def test_section_modulus_and_polar_second_moment_of_area(): d = Symbol('d', positive=True) c = Circle((3, 7), 8) assert c.polar_second_moment_of_area() == 2048 * pi assert c.section_modulus() == (128 * pi, 128 * pi) c = Circle((2, 9), d / 2) assert c.polar_second_moment_of_area( ) == pi * d**3 * Abs(d) / 64 + pi * d * Abs(d)**3 / 64 assert c.section_modulus() == (pi * d**3 / S(32), pi * d**3 / S(32)) a, b = symbols('a, b', positive=True) e = Ellipse((4, 6), a, b) assert e.section_modulus() == (pi * a * b**2 / S(4), pi * a**2 * b / S(4)) assert e.polar_second_moment_of_area( ) == pi * a**3 * b / S(4) + pi * a * b**3 / S(4) e = e.rotate(pi / 2) # no change in polar and section modulus assert e.section_modulus() == (pi * a**2 * b / S(4), pi * a * b**2 / S(4)) assert e.polar_second_moment_of_area( ) == pi * a**3 * b / S(4) + pi * a * b**3 / S(4) e = Ellipse((a, b), 2, 6) assert e.section_modulus() == (18 * pi, 6 * pi) assert e.polar_second_moment_of_area() == 120 * pi e = Ellipse(Point(0, 0), 2, 2) assert e.section_modulus() == (2 * pi, 2 * pi) assert e.section_modulus(Point(2, 2)) == (2 * pi, 2 * pi) assert e.section_modulus((2, 2)) == (2 * pi, 2 * pi)
def test_bounds(): e1 = Ellipse(Point(0, 0), 3, 5) e2 = Ellipse(Point(2, -2), 7, 7) c1 = Circle(Point(2, -2), 7) c2 = Circle(Point(-2, 0), Point(0, 2), Point(2, 0)) assert e1.bounds == (-3, -5, 3, 5) assert e2.bounds == (-5, -9, 9, 5) assert c1.bounds == (-5, -9, 9, 5) assert c2.bounds == (-2, -2, 2, 2)
def test_ellipse_equation_using_slope(): from sympy.abc import x, y e1 = Ellipse(Point(1, 0), 3, 2) assert str(e1.equation(_slope=1)) == str((-x + y + 1)**2/8 + (x + y - 1)**2/18 - 1) e2 = Ellipse(Point(0, 0), 4, 1) assert str(e2.equation(_slope=1)) == str((-x + y)**2/2 + (x + y)**2/32 - 1) e3 = Ellipse(Point(1, 5), 6, 2) assert str(e3.equation(_slope=2)) == str((-2*x + y - 3)**2/20 + (x + 2*y - 11)**2/180 - 1)
def test_Geometry(): assert sstr(Point(0, 0)) == 'Point2D(0, 0)' assert sstr(Circle(Point(0, 0), 3)) == 'Circle(Point2D(0, 0), 3)' assert sstr(Ellipse(Point(1, 2), 3, 4)) == 'Ellipse(Point2D(1, 2), 3, 4)' assert sstr(Triangle(Point(1, 1), Point(7, 8), Point(0, -1))) == \ 'Triangle(Point2D(1, 1), Point2D(7, 8), Point2D(0, -1))' assert sstr(Polygon(Point(5, 6), Point(-2, -3), Point(0, 0), Point(4, 7))) == \ 'Polygon(Point2D(5, 6), Point2D(-2, -3), Point2D(0, 0), Point2D(4, 7))' assert sstr(Triangle(Point(0, 0), Point(1, 0), Point(0, 1)), sympy_integers=True) == \ 'Triangle(Point2D(S(0), S(0)), Point2D(S(1), S(0)), Point2D(S(0), S(1)))' assert sstr(Ellipse(Point(1, 2), 3, 4), sympy_integers=True) == \ 'Ellipse(Point2D(S(1), S(2)), S(3), S(4))'
def test_circumference(): M = Symbol('M') m = Symbol('m') assert Ellipse(Point(0, 0), M, m).circumference == 4 * M * elliptic_e((M ** 2 - m ** 2) / M**2) assert Ellipse(Point(0, 0), 5, 4).circumference == 20 * elliptic_e(S(9) / 25) # circle assert Ellipse(None, 1, None, 0).circumference == 2*pi # test numerically assert abs(Ellipse(None, hradius=5, vradius=3).circumference.evalf(16) - 25.52699886339813) < 1e-10
def test_construction(): e1 = Ellipse(hradius=2, vradius=1, eccentricity=None) assert e1.eccentricity == sqrt(3)/2 e2 = Ellipse(hradius=2, vradius=None, eccentricity=sqrt(3)/2) assert e2.vradius == 1 e3 = Ellipse(hradius=None, vradius=1, eccentricity=sqrt(3)/2) assert e3.hradius == 2 # filter(None, iterator) filters out anything falsey, including 0 # eccentricity would be filtered out in this case and the constructor would throw an error e4 = Ellipse(Point(0, 0), hradius=1, eccentricity=0) assert e4.vradius == 1
def test_evolute(): #ellipse centered at h,k x, y, h, k = symbols('x y h k',real = True) a, b = symbols('a b') e = Ellipse(Point(h, k), a, b) t1 = (e.hradius*(x - e.center.x))**Rational(2, 3) t2 = (e.vradius*(y - e.center.y))**Rational(2, 3) E = t1 + t2 - (e.hradius**2 - e.vradius**2)**Rational(2, 3) assert e.evolute() == E #Numerical Example e = Ellipse(Point(1, 1), 6, 3) t1 = (6*(x - 1))**Rational(2, 3) t2 = (3*(y - 1))**Rational(2, 3) E = t1 + t2 - (27)**Rational(2, 3) assert e.evolute() == E
def test_free_symbols(): a, b, c, d, e, f, s = symbols('a:f,s') assert Point(a,b).free_symbols == set([a, b]) assert Line((a,b),(c,d)).free_symbols == set([a, b, c, d]) assert Ray((a,b),(c,d)).free_symbols == set([a, b, c, d]) assert Ray((a,b),angle=c).free_symbols == set([a, b, c]) assert Segment((a,b),(c,d)).free_symbols == set([a, b, c, d]) assert Line((a,b),slope=c).free_symbols == set([a, b, c]) assert Curve((a*s,b*s),(s,c,d)).free_symbols == set([a, b, c, d]) assert Ellipse((a,b),c,d).free_symbols == set([a, b, c, d]) assert Ellipse((a,b),c, eccentricity=d).free_symbols == set([a, b, c, d]) assert Ellipse((a,b),vradius=c, eccentricity=d).free_symbols == set([a, b, c, d]) assert Circle((a,b),c).free_symbols == set([a, b, c]) assert Circle((a,b),(c,d),(e,f)).free_symbols == set([e, d, c, b, f, a]) assert Polygon((a,b),(c,d),(e,f)).free_symbols == set([e, b, d, f, a, c]) assert RegularPolygon((a,b),c,d,e).free_symbols == set([e, a, b, c, d])
def test_reflect(): b = Symbol('b') m = Symbol('m') l = Line((0, b), slope=m) p = Point(x, y) r = p.reflect(l) dp = l.perpendicular_segment(p).length dr = l.perpendicular_segment(r).length assert test_numerically(dp, dr) t = Triangle((0, 0), (1, 0), (2, 3)) assert t.area == -t.reflect(l).area e = Ellipse((1, 0), 1, 2) assert e.area == -e.reflect(Line((1, 0), slope=0)).area assert e.area == -e.reflect(Line((1, 0), slope=oo)).area raises(NotImplementedError, lambda: e.reflect(Line((1, 0), slope=m))) # test entity overrides c = Circle((x, y), 3) cr = c.reflect(l) assert cr == Circle(r, -3) assert c.area == -cr.area pent = RegularPolygon((1, 2), 1, 5) l = Line((0, pi), slope=sqrt(2)) rpent = pent.reflect(l) poly_pent = Polygon(*pent.vertices) assert rpent.center == pent.center.reflect(l) assert str([w.n(3) for w in rpent.vertices ]) == ('[Point(-0.586, 4.27), Point(-1.69, 4.66), ' 'Point(-2.41, 3.73), Point(-1.74, 2.76), ' 'Point(-0.616, 3.10)]') assert pent.area.equals(-rpent.area)
def test_ellipse_random_point(): e3 = Ellipse(Point(0, 0), y1, y1) rx, ry = Symbol('rx'), Symbol('ry') for ind in range(0, 5): r = e3.random_point() # substitution should give zero*y1**2 assert e3.equation(rx, ry).subs(zip((rx, ry), r.args)).equals(0)
def test_director_circle(): x, y, a, b = symbols('x y a b') e = Ellipse((x, y), a, b) # the general result assert e.director_circle() == Circle((x, y), sqrt(a**2 + b**2)) # a special case where Ellipse is a Circle assert Circle((3, 4), 8).director_circle() == Circle((3, 4), 8*sqrt(2))
def test_auxiliary_circle(): x, y, a, b = symbols('x y a b') e = Ellipse((x, y), a, b) # the general result assert e.auxiliary_circle() == Circle((x, y), Max(a, b)) # a special case where Ellipse is a Circle assert Circle((3, 4), 8).auxiliary_circle() == Circle((3, 4), 8)
def test_parametric_region_list(): point = Point(-5, 12) assert parametric_region_list(point) == [ParametricRegion((-5, 12))] e = Ellipse(Point(2, 8), 2, 6) assert parametric_region_list(e, t) == [ParametricRegion((2*cos(t) + 2, 6*sin(t) + 8), (t, 0, 2*pi))] c = Curve((t, t**3), (t, 5, 3)) assert parametric_region_list(c) == [ParametricRegion((t, t**3), (t, 5, 3))] s = Segment(Point(2, 11, -6), Point(0, 2, 5)) assert parametric_region_list(s, t) == [ParametricRegion((2 - 2*t, 11 - 9*t, 11*t - 6), (t, 0, 1))] s1 = Segment(Point(0, 0), (1, 0)) assert parametric_region_list(s1, t) == [ParametricRegion((t, 0), (t, 0, 1))] s2 = Segment(Point(1, 2, 3), Point(1, 2, 5)) assert parametric_region_list(s2, t) == [ParametricRegion((1, 2, 2*t + 3), (t, 0, 1))] s3 = Segment(Point(12, 56), Point(12, 56)) assert parametric_region_list(s3) == [ParametricRegion((12, 56))] poly = Polygon((1,3), (-3, 8), (2, 4)) assert parametric_region_list(poly, t) == [ParametricRegion((1 - 4*t, 5*t + 3), (t, 0, 1)), ParametricRegion((5*t - 3, 8 - 4*t), (t, 0, 1)), ParametricRegion((2 - t, 4 - t), (t, 0, 1))] p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7,8))) raises(ValueError, lambda: parametric_region_list(p1))
def test_subs(): x = Symbol('x', real=True) y = Symbol('y', real=True) p = Point(x, 2) q = Point(1, 1) r = Point(3, 4) for o in [ p, Segment(p, q), Ray(p, q), Line(p, q), Triangle(p, q, r), RegularPolygon(p, 3, 6), Polygon(p, q, r, Point(5, 4)), Circle(p, 3), Ellipse(p, 3, 4) ]: assert 'y' in str(o.subs(x, y)) assert p.subs({x: 1}) == Point(1, 2) assert Point(1, 2).subs(Point(1, 2), Point(3, 4)) == Point(3, 4) assert Point(1, 2).subs((1, 2), Point(3, 4)) == Point(3, 4) assert Point(1, 2).subs(Point(1, 2), Point(3, 4)) == Point(3, 4) assert Point(1, 2).subs(set([(1, 2)])) == Point(2, 2) raises(ValueError, lambda: Point(1, 2).subs(1)) raises(ValueError, lambda: Point(1, 1).subs( (Point(1, 1), Point(1, 2)), 1, 2))
def test_ellipse_random_point(): y1 = Symbol("y1", real=True) e3 = Ellipse(Point(0, 0), y1, y1) rx, ry = Symbol("rx"), Symbol("ry") for ind in range(0, 5): r = e3.random_point() # substitution should give zero*y1**2 assert e3.equation(rx, ry).subs(zip((rx, ry), r.args)).equals(0)
def test_reflect(): b = Symbol('b') m = Symbol('m') l = Line((0, b), slope=m) t1 = Triangle((0, 0), (1, 0), (2, 3)) assert t1.area == -t1.reflect(l).area e = Ellipse((1, 0), 1, 2) assert e.area == -e.reflect(Line((1, 0), slope=0)).area assert e.area == -e.reflect(Line((1, 0), slope=oo)).area raises(NotImplementedError, lambda: e.reflect(Line((1, 0), slope=m)))
def test_second_moment_of_area(): x, y = symbols('x, y') e = Ellipse(Point(0, 0), 5, 4) I_yy = 2*4*integrate(sqrt(25 - x**2)*x**2, (x, -5, 5))/5 I_xx = 2*5*integrate(sqrt(16 - y**2)*y**2, (y, -4, 4))/4 Y = 3*sqrt(1 - x**2/5**2) I_xy = integrate(integrate(y, (y, -Y, Y))*x, (x, -5, 5)) assert I_yy == e.second_moment_of_area()[1] assert I_xx == e.second_moment_of_area()[0] assert I_xy == e.second_moment_of_area()[2]
def test_ellipse_random_point(): y1 = Symbol('y1', real=True) e3 = Ellipse(Point(0, 0), y1, y1) rx, ry = Symbol('rx'), Symbol('ry') for ind in range(0, 5): r = e3.random_point() # substitution should give zero*y1**2 assert e3.equation(rx, ry).subs(zip((rx, ry), r.args)).equals(0) # test for the case with seed r = e3.random_point(seed=1) assert e3.equation(rx, ry).subs(zip((rx, ry), r.args)).equals(0)
def test_is_tangent(): e1 = Ellipse(Point(0, 0), 3, 5) c1 = Circle(Point(2, -2), 7) assert e1.is_tangent(Point(0, 0)) is False assert e1.is_tangent(Point(3, 0)) is False assert e1.is_tangent(e1) is True assert e1.is_tangent(Ellipse((0, 0), 1, 2)) is False assert e1.is_tangent(Ellipse((0, 0), 3, 2)) is True assert c1.is_tangent(Ellipse((2, -2), 7, 1)) is True assert c1.is_tangent(Circle((11, -2), 2)) is True assert c1.is_tangent(Circle((7, -2), 2)) is True assert c1.is_tangent(Ray((-5, -2), (-15, -20))) is False assert c1.is_tangent(Ray((-3, -2), (-15, -20))) is False assert c1.is_tangent(Ray((-3, -22), (15, 20))) is False assert c1.is_tangent(Ray((9, 20), (9, -20))) is True assert e1.is_tangent(Segment((2, 2), (-7, 7))) is False assert e1.is_tangent(Segment((0, 0), (1, 2))) is False assert c1.is_tangent(Segment((0, 0), (-5, -2))) is False assert e1.is_tangent(Segment((3, 0), (12, 12))) is False assert e1.is_tangent(Segment((12, 12), (3, 0))) is False assert e1.is_tangent(Segment((-3, 0), (3, 0))) is False assert e1.is_tangent(Segment((-3, 5), (3, 5))) is True assert e1.is_tangent(Line((10, 0), (10, 10))) is False assert e1.is_tangent(Line((0, 0), (1, 1))) is False assert e1.is_tangent(Line((-3, 0), (-2.99, -0.001))) is False assert e1.is_tangent(Line((-3, 0), (-3, 1))) is True assert e1.is_tangent(Polygon((0, 0), (5, 5), (5, -5))) is False assert e1.is_tangent(Polygon((-100, -50), (-40, -334), (-70, -52))) is False assert e1.is_tangent(Polygon((-3, 0), (3, 0), (0, 1))) is False assert e1.is_tangent(Polygon((-3, 0), (3, 0), (0, 5))) is False assert e1.is_tangent(Polygon((-3, 0), (0, -5), (3, 0), (0, 5))) is False assert e1.is_tangent(Polygon((-3, -5), (-3, 5), (3, 5), (3, -5))) is True assert c1.is_tangent(Polygon((-3, -5), (-3, 5), (3, 5), (3, -5))) is False assert e1.is_tangent(Polygon((0, 0), (3, 0), (7, 7), (0, 5))) is False assert e1.is_tangent(Polygon((3, 12), (3, -12), (6, 5))) is True assert e1.is_tangent(Polygon((3, 12), (3, -12), (0, -5), (0, 5))) is False assert e1.is_tangent(Polygon((3, 0), (5, 7), (6, -5))) is False raises(TypeError, lambda: e1.is_tangent(Point(0, 0, 0))) raises(TypeError, lambda: e1.is_tangent(Rational(5)))
def test_second_moment_of_area(): x, y = symbols('x, y') e = Ellipse(Point(0, 0), 5, 4) I_yy = 2*4*integrate(sqrt(25 - x**2)*x**2, (x, -5, 5))/5 I_xx = 2*5*integrate(sqrt(16 - y**2)*y**2, (y, -4, 4))/4 Y = 3*sqrt(1 - x**2/5**2) I_xy = integrate(integrate(y, (y, -Y, Y))*x, (x, -5, 5)) assert I_yy == e.second_moment_of_area()[1] assert I_xx == e.second_moment_of_area()[0] assert I_xy == e.second_moment_of_area()[2] #checking for other point t1 = e.second_moment_of_area(Point(6,5)) t2 = (580*pi, 845*pi, 600*pi) assert t1==t2
def test_subs(): p = Point(x, 2) q = Point(1, 1) r = Point(3, 4) for o in [p, Segment(p, q), Ray(p, q), Line(p, q), Triangle(p, q, r), RegularPolygon(p, 3, 6), Polygon(p, q, r, Point(5,4)), Circle(p, 3), Ellipse(p, 3, 4)]: assert 'y' in str(o.subs(x, y))
def test_geometry_transforms(): from sympy import Tuple c = Curve((x, x**2), (x, 0, 1)) pts = [Point(0, 0), Point(S(1) / 2, S(1) / 4), Point(1, 1)] cout = Curve((2 * x - 4, 3 * x**2 - 10), (x, 0, 1)) pts_out = [Point(-4, -10), Point(-3, -S(37) / 4), Point(-2, -7)] assert c.scale(2, 3, (4, 5)) == cout assert [c.subs(x, xi / 2) for xi in Tuple(0, 1, 2)] == pts assert [cout.subs(x, xi / 2) for xi in Tuple(0, 1, 2)] == pts_out assert Triangle(*pts).scale(2, 3, (4, 5)) == Triangle(*pts_out) assert Ellipse((0, 0), 2, 3).scale(2, 3, (4, 5)) == \ Ellipse(Point(-4, -10), 4, 9) assert Circle((0, 0), 2).scale(2, 3, (4, 5)) == \ Ellipse(Point(-4, -10), 4, 6) assert Ellipse((0, 0), 2, 3).scale(3, 3, (4, 5)) == \ Ellipse(Point(-8, -10), 6, 9) assert Circle((0, 0), 2).scale(3, 3, (4, 5)) == \ Circle(Point(-8, -10), 6) assert Circle(Point(-8, -10), 6).scale(S(1)/3, S(1)/3, (4, 5)) == \ Circle((0, 0), 2) assert Curve((x + y, 3*x), (x, 0, 1)).subs(y, S.Half) == \ Curve((x + S(1)/2, 3*x), (x, 0, 1)) assert Curve((x, 3*x), (x, 0, 1)).translate(4, 5) == \ Curve((x + 4, 3*x + 5), (x, 0, 1)) assert Circle((0, 0), 2).translate(4, 5) == \ Circle((4, 5), 2) assert Circle((0, 0), 2).scale(3, 3) == \ Circle((0, 0), 6) assert Point(1, 1).scale(2, 3, (4, 5)) == \ Point(-2, -7) assert Point(1, 1).translate(4, 5) == \ Point(5, 6) assert scale(1, 2, (3, 4)).tolist() == \ [[1, 0, 0], [0, 2, 0], [0, -4, 1]] assert RegularPolygon((0, 0), 1, 4).scale(2, 3, (4, 5)) == \ Polygon(Point(-2, -10), Point(-4, -7), Point(-6, -10), Point(-4, -13))
def test_geometry_EvalfMixin(): x = pi t = Symbol('t') for g in [ Point(x, x), Plane(Point(0, x, 0), (0, 0, x)), Curve((x * t, x), (t, 0, x)), Ellipse((x, x), x, -x), Circle((x, x), x), Line((0, x), (x, 0)), Segment((0, x), (x, 0)), Ray((0, x), (x, 0)), Parabola((0, x), Line((-x, 0), (x, 0))), Polygon((0, 0), (0, x), (x, 0), (x, x)), RegularPolygon((0, x), x, 4, x), Triangle((0, 0), (x, 0), (x, x)), ]: assert str(g).replace('pi', '3.1') == str(g.n(2))
def test_geometry(): def do_test(*g, s=GeometrySeries, **kwargs): s1 = _build_series(*g, pt="g", **kwargs) assert isinstance(s1, s) # since the range could be None, it is imperative to test that label # receive the correct value. assert s1.label == str(g[0]) s2 = _build_series(*g, **kwargs) assert isinstance(s2, s) assert s2.label == str(g[0]) assert np.array_equal(s1.get_data(), s2.get_data(), equal_nan=True) x, y, z = symbols("x, y, z") do_test(Point2D(1, 2)) do_test(Point3D(1, 2, 3)) do_test(Ray((1, 2), (3, 4))) do_test(Segment((1, 2), (3, 4))) do_test(Line((1, 2), (3, 4)), (x, -5, 5)) do_test(Ray3D((1, 2, 3), (3, 4, 5))) do_test(Segment3D((1, 2, 3), (3, 4, 5))) do_test(Line3D((1, 2, 3), (3, 4, 5))) do_test(Polygon((1, 2), 3, n=10)) do_test(Circle((1, 2), 3)) do_test(Ellipse((1, 2), hradius=3, vradius=2)) do_test(Plane((0, 0, 0), (1, 1, 1)), (x, -5, 5), (y, -4, 4), (z, -3, 3), s=PlaneSeries) # Interactive series. Note that GeometryInteractiveSeries is an instance of # GeometrySeries do_test(Point2D(x, y), params={x: 1, y: 2}) do_test( Plane((x, y, z), (1, 1, 1)), (x, -5, 5), (y, -4, 4), (z, -3, 3), params={ x: 1, y: 2, z: 3 }, s=PlaneInteractiveSeries, )
def test_transform(): 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) assert Ellipse((0, 0), 2, 3).scale(2, 3, (4, 5)) == Ellipse(Point(-4, -10), 4, 9) assert Circle((0, 0), 2).scale(2, 3, (4, 5)) == Ellipse(Point(-4, -10), 4, 6) assert Ellipse((0, 0), 2, 3).scale(3, 3, (4, 5)) == Ellipse(Point(-8, -10), 6, 9) assert Circle((0, 0), 2).scale(3, 3, (4, 5)) == Circle(Point(-8, -10), 6) assert Circle(Point(-8, -10), 6).scale(Rational(1, 3), Rational(1, 3), (4, 5)) == Circle((0, 0), 2) assert Circle((0, 0), 2).translate(4, 5) == Circle((4, 5), 2) assert Circle((0, 0), 2).scale(3, 3) == Circle((0, 0), 6)
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, "Circle(Point(0,0), Point(1,1), Point(2,2))") raises(ValueError, "Ellipse(None, None, None, 1)") raises(GeometryError, "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 # 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) in [[(1, 0), (0, 1)], [(0, 1), (1, 0)]] assert intersection(c1, c3) == [(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) ans = list(reversed(ans)) 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
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)
def test_parameter_value(): t = Symbol('t') e = Ellipse(Point(0, 0), 3, 5) assert e.parameter_value((3, 0), t) == {t: 0} raises(ValueError, lambda: e.parameter_value((4, 0), t))
def test_Geometry(): sT(Point(0, 0), "Point(Integer(0), Integer(0))") sT(Ellipse(Point(0, 0), 5, 1), "Ellipse(Point(Integer(0), Integer(0)), Integer(5), Integer(1))")
def test_Geometry(): sT(Point(0, 0), "Point(Zero, Zero)") sT(Ellipse(Point(0, 0), 5, 1), "Ellipse(Point(Zero, Zero), Integer(5), One)")