]
    assert intersection(s1, l1) == [s1]

    entity1 = Segment(Point(-10,10), Point(10,10))
    entity2 = Segment(Point(-5,-5), Point(-5,5))
    assert intersection(entity1, entity2) == []
def test_polygon():
    t = Triangle(Point(0, 0), Point(2, 0), Point(3, 3))
    assert Polygon(Point(0, 0), Point(1, 0), Point(2, 0), Point(3, 3)) == t
    assert Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(0, 0)) == t
    assert Polygon(Point(2, 0), Point(3, 3), Point(0, 0), Point(1, 0)) == t

    p1 = Polygon(
        Point(0, 0), Point(3,-1),
        Point(6, 0), Point(4, 5),
        Point(2, 3), Point(0, 3))
    p2 = Polygon(
        Point(6, 0), Point(3,-1),
        Point(0, 0), Point(0, 3),
        Point(2, 3), Point(4, 5))
    p3 = Polygon(
        Point(0, 0), Point(3, 0),
        Point(5, 2), Point(4, 4))
    p4 = Polygon(
        Point(0, 0), Point(4, 4),
        Point(5, 2), Point(3, 0))

    #
    # General polygon
    #
    assert p1 == p2
    assert len(p1) == 6
    assert len(p1.sides) == 6
    assert p1.perimeter == 5+2*sqrt(10)+sqrt(29)+sqrt(8)
    assert p1.area == 22
    assert not p1.is_convex()
    assert p3.is_convex()
    assert p4.is_convex()  # ensure convex for both CW and CCW point specification

    #
    # Regular polygon
    #
    p1 = RegularPolygon(Point(0, 0), 10, 5)
    p2 = RegularPolygon(Point(0, 0), 5, 5)

    assert p1 != p2
    assert p1.interior_angle == 3*pi/5
    assert p1.exterior_angle == 2*pi/5
    assert p2.apothem == 5*cos(pi/5)
    assert p2.circumcircle == Circle(Point(0, 0), 5)
    assert p2.incircle == Circle(Point(0, 0), p2.apothem)
    assert p1.is_convex()
    assert p1.rotation == 0
    p1.spin(pi/3)
    assert p1.rotation == pi/3
    assert p1[0] == Point(5, 5*sqrt(3))
    # while spin works in place (notice that rotation is 2pi/3 below)
    # rotate returns a new object
    p1_old = p1
    assert p1.rotate(pi/3) == RegularPolygon(Point(0, 0), 10, 5, 2*pi/3)
    assert p1 == p1_old

    #
    # Angles
    #
    angles = p4.angles
    assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))

    angles = p3.angles
    assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))

    #
    # Triangle
    #
    p1 = Point(0, 0)
    p2 = Point(5, 0)
    p3 = Point(0, 5)
    t1 = Triangle(p1, p2, p3)
    t2 = Triangle(p1, p2, Point(Rational(5,2), sqrt(Rational(75,4))))
    t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
    s1 = t1.sides

    # Basic stuff
    assert Triangle(p1, p1, p1) == p1
    assert Triangle(p2, p2*2, p2*3) == Segment(p2, p2*3)
    assert t1.area == Rational(25,2)
    assert t1.is_right()
    assert t2.is_right() == False
    assert t3.is_right()
    assert p1 in t1
    assert t1.sides[0] in t1
    assert Segment((0, 0), (1, 0)) in t1
    assert Point(5, 5) not in t2
    assert t1.is_convex()
    assert feq(t1.angles[p1].evalf(), pi.evalf()/2)

    assert t1.is_equilateral() == False
    assert t2.is_equilateral()
    assert t3.is_equilateral() == False
    assert are_similar(t1, t2) == False
    assert are_similar(t1, t3)
    assert are_similar(t2, t3) == False

    # Bisectors
    bisectors = t1.bisectors()
    assert bisectors[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    ic = (250 - 125*sqrt(2)) / 50
    assert t1.incenter == Point(ic, ic)

    # Inradius
    assert t1.inradius == 5 - 5*sqrt(2)/2
    assert t2.inradius == 5*sqrt(3)/6
    assert t3.inradius == x1**2/((2 + sqrt(2))*Abs(x1))

    # Medians + Centroid
    m = t1.medians
    assert t1.centroid == Point(Rational(5,3), Rational(5,3))
    assert m[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    assert t3.medians[p1] == Segment(p1, Point(x1/2, x1/2))
    assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]

    # Perpendicular
    altitudes = t1.altitudes
    assert altitudes[p1] == Segment(p1, Point(Rational(5,2), Rational(5,2)))
    assert altitudes[p2] == s1[0]
    assert altitudes[p3] == s1[2]

    # Ensure
    assert len(intersection(*bisectors.values())) == 1
    assert len(intersection(*altitudes.values())) == 1
    assert len(intersection(*m.values())) == 1

    # Distance
    p1 = Polygon(
        Point(0, 0), Point(1, 0),
        Point(1, 1), Point(0, 1))
    p2 = Polygon(
        Point(0, Rational(5)/4), Point(1, Rational(5)/4),
        Point(1, Rational(9)/4), Point(0,  Rational(9)/4))
    p3 = Polygon(
        Point(1, 2), Point(2, 2),
        Point(2, 1))
    p4 = Polygon(
        Point(1, 1), Point(Rational(6)/5, 1),
        Point(1, Rational(6)/5))
    pt1 = Point(half, half)
    pt2 = Point(1, 1)

    '''Polygon to Point'''
    assert p1.distance(pt1) == half
    assert p1.distance(pt2) == 0
    assert p2.distance(pt1) == Rational(3)/4
    assert p3.distance(pt2) == sqrt(2)/2
Beispiel #30
0
def test_ellipse_geom():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    t = Symbol('t', real=True)
    y1 = Symbol('y1', real=True)
    half = Rational(1, 2)
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p4 = Point(0, 1)

    e1 = Ellipse(p1, 1, 1)
    e2 = Ellipse(p2, half, 1)
    e3 = Ellipse(p1, y1, y1)
    c1 = Circle(p1, 1)
    c2 = Circle(p2, 1)
    c3 = Circle(Point(sqrt(2), sqrt(2)), 1)
    l1 = Line(p1, p2)

    # Test creation with three points
    cen, rad = Point(3*half, 2), 5*half
    assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
    assert Circle(Point(0, 0), Point(1, 1), Point(2, 2)) == Segment2D(Point2D(0, 0), Point2D(2, 2))

    raises(ValueError, lambda: Ellipse(None, None, None, 1))
    raises(GeometryError, lambda: Circle(Point(0, 0)))

    # Basic Stuff
    assert Ellipse(None, 1, 1).center == Point(0, 0)
    assert e1 == c1
    assert e1 != e2
    assert e1 != l1
    assert p4 in e1
    assert p2 not in e2
    assert e1.area == pi
    assert e2.area == pi/2
    assert e3.area == pi*y1*abs(y1)
    assert c1.area == e1.area
    assert c1.circumference == e1.circumference
    assert e3.circumference == 2*pi*y1
    assert e1.plot_interval() == e2.plot_interval() == [t, -pi, pi]
    assert e1.plot_interval(x) == e2.plot_interval(x) == [x, -pi, pi]

    assert c1.minor == 1
    assert c1.major == 1
    assert c1.hradius == 1
    assert c1.vradius == 1

    assert Ellipse((1, 1), 0, 0) == Point(1, 1)
    assert Ellipse((1, 1), 1, 0) == Segment(Point(0, 1), Point(2, 1))
    assert Ellipse((1, 1), 0, 1) == Segment(Point(1, 0), Point(1, 2))

    # Private Functions
    assert hash(c1) == hash(Circle(Point(1, 0), Point(0, 1), Point(0, -1)))
    assert c1 in e1
    assert (Line(p1, p2) in e1) is False
    assert e1.__cmp__(e1) == 0
    assert e1.__cmp__(Point(0, 0)) > 0

    # Encloses
    assert e1.encloses(Segment(Point(-0.5, -0.5), Point(0.5, 0.5))) is True
    assert e1.encloses(Line(p1, p2)) is False
    assert e1.encloses(Ray(p1, p2)) is False
    assert e1.encloses(e1) is False
    assert e1.encloses(
        Polygon(Point(-0.5, -0.5), Point(-0.5, 0.5), Point(0.5, 0.5))) is True
    assert e1.encloses(RegularPolygon(p1, 0.5, 3)) is True
    assert e1.encloses(RegularPolygon(p1, 5, 3)) is False
    assert e1.encloses(RegularPolygon(p2, 5, 3)) is False

    assert e2.arbitrary_point() in e2

    # Foci
    f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
    ef = Ellipse(Point(0, 0), 4, 2)
    assert ef.foci in [(f1, f2), (f2, f1)]

    # Tangents
    v = sqrt(2) / 2
    p1_1 = Point(v, v)
    p1_2 = p2 + Point(half, 0)
    p1_3 = p2 + Point(0, 1)
    assert e1.tangent_lines(p4) == c1.tangent_lines(p4)
    assert e2.tangent_lines(p1_2) == [Line(Point(S(3)/2, 1), Point(S(3)/2, S(1)/2))]
    assert e2.tangent_lines(p1_3) == [Line(Point(1, 2), Point(S(5)/4, 2))]
    assert c1.tangent_lines(p1_1) != [Line(p1_1, Point(0, sqrt(2)))]
    assert c1.tangent_lines(p1) == []
    assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
    assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
    assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
    assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) is False
    assert c1.is_tangent(e1) is True
    assert c1.is_tangent(Ellipse(Point(2, 0), 1, 1)) is True
    assert c1.is_tangent(
        Polygon(Point(1, 1), Point(1, -1), Point(2, 0))) is True
    assert c1.is_tangent(
        Polygon(Point(1, 1), Point(1, 0), Point(2, 0))) is False
    assert Circle(Point(5, 5), 3).is_tangent(Circle(Point(0, 5), 1)) is False

    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(0, 0)) == \
        [Line(Point(0, 0), Point(S(77)/25, S(132)/25)),
     Line(Point(0, 0), Point(S(33)/5, S(22)/5))]
    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(3, 4)) == \
        [Line(Point(3, 4), Point(4, 4)), Line(Point(3, 4), Point(3, 5))]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(3, 3)) == \
        [Line(Point(3, 3), Point(4, 3)), Line(Point(3, 3), Point(3, 4))]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(5 - 2*sqrt(2), 5)) == \
        [Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 - sqrt(2))),
     Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 + sqrt(2))), ]

    # for numerical calculations, we shouldn't demand exact equality,
    # so only test up to the desired precision
    def lines_close(l1, l2, prec):
        """ tests whether l1 and 12 are within 10**(-prec)
        of each other """
        return abs(l1.p1 - l2.p1) < 10**(-prec) and abs(l1.p2 - l2.p2) < 10**(-prec)
    def line_list_close(ll1, ll2, prec):
        return all(lines_close(l1, l2, prec) for l1, l2 in zip(ll1, ll2))

    e = Ellipse(Point(0, 0), 2, 1)
    assert e.normal_lines(Point(0, 0)) == \
        [Line(Point(0, 0), Point(0, 1)), Line(Point(0, 0), Point(1, 0))]
    assert e.normal_lines(Point(1, 0)) == \
        [Line(Point(0, 0), Point(1, 0))]
    assert e.normal_lines((0, 1)) == \
        [Line(Point(0, 0), Point(0, 1))]
    assert line_list_close(e.normal_lines(Point(1, 1), 2), [
        Line(Point(-S(51)/26, -S(1)/5), Point(-S(25)/26, S(17)/83)),
        Line(Point(S(28)/29, -S(7)/8), Point(S(57)/29, -S(9)/2))], 2)
    # test the failure of Poly.intervals and checks a point on the boundary
    p = Point(sqrt(3), S.Half)
    assert p in e
    assert line_list_close(e.normal_lines(p, 2), [
        Line(Point(-S(341)/171, -S(1)/13), Point(-S(170)/171, S(5)/64)),
        Line(Point(S(26)/15, -S(1)/2), Point(S(41)/15, -S(43)/26))], 2)
    # be sure to use the slope that isn't undefined on boundary
    e = Ellipse((0, 0), 2, 2*sqrt(3)/3)
    assert line_list_close(e.normal_lines((1, 1), 2), [
        Line(Point(-S(64)/33, -S(20)/71), Point(-S(31)/33, S(2)/13)),
        Line(Point(1, -1), Point(2, -4))], 2)
    # general ellipse fails except under certain conditions
    e = Ellipse((0, 0), x, 1)
    assert e.normal_lines((x + 1, 0)) == [Line(Point(0, 0), Point(1, 0))]
    raises(NotImplementedError, lambda: e.normal_lines((x + 1, 1)))
    # Properties
    major = 3
    minor = 1
    e4 = Ellipse(p2, minor, major)
    assert e4.focus_distance == sqrt(major**2 - minor**2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major*(1 - ecc)
    assert e4.apoapsis == major*(1 + ecc)
    assert e4.semilatus_rectum == major*(1 - ecc ** 2)
    # independent of orientation
    e4 = Ellipse(p2, major, minor)
    assert e4.focus_distance == sqrt(major**2 - minor**2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major*(1 - ecc)
    assert e4.apoapsis == major*(1 + ecc)

    # Intersection
    l1 = Line(Point(1, -5), Point(1, 5))
    l2 = Line(Point(-5, -1), Point(5, -1))
    l3 = Line(Point(-1, -1), Point(1, 1))
    l4 = Line(Point(-10, 0), Point(0, 10))
    pts_c1_l3 = [Point(sqrt(2)/2, sqrt(2)/2), Point(-sqrt(2)/2, -sqrt(2)/2)]

    assert intersection(e2, l4) == []
    assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
    assert intersection(c1, l1) == [Point(1, 0)]
    assert intersection(c1, l2) == [Point(0, -1)]
    assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
    assert intersection(c1, c2) == [Point(0, 1), Point(1, 0)]
    assert intersection(c1, c3) == [Point(sqrt(2)/2, sqrt(2)/2)]
    assert e1.intersection(l1) == [Point(1, 0)]
    assert e2.intersection(l4) == []
    assert e1.intersection(Circle(Point(0, 2), 1)) == [Point(0, 1)]
    assert e1.intersection(Circle(Point(5, 0), 1)) == []
    assert e1.intersection(Ellipse(Point(2, 0), 1, 1)) == [Point(1, 0)]
    assert e1.intersection(Ellipse(Point(5, 0), 1, 1)) == []
    assert e1.intersection(Point(2, 0)) == []
    assert e1.intersection(e1) == e1
    assert intersection(Ellipse(Point(0, 0), 2, 1), Ellipse(Point(3, 0), 1, 2)) == [Point(2, 0)]
    assert intersection(Circle(Point(0, 0), 2), Circle(Point(3, 0), 1)) == [Point(2, 0)]
    assert intersection(Circle(Point(0, 0), 2), Circle(Point(7, 0), 1)) == []
    assert intersection(Ellipse(Point(0, 0), 5, 17), Ellipse(Point(4, 0), 1, 0.2)) == [Point(5, 0)]
    assert intersection(Ellipse(Point(0, 0), 5, 17), Ellipse(Point(4, 0), 0.999, 0.2)) == []
    assert Circle((0, 0), S(1)/2).intersection(
        Triangle((-1, 0), (1, 0), (0, 1))) == [
        Point(-S(1)/2, 0), Point(S(1)/2, 0)]
    raises(TypeError, lambda: intersection(e2, Line((0, 0, 0), (0, 0, 1))))
    raises(TypeError, lambda: intersection(e2, Rational(12)))
    # some special case intersections
    csmall = Circle(p1, 3)
    cbig = Circle(p1, 5)
    cout = Circle(Point(5, 5), 1)
    # one circle inside of another
    assert csmall.intersection(cbig) == []
    # separate circles
    assert csmall.intersection(cout) == []
    # coincident circles
    assert csmall.intersection(csmall) == csmall

    v = sqrt(2)
    t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
    points = intersection(t1, c1)
    assert len(points) == 4
    assert Point(0, 1) in points
    assert Point(0, -1) in points
    assert Point(v/2, v/2) in points
    assert Point(v/2, -v/2) in points

    circ = Circle(Point(0, 0), 5)
    elip = Ellipse(Point(0, 0), 5, 20)
    assert intersection(circ, elip) in \
        [[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
    assert elip.tangent_lines(Point(0, 0)) == []
    elip = Ellipse(Point(0, 0), 3, 2)
    assert elip.tangent_lines(Point(3, 0)) == \
        [Line(Point(3, 0), Point(3, -12))]

    e1 = Ellipse(Point(0, 0), 5, 10)
    e2 = Ellipse(Point(2, 1), 4, 8)
    a = S(53)/17
    c = 2*sqrt(3991)/17
    ans = [Point(a - c/8, a/2 + c), Point(a + c/8, a/2 - c)]
    assert e1.intersection(e2) == ans
    e2 = Ellipse(Point(x, y), 4, 8)
    c = sqrt(3991)
    ans = [Point(-c/68 + a, 2*c/17 + a/2), Point(c/68 + a, -2*c/17 + a/2)]
    assert [p.subs({x: 2, y:1}) for p in e1.intersection(e2)] == ans

    # Combinations of above
    assert e3.is_tangent(e3.tangent_lines(p1 + Point(y1, 0))[0])

    e = Ellipse((1, 2), 3, 2)
    assert e.tangent_lines(Point(10, 0)) == \
        [Line(Point(10, 0), Point(1, 0)),
        Line(Point(10, 0), Point(S(14)/5, S(18)/5))]

    # encloses_point
    e = Ellipse((0, 0), 1, 2)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(
        e.center + Point(e.hradius + Rational(1, 10), 0)) is False
    e = Ellipse((0, 0), 2, 1)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(
        e.center + Point(e.hradius + Rational(1, 10), 0)) is False
    assert c1.encloses_point(Point(1, 0)) is False
    assert c1.encloses_point(Point(0.3, 0.4)) is True

    assert e.scale(2, 3) == Ellipse((0, 0), 4, 3)
    assert e.scale(3, 6) == Ellipse((0, 0), 6, 6)
    assert e.rotate(pi) == e
    assert e.rotate(pi, (1, 2)) == Ellipse(Point(2, 4), 2, 1)
    raises(NotImplementedError, lambda: e.rotate(pi/3))

    # Circle rotation tests (Issue #11743)
    # Link - https://github.com/sympy/sympy/issues/11743
    cir = Circle(Point(1, 0), 1)
    assert cir.rotate(pi/2) == Circle(Point(0, 1), 1)
    assert cir.rotate(pi/3) == Circle(Point(S(1)/2, sqrt(3)/2), 1)
    assert cir.rotate(pi/3, Point(1, 0)) == Circle(Point(1, 0), 1)
    assert cir.rotate(pi/3, Point(0, 1)) == Circle(Point(S(1)/2 + sqrt(3)/2, S(1)/2 + sqrt(3)/2), 1)
Beispiel #31
0
def test_ellipse():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x2)
    p4 = Point(0, 1)
    p5 = Point(-1, 0)

    e1 = Ellipse(p1, 1, 1)
    e2 = Ellipse(p2, half, 1)
    e3 = Ellipse(p1, y1, y1)
    c1 = Circle(p1, 1)
    c2 = Circle(p2,1)
    c3 = Circle(Point(sqrt(2),sqrt(2)),1)

    # Test creation with three points
    cen,rad = Point(3*half, 2), 5*half
    assert Circle(Point(0,0), Point(3,0), Point(0,4)) == Circle(cen, rad)
    raises(GeometryError, "Circle(Point(0,0), Point(1,1), Point(2,2))")

    # Basic Stuff
    assert e1 == c1
    assert e1 != e2
    assert p4 in e1
    assert p2 not in e2
    assert e1.area == pi
    assert e2.area == pi/2
    assert e3.area == pi*(y1**2)
    assert c1.area == e1.area
    assert c1.circumference == 2*pi

    assert e2.arbitrary_point() in e2
    for ind in xrange(0, 5):
        assert e3.random_point() in e3

    # Foci
    f1,f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
    ef = Ellipse(Point(0, 0), 4, 2)
    assert ef.foci in [(f1, f2), (f2, f1)]

    # Tangents
    v = sqrt(2) / 2
    p1_1 = Point(v, v)
    p1_2 = p2 + Point(half, 0)
    p1_3 = p2 + Point(0, 1)
    assert e1.tangent_line(p4) == c1.tangent_line(p4)
    assert e2.tangent_line(p1_2) == Line(p1_2, p2 + Point(half, 1))
    assert e2.tangent_line(p1_3) == Line(p1_3, p2 + Point(half, 1))
    assert c1.tangent_line(p1_1) == Line(p1_1, Point(0, sqrt(2)))
    assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
    assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
    assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
    assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) == False

    # Intersection
    l1 = Line(Point(1, -5), Point(1, 5))
    l2 = Line(Point(-5, -1), Point(5, -1))
    l3 = Line(Point(-1, -1), Point(1, 1))
    l4 = Line(Point(-10, 0), Point(0, 10))
    pts_c1_l3 = [Point(sqrt(2)/2, sqrt(2)/2), Point(-sqrt(2)/2, -sqrt(2)/2)]

    assert intersection(e2, l4) == []
    assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
    assert intersection(c1, l1) == [Point(1, 0)]
    assert intersection(c1, l2) == [Point(0, -1)]
    assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
    assert intersection(c1, c2) in [[(1,0), (0,1)],[(0,1),(1,0)]]
    assert intersection(c1, c3) == [(sqrt(2)/2, sqrt(2)/2)]

    v = sqrt(2)
    t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
    points = intersection(t1, c1)
    assert len(points) == 4
    assert Point(0, 1) in points
    assert Point(0, -1) in points
    assert Point(v/2, v/2) in points
    assert Point(v/2, -v/2) in points

    e1 = Circle(Point(0, 0), 5)
    e2 = Ellipse(Point(0, 0), 5, 20)
    assert intersection(e1, e2) in \
        [[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]

    # Combinations of above
    assert e3.is_tangent(e3.tangent_line(p1 + Point(y1, 0)))
Beispiel #32
0
def test_ellipse():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p4 = Point(0, 1)

    e1 = Ellipse(p1, 1, 1)
    e2 = Ellipse(p2, half, 1)
    e3 = Ellipse(p1, y1, y1)
    c1 = Circle(p1, 1)
    c2 = Circle(p2, 1)
    c3 = Circle(Point(sqrt(2), sqrt(2)), 1)

    # Test creation with three points
    cen, rad = Point(3 * half, 2), 5 * half
    assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
    raises(GeometryError, lambda: Circle(Point(0, 0), Point(1, 1), Point(2, 2)))

    raises(ValueError, lambda: Ellipse(None, None, None, 1))
    raises(GeometryError, lambda: Circle(Point(0, 0)))

    # Basic Stuff
    assert Ellipse(None, 1, 1).center == Point(0, 0)
    assert e1 == c1
    assert e1 != e2
    assert p4 in e1
    assert p2 not in e2
    assert e1.area == pi
    assert e2.area == pi / 2
    assert e3.area == pi * (y1 ** 2)
    assert c1.area == e1.area
    assert c1.circumference == e1.circumference
    assert e3.circumference == 2 * pi * y1
    assert e1.plot_interval() == e2.plot_interval() == [t, -pi, pi]
    assert e1.plot_interval(x) == e2.plot_interval(x) == [x, -pi, pi]
    assert Ellipse(None, 1, None, 1).circumference == 2 * pi
    assert c1.minor == 1
    assert c1.major == 1
    assert c1.hradius == 1
    assert c1.vradius == 1

    # Private Functions
    assert hash(c1) == hash(Circle(Point(1, 0), Point(0, 1), Point(0, -1)))
    assert c1 in e1
    assert (Line(p1, p2) in e1) == False
    assert e1.__cmp__(e1) == 0
    assert e1.__cmp__(Point(0, 0)) > 0

    # Encloses
    assert e1.encloses(Segment(Point(-0.5, -0.5), Point(0.5, 0.5))) == True
    assert e1.encloses(Line(p1, p2)) == False
    assert e1.encloses(Ray(p1, p2)) == False
    assert e1.encloses(e1) == False
    assert e1.encloses(Polygon(Point(-0.5, -0.5), Point(-0.5, 0.5), Point(0.5, 0.5))) == True
    assert e1.encloses(RegularPolygon(p1, 0.5, 3)) == True
    assert e1.encloses(RegularPolygon(p1, 5, 3)) == False
    assert e1.encloses(RegularPolygon(p2, 5, 3)) == False

    # with generic symbols, the hradius is assumed to contain the major radius
    M = Symbol("M")
    m = Symbol("m")
    c = Ellipse(p1, M, m).circumference
    _x = c.atoms(Dummy).pop()
    assert c == 4 * M * C.Integral(sqrt((1 - _x ** 2 * (M ** 2 - m ** 2) / M ** 2) / (1 - _x ** 2)), (_x, 0, 1))

    assert e2.arbitrary_point() in e2

    # Foci
    f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
    ef = Ellipse(Point(0, 0), 4, 2)
    assert ef.foci in [(f1, f2), (f2, f1)]

    # Tangents
    v = sqrt(2) / 2
    p1_1 = Point(v, v)
    p1_2 = p2 + Point(half, 0)
    p1_3 = p2 + Point(0, 1)
    assert e1.tangent_lines(p4) == c1.tangent_lines(p4)
    assert e2.tangent_lines(p1_2) == [Line(p1_2, p2 + Point(half, 1))]
    assert e2.tangent_lines(p1_3) == [Line(p1_3, p2 + Point(half, 1))]
    assert c1.tangent_lines(p1_1) == [Line(p1_1, Point(0, sqrt(2)))]
    assert c1.tangent_lines(p1) == []
    assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
    assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
    assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
    assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) == False
    assert c1.is_tangent(e1) == False
    assert c1.is_tangent(Ellipse(Point(2, 0), 1, 1)) == True
    assert c1.is_tangent(Polygon(Point(1, 1), Point(1, -1), Point(2, 0))) == True
    assert c1.is_tangent(Polygon(Point(1, 1), Point(1, 0), Point(2, 0))) == False

    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(0, 0)) == [
        Line(Point(0, 0), Point(S(77) / 25, S(132) / 25)),
        Line(Point(0, 0), Point(S(33) / 5, S(22) / 5)),
    ]
    assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(3, 4)) == [
        Line(Point(3, 4), Point(4, 4)),
        Line(Point(3, 4), Point(3, 5)),
    ]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(3, 3)) == [
        Line(Point(3, 3), Point(4, 3)),
        Line(Point(3, 3), Point(3, 4)),
    ]
    assert Circle(Point(5, 5), 2).tangent_lines(Point(5 - 2 * sqrt(2), 5)) == [
        Line(Point(5 - 2 * sqrt(2), 5), Point(5 - sqrt(2), 5 - sqrt(2))),
        Line(Point(5 - 2 * sqrt(2), 5), Point(5 - sqrt(2), 5 + sqrt(2))),
    ]

    # Properties
    major = 3
    minor = 1
    e4 = Ellipse(p2, minor, major)
    assert e4.focus_distance == sqrt(major ** 2 - minor ** 2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major * (1 - ecc)
    assert e4.apoapsis == major * (1 + ecc)
    # independent of orientation
    e4 = Ellipse(p2, major, minor)
    assert e4.focus_distance == sqrt(major ** 2 - minor ** 2)
    ecc = e4.focus_distance / major
    assert e4.eccentricity == ecc
    assert e4.periapsis == major * (1 - ecc)
    assert e4.apoapsis == major * (1 + ecc)

    # Intersection
    l1 = Line(Point(1, -5), Point(1, 5))
    l2 = Line(Point(-5, -1), Point(5, -1))
    l3 = Line(Point(-1, -1), Point(1, 1))
    l4 = Line(Point(-10, 0), Point(0, 10))
    pts_c1_l3 = [Point(sqrt(2) / 2, sqrt(2) / 2), Point(-sqrt(2) / 2, -sqrt(2) / 2)]

    assert intersection(e2, l4) == []
    assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
    assert intersection(c1, l1) == [Point(1, 0)]
    assert intersection(c1, l2) == [Point(0, -1)]
    assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
    assert intersection(c1, c2) == [Point(0, 1), Point(1, 0)]
    assert intersection(c1, c3) == [Point(sqrt(2) / 2, sqrt(2) / 2)]
    assert e1.intersection(l1) == [Point(1, 0)]
    assert e2.intersection(l4) == []
    assert e1.intersection(Circle(Point(0, 2), 1)) == [Point(0, 1)]
    assert e1.intersection(Circle(Point(5, 0), 1)) == []
    assert e1.intersection(Ellipse(Point(2, 0), 1, 1)) == [Point(1, 0)]
    assert e1.intersection(Ellipse(Point(5, 0), 1, 1)) == []
    assert e1.intersection(Point(2, 0)) == []
    assert e1.intersection(e1) == e1

    # some special case intersections
    csmall = Circle(p1, 3)
    cbig = Circle(p1, 5)
    cout = Circle(Point(5, 5), 1)
    # one circle inside of another
    assert csmall.intersection(cbig) == []
    # separate circles
    assert csmall.intersection(cout) == []
    # coincident circles
    assert csmall.intersection(csmall) == csmall

    v = sqrt(2)
    t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
    points = intersection(t1, c1)
    assert len(points) == 4
    assert Point(0, 1) in points
    assert Point(0, -1) in points
    assert Point(v / 2, v / 2) in points
    assert Point(v / 2, -v / 2) in points

    circ = Circle(Point(0, 0), 5)
    elip = Ellipse(Point(0, 0), 5, 20)
    assert intersection(circ, elip) in [[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
    assert elip.tangent_lines(Point(0, 0)) == []
    elip = Ellipse(Point(0, 0), 3, 2)
    assert elip.tangent_lines(Point(3, 0)) == [Line(Point(3, 0), Point(3, -12))]

    e1 = Ellipse(Point(0, 0), 5, 10)
    e2 = Ellipse(Point(2, 1), 4, 8)
    a = S(53) / 17
    c = 2 * sqrt(3991) / 17
    ans = [Point(a - c / 8, a / 2 + c), Point(a + c / 8, a / 2 - c)]
    assert e1.intersection(e2) == ans
    e2 = Ellipse(Point(x, y), 4, 8)
    assert [p.subs({x: 2, y: 1}) for p in e1.intersection(e2)] == ans

    # Combinations of above
    assert e3.is_tangent(e3.tangent_lines(p1 + Point(y1, 0))[0])

    e = Ellipse((1, 2), 3, 2)
    assert e.tangent_lines(Point(10, 0)) == [
        Line(Point(10, 0), Point(1, 0)),
        Line(Point(10, 0), Point(S(14) / 5, S(18) / 5)),
    ]

    # encloses_point
    e = Ellipse((0, 0), 1, 2)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(e.center + Point(e.hradius + Rational(1, 10), 0)) is False
    e = Ellipse((0, 0), 2, 1)
    assert e.encloses_point(e.center)
    assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
    assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
    assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
    assert e.encloses_point(e.center + Point(e.hradius + Rational(1, 10), 0)) is False
    assert c1.encloses_point(Point(1, 0)) is False
    assert c1.encloses_point(Point(0.3, 0.4)) is True

    assert e.scale(2, 3) == Ellipse((0, 0), 4, 3)
    assert e.scale(3, 6) == Ellipse((0, 0), 6, 6)
    assert e.rotate(pi / 3) == e
    assert e.rotate(pi / 3, (1, 2)) == Ellipse(Point(S(1) / 2 + sqrt(3), -sqrt(3) / 2 + 1), 2, 1)
Beispiel #33
0
def test_polygon():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    x1 = Symbol('x1', real=True)
    half = Rational(1, 2)
    a, b, c = Point(0, 0), Point(2, 0), Point(3, 3)
    t = Triangle(a, b, c)
    assert Polygon(a, Point(1, 0), b, c) == t
    assert Polygon(Point(1, 0), b, c, a) == t
    assert Polygon(b, c, a, Point(1, 0)) == t
    # 2 "remove folded" tests
    assert Polygon(a, Point(3, 0), b, c) == t
    assert Polygon(a, b, Point(3, -1), b, c) == t
    raises(GeometryError, lambda: Polygon((0, 0), (1, 0), (0, 1), (1, 1)))
    # remove multiple collinear points
    assert Polygon(Point(-4, 15), Point(-11, 15), Point(-15, 15),
        Point(-15, 33/5), Point(-15, -87/10), Point(-15, -15),
        Point(-42/5, -15), Point(-2, -15), Point(7, -15), Point(15, -15),
        Point(15, -3), Point(15, 10), Point(15, 15)) == \
        Polygon(Point(-15,-15), Point(15,-15), Point(15,15), Point(-15,15))


    p1 = Polygon(
        Point(0, 0), Point(3, -1),
        Point(6, 0), Point(4, 5),
        Point(2, 3), Point(0, 3))
    p2 = Polygon(
        Point(6, 0), Point(3, -1),
        Point(0, 0), Point(0, 3),
        Point(2, 3), Point(4, 5))
    p3 = Polygon(
        Point(0, 0), Point(3, 0),
        Point(5, 2), Point(4, 4))
    p4 = Polygon(
        Point(0, 0), Point(4, 4),
        Point(5, 2), Point(3, 0))
    p5 = Polygon(
        Point(0, 0), Point(4, 4),
        Point(0, 4))
    p6 = Polygon(
        Point(-11, 1), Point(-9, 6.6),
        Point(-4, -3), Point(-8.4, -8.7))
    r = Ray(Point(-9,6.6), Point(-9,5.5))
    #
    # General polygon
    #
    assert p1 == p2
    assert len(p1.args) == 6
    assert len(p1.sides) == 6
    assert p1.perimeter == 5 + 2*sqrt(10) + sqrt(29) + sqrt(8)
    assert p1.area == 22
    assert not p1.is_convex()
    # ensure convex for both CW and CCW point specification
    assert p3.is_convex()
    assert p4.is_convex()
    dict5 = p5.angles
    assert dict5[Point(0, 0)] == pi / 4
    assert dict5[Point(0, 4)] == pi / 2
    assert p5.encloses_point(Point(x, y)) is None
    assert p5.encloses_point(Point(1, 3))
    assert p5.encloses_point(Point(0, 0)) is False
    assert p5.encloses_point(Point(4, 0)) is False
    assert p1.encloses(Circle(Point(2.5,2.5),5)) is False
    assert p1.encloses(Ellipse(Point(2.5,2),5,6)) is False
    p5.plot_interval('x') == [x, 0, 1]
    assert p5.distance(
        Polygon(Point(10, 10), Point(14, 14), Point(10, 14))) == 6 * sqrt(2)
    assert p5.distance(
        Polygon(Point(1, 8), Point(5, 8), Point(8, 12), Point(1, 12))) == 4
    warnings.filterwarnings(
        "error", message="Polygons may intersect producing erroneous output")
    raises(UserWarning,
           lambda: Polygon(Point(0, 0), Point(1, 0),
           Point(1, 1)).distance(
           Polygon(Point(0, 0), Point(0, 1), Point(1, 1))))
    warnings.filterwarnings(
        "ignore", message="Polygons may intersect producing erroneous output")
    assert hash(p5) == hash(Polygon(Point(0, 0), Point(4, 4), Point(0, 4)))
    assert p5 == Polygon(Point(4, 4), Point(0, 4), Point(0, 0))
    assert Polygon(Point(4, 4), Point(0, 4), Point(0, 0)) in p5
    assert p5 != Point(0, 4)
    assert Point(0, 1) in p5
    assert p5.arbitrary_point('t').subs(Symbol('t', real=True), 0) == \
        Point(0, 0)
    raises(ValueError, lambda: Polygon(
        Point(x, 0), Point(0, y), Point(x, y)).arbitrary_point('x'))
    assert p6.intersection(r) == [Point(-9, 33/5), Point(-9, -84/13)]
    #
    # Regular polygon
    #
    p1 = RegularPolygon(Point(0, 0), 10, 5)
    p2 = RegularPolygon(Point(0, 0), 5, 5)
    raises(GeometryError, lambda: RegularPolygon(Point(0, 0), Point(0,
           1), Point(1, 1)))
    raises(GeometryError, lambda: RegularPolygon(Point(0, 0), 1, 2))
    raises(ValueError, lambda: RegularPolygon(Point(0, 0), 1, 2.5))

    assert p1 != p2
    assert p1.interior_angle == 3*pi/5
    assert p1.exterior_angle == 2*pi/5
    assert p2.apothem == 5*cos(pi/5)
    assert p2.circumcenter == p1.circumcenter == Point(0, 0)
    assert p1.circumradius == p1.radius == 10
    assert p2.circumcircle == Circle(Point(0, 0), 5)
    assert p2.incircle == Circle(Point(0, 0), p2.apothem)
    assert p2.inradius == p2.apothem == (5 * (1 + sqrt(5)) / 4)
    p2.spin(pi / 10)
    dict1 = p2.angles
    assert dict1[Point(0, 5)] == 3 * pi / 5
    assert p1.is_convex()
    assert p1.rotation == 0
    assert p1.encloses_point(Point(0, 0))
    assert p1.encloses_point(Point(11, 0)) is False
    assert p2.encloses_point(Point(0, 4.9))
    p1.spin(pi/3)
    assert p1.rotation == pi/3
    assert p1.vertices[0] == Point(5, 5*sqrt(3))
    for var in p1.args:
        if isinstance(var, Point):
            assert var == Point(0, 0)
        else:
            assert var == 5 or var == 10 or var == pi / 3
    assert p1 != Point(0, 0)
    assert p1 != p5

    # while spin works in place (notice that rotation is 2pi/3 below)
    # rotate returns a new object
    p1_old = p1
    assert p1.rotate(pi/3) == RegularPolygon(Point(0, 0), 10, 5, 2*pi/3)
    assert p1 == p1_old

    assert p1.area == (-250*sqrt(5) + 1250)/(4*tan(pi/5))
    assert p1.length == 20*sqrt(-sqrt(5)/8 + 5/8)
    assert p1.scale(2, 2) == \
        RegularPolygon(p1.center, p1.radius*2, p1._n, p1.rotation)
    assert RegularPolygon((0, 0), 1, 4).scale(2, 3) == \
        Polygon(Point(2, 0), Point(0, 3), Point(-2, 0), Point(0, -3))

    assert repr(p1) == str(p1)

    #
    # Angles
    #
    angles = p4.angles
    assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))

    angles = p3.angles
    assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
    assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
    assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
    assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))

    #
    # Triangle
    #
    p1 = Point(0, 0)
    p2 = Point(5, 0)
    p3 = Point(0, 5)
    t1 = Triangle(p1, p2, p3)
    t2 = Triangle(p1, p2, Point(Rational(5, 2), sqrt(Rational(75, 4))))
    t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
    s1 = t1.sides
    assert Triangle(p1, p2, p1) == Polygon(p1, p2, p1) == Segment(p1, p2)
    raises(GeometryError, lambda: Triangle(Point(0, 0)))

    # Basic stuff
    assert Triangle(p1, p1, p1) == p1
    assert Triangle(p2, p2*2, p2*3) == Segment(p2, p2*3)
    assert t1.area == Rational(25, 2)
    assert t1.is_right()
    assert t2.is_right() is False
    assert t3.is_right()
    assert p1 in t1
    assert t1.sides[0] in t1
    assert Segment((0, 0), (1, 0)) in t1
    assert Point(5, 5) not in t2
    assert t1.is_convex()
    assert feq(t1.angles[p1].evalf(), pi.evalf()/2)

    assert t1.is_equilateral() is False
    assert t2.is_equilateral()
    assert t3.is_equilateral() is False
    assert are_similar(t1, t2) is False
    assert are_similar(t1, t3)
    assert are_similar(t2, t3) is False
    assert t1.is_similar(Point(0, 0)) is False

    # Bisectors
    bisectors = t1.bisectors()
    assert bisectors[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    ic = (250 - 125*sqrt(2)) / 50
    assert t1.incenter == Point(ic, ic)

    # Inradius
    assert t1.inradius == t1.incircle.radius == 5 - 5*sqrt(2)/2
    assert t2.inradius == t2.incircle.radius == 5*sqrt(3)/6
    assert t3.inradius == t3.incircle.radius == x1**2/((2 + sqrt(2))*Abs(x1))

    # Circumcircle
    assert t1.circumcircle.center == Point(2.5, 2.5)

    # Medians + Centroid
    m = t1.medians
    assert t1.centroid == Point(Rational(5, 3), Rational(5, 3))
    assert m[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    assert t3.medians[p1] == Segment(p1, Point(x1/2, x1/2))
    assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]
    assert t1.medial == Triangle(Point(2.5, 0), Point(0, 2.5), Point(2.5, 2.5))

    # Nine-point circle
    assert t1.nine_point_circle == Circle(Point(2.5, 0), Point(0, 2.5), Point(2.5, 2.5))
    assert t1.nine_point_circle == Circle(Point(0, 0), Point(0, 2.5), Point(2.5, 2.5))

    # Perpendicular
    altitudes = t1.altitudes
    assert altitudes[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
    assert altitudes[p2] == s1[0]
    assert altitudes[p3] == s1[2]
    assert t1.orthocenter == p1
    t = S('''Triangle(
    Point(100080156402737/5000000000000, 79782624633431/500000000000),
    Point(39223884078253/2000000000000, 156345163124289/1000000000000),
    Point(31241359188437/1250000000000, 338338270939941/1000000000000000))''')
    assert t.orthocenter == S('''Point(-780660869050599840216997'''
    '''79471538701955848721853/80368430960602242240789074233100000000000000,'''
    '''20151573611150265741278060334545897615974257/16073686192120448448157'''
    '''8148466200000000000)''')

    # Ensure
    assert len(intersection(*bisectors.values())) == 1
    assert len(intersection(*altitudes.values())) == 1
    assert len(intersection(*m.values())) == 1

    # Distance
    p1 = Polygon(
        Point(0, 0), Point(1, 0),
        Point(1, 1), Point(0, 1))
    p2 = Polygon(
        Point(0, Rational(5)/4), Point(1, Rational(5)/4),
        Point(1, Rational(9)/4), Point(0, Rational(9)/4))
    p3 = Polygon(
        Point(1, 2), Point(2, 2),
        Point(2, 1))
    p4 = Polygon(
        Point(1, 1), Point(Rational(6)/5, 1),
        Point(1, Rational(6)/5))
    pt1 = Point(half, half)
    pt2 = Point(1, 1)

    '''Polygon to Point'''
    assert p1.distance(pt1) == half
    assert p1.distance(pt2) == 0
    assert p2.distance(pt1) == Rational(3)/4
    assert p3.distance(pt2) == sqrt(2)/2

    '''Polygon to Polygon'''
    # p1.distance(p2) emits a warning
    # First, test the warning
    warnings.filterwarnings("error",
        message="Polygons may intersect producing erroneous output")
    raises(UserWarning, lambda: p1.distance(p2))
    # now test the actual output
    warnings.filterwarnings("ignore",
        message="Polygons may intersect producing erroneous output")
    assert p1.distance(p2) == half/2

    assert p1.distance(p3) == sqrt(2)/2
    assert p3.distance(p4) == (sqrt(2)/2 - sqrt(Rational(2)/25)/2)
Beispiel #34
0
def test_util():
    # coverage for some leftover functions in sympy.geometry.util
    assert intersection(Point(0, 0)) == []
    raises(ValueError, lambda: intersection(Point(0, 0), 3))
    raises(ValueError, lambda: convex_hull(Point(0, 0), 3))
Beispiel #35
0
def test_intersection_2d():
    p1 = Point(0, 0)
    p2 = Point(1, 1)
    p3 = Point(x1, x1)
    p4 = Point(y1, y1)

    l1 = Line(p1, p2)
    l3 = Line(Point(0, 0), Point(3, 4))

    r1 = Ray(Point(1, 1), Point(2, 2))
    r2 = Ray(Point(0, 0), Point(3, 4))
    r4 = Ray(p1, p2)
    r6 = Ray(Point(0, 1), Point(1, 2))
    r7 = Ray(Point(0.5, 0.5), Point(1, 1))

    s1 = Segment(p1, p2)
    s2 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
    s3 = Segment(Point(0, 0), Point(3, 4))

    assert intersection(l1, p1) == [p1]
    assert intersection(l1, Point(x1, 1 + x1)) == []
    assert intersection(l1, Line(p3, p4)) in [[l1], [Line(p3, p4)]]
    assert intersection(l1, l1.parallel_line(Point(x1, 1 + x1))) == []
    assert intersection(l3, l3) == [l3]
    assert intersection(l3, r2) == [r2]
    assert intersection(l3, s3) == [s3]
    assert intersection(s3, l3) == [s3]
    assert intersection(Segment(Point(-10, 10), Point(10, 10)),
                        Segment(Point(-5, -5), Point(-5, 5))) == []
    assert intersection(r2, l3) == [r2]
    assert intersection(r1,
                        Ray(Point(2, 2),
                            Point(0,
                                  0))) == [Segment(Point(1, 1), Point(2, 2))]
    assert intersection(r1, Ray(Point(1, 1), Point(-1, -1))) == [Point(1, 1)]
    assert intersection(r1, Segment(Point(0, 0), Point(
        2, 2))) == [Segment(Point(1, 1), Point(2, 2))]

    assert r4.intersection(s2) == [s2]
    assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
    assert r4.intersection(Segment(Point(-1, -1), Point(
        0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert r4.intersection(Ray(p2, p1)) == [s1]
    assert Ray(p2, p1).intersection(r6) == []
    assert r4.intersection(r7) == r7.intersection(r4) == [r7]
    assert Ray3D((0, 0), (3, 0)).intersection(Ray3D(
        (1, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
    assert Ray3D((1, 0), (3, 0)).intersection(Ray3D(
        (0, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
    assert Ray(Point(0, 0), Point(0, 4)).intersection(Ray(Point(0, 1), Point(0, -1))) == \
           [Segment(Point(0, 0), Point(0, 1))]

    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (1, 0), (2, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((1, 0), (2, 0)).intersection(Segment3D(
        (0, 0), (3, 0))) == [Segment3D((1, 0), (2, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (3, 0), (4, 0))) == [Point3D((3, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (2, 0), (5, 0))) == [Segment3D((2, 0), (3, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (-2, 0), (1, 0))) == [Segment3D((0, 0), (1, 0))]
    assert Segment3D((0, 0), (3, 0)).intersection(Segment3D(
        (-2, 0), (0, 0))) == [Point3D(0, 0)]
    assert s1.intersection(Segment(Point(1, 1), Point(2, 2))) == [Point(1, 1)]
    assert s1.intersection(Segment(Point(0.5, 0.5), Point(
        1.5, 1.5))) == [Segment(Point(0.5, 0.5), p2)]
    assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
    assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
    assert s1.intersection(Segment(Point(-1, -1), Point(
        0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
    assert s1.intersection(Line(Point(1, 0), Point(2, 1))) == []
    assert s1.intersection(s2) == [s2]
    assert s2.intersection(s1) == [s2]

    assert asa(120, 8, 52) == \
           Triangle(
               Point(0, 0),
               Point(8, 0),
               Point(-4 * cos(19 * pi / 90) / sin(2 * pi / 45),
                     4 * sqrt(3) * cos(19 * pi / 90) / sin(2 * pi / 45)))
    assert Line((0, 0), (1, 1)).intersection(Ray((1, 0),
                                                 (1, 2))) == [Point(1, 1)]
    assert Line((0, 0), (1, 1)).intersection(Segment((1, 0),
                                                     (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (1, 1)).intersection(Ray((1, 0),
                                                (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (1, 1)).intersection(Segment((1, 0),
                                                    (1, 2))) == [Point(1, 1)]
    assert Ray((0, 0), (10, 10)).contains(Segment((1, 1), (2, 2))) is True
    assert Segment((1, 1), (2, 2)) in Line((0, 0), (10, 10))

    # 16628 - this should be fast
    p0 = Point2D(S(249) / 5, S(497999) / 10000)
    p1 = Point2D(
        (-58977084786 * sqrt(405639795226) + 2030690077184193 + 20112207807 *
         sqrt(630547164901) + 99600 * sqrt(255775022850776494562626)) /
        (2000 * sqrt(255775022850776494562626) +
         1991998000 * sqrt(405639795226) + 1991998000 * sqrt(630547164901) +
         1622561172902000),
        (-498000 * sqrt(255775022850776494562626) - 995999 * sqrt(630547164901)
         + 90004251917891999 + 496005510002 * sqrt(405639795226)) /
        (10000 * sqrt(255775022850776494562626) +
         9959990000 * sqrt(405639795226) + 9959990000 * sqrt(630547164901) +
         8112805864510000))
    p2 = Point2D(S(497) / 10, -S(497) / 10)
    p3 = Point2D(-S(497) / 10, -S(497) / 10)
    l = Line(p0, p1)
    s = Segment(p2, p3)
    n = (-52673223862 * sqrt(405639795226) - 15764156209307469 -
         9803028531 * sqrt(630547164901) +
         33200 * sqrt(255775022850776494562626))
    d = sqrt(405639795226) + 315274080450 + 498000 * sqrt(630547164901) + sqrt(
        255775022850776494562626)
    assert intersection(l, s) == [Point2D(n / d * S(3) / 2000, -S(497) / 10)]