Пример #1
0
    def reflect(self, prev_position: Point, circle: Circle) -> None:
        """
        Reflects the photon on the given circle.

         - calculates distance covered and subtracts it from self.distance
         - calculates new direction vector
         - jumps to the new position and prints it

        :rtype: None
        """
        if prev_position.distance(self.position) > self.distance:
            self.position = prev_position.translate(
                self.ray.direction.x * self.distance,
                self.ray.direction.y * self.distance)
            self.distance = 0
            return

        self.distance -= prev_position.distance(self.position).evalf(PRECISION)
        self.checked_circles = [circle]

        tangent = circle.tangent_lines(self.position)[0]
        self.position = Point(self.position.x.evalf(PRECISION),
                              self.position.y.evalf(PRECISION))
        self.imprecise_position = self.position

        direction = self.position + self.ray.reflect(tangent).direction
        direction = Point(direction.x.evalf(PRECISION),
                          direction.y.evalf(PRECISION))

        self.ray = Ray(self.position, direction)

        self.jump_to_position()
        dot()
        self.print_position()
Пример #2
0
def test_point():
    p1 = Point(x1, x2)
    p2 = Point(y1, y2)
    p3 = Point(0, 0)
    p4 = Point(1, 1)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point(y1 - x1, y2 - x2)
    assert p4 * 5 == Point(5, 5)
    assert -p2 == Point(-y1, -y2)

    assert Point.midpoint(p3, p4) == Point(half, half)
    assert Point.midpoint(p1, p4) == Point(half + half * x1, half + half * x2)
    assert Point.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point.distance(p3, p4) == sqrt(2)
    assert Point.distance(p1, p1) == 0
    assert Point.distance(p3, p2) == sqrt(p2.x ** 2 + p2.y ** 2)

    p1_1 = Point(x1, x1)
    p1_2 = Point(y2, y2)
    p1_3 = Point(x1 + 1, x1)
    assert Point.is_collinear(p3)
    assert Point.is_collinear(p3, p4)
    assert Point.is_collinear(p3, p4, p1_1, p1_2)
    assert Point.is_collinear(p3, p4, p1_1, p1_3) == False

    assert p3.intersection(Point(0, 0)) == [p3]
    assert p3.intersection(p4) == []

    x_pos = Symbol("x", real=True, positive=True)
    p2_1 = Point(x_pos, 0)
    p2_2 = Point(0, x_pos)
    p2_3 = Point(-x_pos, 0)
    p2_4 = Point(0, -x_pos)
    p2_5 = Point(x_pos, 5)
    assert Point.is_concyclic(p2_1)
    assert Point.is_concyclic(p2_1, p2_2)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_5) == False
    assert Point.is_concyclic(p4, p4 * 2, p4 * 3) == False

    assert p4.scale(2, 3) == Point(2, 3)
    assert p3.scale(2, 3) == p3

    assert p4.rotate(pi, Point(0.5, 0.5)) == p3
    assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
    assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)

    assert p4 * 5 == Point(5, 5)
    assert p4 / 5 == Point(0.2, 0.2)

    raises(ValueError, lambda: Point(0, 0) + 10)

    # Point differences should be simplified
    assert Point(x * (x - 1), y) - Point(x ** 2 - x, y + 1) == Point(0, -1)

    a, b = Rational(1, 2), Rational(1, 3)
    assert Point(a, b).evalf(2) == Point(a.n(2), b.n(2))
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point(1, 0)
    assert p.rotate(pi / 2) == Point(0, 1)
    assert p.rotate(pi / 2, p) == p
    p = Point(1, 1)
    assert p.scale(2, 3) == Point(2, 3)
    assert p.translate(1, 2) == Point(2, 3)
    assert p.translate(1) == Point(2, 1)
    assert p.translate(y=1) == Point(1, 2)
    assert p.translate(*p.args) == Point(2, 2)
Пример #3
0
def test_point():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    x1 = Symbol('x1', real=True)
    x2 = Symbol('x2', real=True)
    y1 = Symbol('y1', real=True)
    y2 = Symbol('y2', real=True)
    half = S.Half
    p1 = Point(x1, x2)
    p2 = Point(y1, y2)
    p3 = Point(0, 0)
    p4 = Point(1, 1)
    p5 = Point(0, 1)
    line = Line(Point(1, 0), slope=1)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point(y1 - x1, y2 - x2)
    assert -p2 == Point(-y1, -y2)
    raises(ValueError, lambda: Point(3, I))
    raises(ValueError, lambda: Point(2 * I, I))
    raises(ValueError, lambda: Point(3 + I, I))

    assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
    assert Point.midpoint(p3, p4) == Point(half, half)
    assert Point.midpoint(p1, p4) == Point(half + half * x1, half + half * x2)
    assert Point.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point.distance(p3, p4) == sqrt(2)
    assert Point.distance(p1, p1) == 0
    assert Point.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2)

    # distance should be symmetric
    assert p1.distance(line) == line.distance(p1)
    assert p4.distance(line) == line.distance(p4)

    assert Point.taxicab_distance(p4, p3) == 2

    assert Point.canberra_distance(p4, p5) == 1

    p1_1 = Point(x1, x1)
    p1_2 = Point(y2, y2)
    p1_3 = Point(x1 + 1, x1)
    assert Point.is_collinear(p3)

    with warns(UserWarning):
        assert Point.is_collinear(p3, Point(p3, dim=4))
    assert p3.is_collinear()
    assert Point.is_collinear(p3, p4)
    assert Point.is_collinear(p3, p4, p1_1, p1_2)
    assert Point.is_collinear(p3, p4, p1_1, p1_3) is False
    assert Point.is_collinear(p3, p3, p4, p5) is False

    raises(TypeError, lambda: Point.is_collinear(line))
    raises(TypeError, lambda: p1_1.is_collinear(line))

    assert p3.intersection(Point(0, 0)) == [p3]
    assert p3.intersection(p4) == []

    x_pos = Symbol('x', real=True, positive=True)
    p2_1 = Point(x_pos, 0)
    p2_2 = Point(0, x_pos)
    p2_3 = Point(-x_pos, 0)
    p2_4 = Point(0, -x_pos)
    p2_5 = Point(x_pos, 5)
    assert Point.is_concyclic(p2_1)
    assert Point.is_concyclic(p2_1, p2_2)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
    for pts in permutations((p2_1, p2_2, p2_3, p2_5)):
        assert Point.is_concyclic(*pts) is False
    assert Point.is_concyclic(p4, p4 * 2, p4 * 3) is False
    assert Point(0, 0).is_concyclic((1, 1), (2, 2), (2, 1)) is False

    assert p4.scale(2, 3) == Point(2, 3)
    assert p3.scale(2, 3) == p3

    assert p4.rotate(pi, Point(0.5, 0.5)) == p3
    assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
    assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)

    assert p4 * 5 == Point(5, 5)
    assert p4 / 5 == Point(0.2, 0.2)
    assert 5 * p4 == Point(5, 5)

    raises(ValueError, lambda: Point(0, 0) + 10)

    # Point differences should be simplified
    assert Point(x * (x - 1), y) - Point(x**2 - x, y + 1) == Point(0, -1)

    a, b = S.Half, Rational(1, 3)
    assert Point(a, b).evalf(2) == \
        Point(a.n(2), b.n(2), evaluate=False)
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point(1, 0)
    assert p.rotate(pi / 2) == Point(0, 1)
    assert p.rotate(pi / 2, p) == p
    p = Point(1, 1)
    assert p.scale(2, 3) == Point(2, 3)
    assert p.translate(1, 2) == Point(2, 3)
    assert p.translate(1) == Point(2, 1)
    assert p.translate(y=1) == Point(1, 2)
    assert p.translate(*p.args) == Point(2, 2)

    # Check invalid input for transform
    raises(ValueError, lambda: p3.transform(p3))
    raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]])))
Пример #4
0
def test_point():
    p1 = Point(x1, x2)
    p2 = Point(y1, y2)
    p3 = Point(0, 0)
    p4 = Point(1, 1)
    p5 = Point(0, 1)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point(y1 - x1, y2 - x2)
    assert p4 * 5 == Point(5, 5)
    assert -p2 == Point(-y1, -y2)

    assert Point.midpoint(p3, p4) == Point(half, half)
    assert Point.midpoint(p1, p4) == Point(half + half * x1, half + half * x2)
    assert Point.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point.distance(p3, p4) == sqrt(2)
    assert Point.distance(p1, p1) == 0
    assert Point.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2)

    p1_1 = Point(x1, x1)
    p1_2 = Point(y2, y2)
    p1_3 = Point(x1 + 1, x1)
    assert Point.is_collinear(p3)
    assert Point.is_collinear(p3, p4)
    assert Point.is_collinear(p3, p4, p1_1, p1_2)
    assert Point.is_collinear(p3, p4, p1_1, p1_3) is False
    assert Point.is_collinear(p3, p3, p4, p5) is False

    assert p3.intersection(Point(0, 0)) == [p3]
    assert p3.intersection(p4) == []

    x_pos = Symbol('x', real=True, positive=True)
    p2_1 = Point(x_pos, 0)
    p2_2 = Point(0, x_pos)
    p2_3 = Point(-x_pos, 0)
    p2_4 = Point(0, -x_pos)
    p2_5 = Point(x_pos, 5)
    assert Point.is_concyclic(p2_1)
    assert Point.is_concyclic(p2_1, p2_2)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_5) is False
    assert Point.is_concyclic(p4, p4 * 2, p4 * 3) is False

    assert p4.scale(2, 3) == Point(2, 3)
    assert p3.scale(2, 3) == p3

    assert p4.rotate(pi, Point(0.5, 0.5)) == p3
    assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
    assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)

    assert p4 * 5 == Point(5, 5)
    assert p4 / 5 == Point(0.2, 0.2)

    raises(ValueError, lambda: Point(0, 0) + 10)

    # Point differences should be simplified
    assert Point(x * (x - 1), y) - Point(x**2 - x, y + 1) == Point(0, -1)

    a, b = Rational(1, 2), Rational(1, 3)
    assert Point(a, b).evalf(2) == \
        Point(a.n(2), b.n(2))
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point(1, 0)
    assert p.rotate(pi / 2) == Point(0, 1)
    assert p.rotate(pi / 2, p) == p
    p = Point(1, 1)
    assert p.scale(2, 3) == Point(2, 3)
    assert p.translate(1, 2) == Point(2, 3)
    assert p.translate(1) == Point(2, 1)
    assert p.translate(y=1) == Point(1, 2)
    assert p.translate(*p.args) == Point(2, 2)
Пример #5
0
class Photon:
    position: Point
    imprecise_position: Point
    ray: Ray
    checked_circles: List[Circle]
    distance: Rational

    def __init__(self, position: Point, angle: float,
                 distance: Rational) -> None:
        """
        Creates a new photon with the given current position, angle and distance to cover.

        :param position: starting position of the photon
        :param angle: starting angle of the photon
        :param distance: distance to cover
        :rtype: None
        """

        # Current position which one changes when the src hits a circle.
        self.position = position

        # Imprecise current position used for ray tracing.
        self.imprecise_position = Point(position, evaluate=False)

        # Current heading of the src.
        self.ray = Ray(position, angle=angle)

        # Circles we have already checked before, after the last reflection, so we don't have to do it again.
        self.checked_circles = []

        # Distance left to cover.
        self.distance = distance

    def jump_to_position(self) -> None:
        """
        Draws a line from the current turtle position to self.position.
        Sets current turtle position to self.position.

        :rtype: None
        """

        set_position(float(self.position.x), float(self.position.y))

    def invisible_jump_to_position(self) -> None:
        """
        Sets current turtle position to self.position without drawing a line.

        :rtype: None
        """

        inv_set_position(float(self.position.x), float(self.position.y))

    def raytrace(self) -> None:
        """
        Moves imprecise_position STEP distance in the direction of the photon.

        :rtype: None
        """

        self.imprecise_position = self.imprecise_position.translate(
            (STEP * self.ray.direction.x).evalf(PRECISION),
            (STEP * self.ray.direction.y).evalf(PRECISION))

    def check_nearby_circles(self) -> None:
        """
        Checks the circles around the photon to see if it has collided with any of them.

        If it has, it sets the new position and calls self.reflect with the previous position and the collided circle.
        Otherwise, it adds the circle to checked_circles.

        :rtype: None
        """

        for (x, y) in self.possible_centers():
            center = Point(x, y)
            circle = Circle(center, RADIUS)

            # If we already checked that circle after the last reflection.
            if circle in self.checked_circles:
                continue

            intersect = intersection(self.ray, circle)

            prev_position = self.position

            # If intersection returns one Point, set position to it.
            if len(intersect) == 1:
                self.position = intersect[0]

                self.reflect(prev_position, circle)
                break

            # If intersection returns two Points, find the closer one and set position to it.
            elif len(intersect) == 2:
                if (self.position.distance(intersect[0]).evalf(PRECISION)) < \
                        (self.position.distance(intersect[1]).evalf(PRECISION)):
                    self.position = intersect[0]
                else:
                    self.position = intersect[1]

                self.reflect(prev_position, circle)
                break

            # Otherwise we have not collided with that circle, so we add it to the circles list.
            else:
                self.checked_circles.append(circle)

    def reflect(self, prev_position: Point, circle: Circle) -> None:
        """
        Reflects the photon on the given circle.

         - calculates distance covered and subtracts it from self.distance
         - calculates new direction vector
         - jumps to the new position and prints it

        :rtype: None
        """
        if prev_position.distance(self.position) > self.distance:
            self.position = prev_position.translate(
                self.ray.direction.x * self.distance,
                self.ray.direction.y * self.distance)
            self.distance = 0
            return

        self.distance -= prev_position.distance(self.position).evalf(PRECISION)
        self.checked_circles = [circle]

        tangent = circle.tangent_lines(self.position)[0]
        self.position = Point(self.position.x.evalf(PRECISION),
                              self.position.y.evalf(PRECISION))
        self.imprecise_position = self.position

        direction = self.position + self.ray.reflect(tangent).direction
        direction = Point(direction.x.evalf(PRECISION),
                          direction.y.evalf(PRECISION))

        self.ray = Ray(self.position, direction)

        self.jump_to_position()
        dot()
        self.print_position()

    def possible_centers(self) -> Generator[Tuple[float, float], None, None]:
        """
        Returns the possible centers of the circles around the photon as a generator.

        :return: a generator for the possible circle centers
        :rtype: Generator[Tuple[float, float]]
        """

        # The only circles we have to check are the ones with these center coordinates.
        possible_center_x = [
            floor(self.imprecise_position.x),
            ceil(self.imprecise_position.x)
        ]

        possible_center_y = [
            floor(self.imprecise_position.y),
            ceil(self.imprecise_position.y)
        ]

        for center_x in possible_center_x:
            for center_y in possible_center_y:
                yield (center_x, center_y)

    def print_position(self) -> None:
        """
        Print coordinates of the current position.

        :rtype: None
        """

        x = self.position.x.evalf(PRECISION)
        y = self.position.y.evalf(PRECISION)

        print(f'{x: 1.15f} {y: 1.15f}')
Пример #6
0
def test_point():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    x1 = Symbol('x1', real=True)
    x2 = Symbol('x2', real=True)
    y1 = Symbol('y1', real=True)
    y2 = Symbol('y2', real=True)
    half = Rational(1, 2)
    p1 = Point(x1, x2)
    p2 = Point(y1, y2)
    p3 = Point(0, 0)
    p4 = Point(1, 1)
    p5 = Point(0, 1)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point(y1 - x1, y2 - x2)
    assert p4*5 == Point(5, 5)
    assert -p2 == Point(-y1, -y2)
    raises(ValueError, lambda: Point(3, I))
    raises(ValueError, lambda: Point(2*I, I))
    raises(ValueError, lambda: Point(3 + I, I))

    assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
    assert Point.midpoint(p3, p4) == Point(half, half)
    assert Point.midpoint(p1, p4) == Point(half + half*x1, half + half*x2)
    assert Point.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2

    assert Point.distance(p3, p4) == sqrt(2)
    assert Point.distance(p1, p1) == 0
    assert Point.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2)

    assert Point.taxicab_distance(p4, p3) == 2

    p1_1 = Point(x1, x1)
    p1_2 = Point(y2, y2)
    p1_3 = Point(x1 + 1, x1)
    assert Point.is_collinear(p3)
    assert Point.is_collinear(p3, p4)
    assert Point.is_collinear(p3, p4, p1_1, p1_2)
    assert Point.is_collinear(p3, p4, p1_1, p1_3) is False
    assert Point.is_collinear(p3, p3, p4, p5) is False
    line = Line(Point(1,0), slope = 1)
    raises(TypeError, lambda: Point.is_collinear(line))
    raises(TypeError, lambda: p1_1.is_collinear(line))

    assert p3.intersection(Point(0, 0)) == [p3]
    assert p3.intersection(p4) == []

    assert p1.dot(p4) == x1 + x2
    assert p3.dot(p4) == 0
    assert p4.dot(p5) == 1

    x_pos = Symbol('x', real=True, positive=True)
    p2_1 = Point(x_pos, 0)
    p2_2 = Point(0, x_pos)
    p2_3 = Point(-x_pos, 0)
    p2_4 = Point(0, -x_pos)
    p2_5 = Point(x_pos, 5)
    assert Point.is_concyclic(p2_1)
    assert Point.is_concyclic(p2_1, p2_2)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_5) is False
    assert Point.is_concyclic(p4, p4 * 2, p4 * 3) is False

    assert p4.scale(2, 3) == Point(2, 3)
    assert p3.scale(2, 3) == p3

    assert p4.rotate(pi, Point(0.5, 0.5)) == p3
    assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
    assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)

    assert p4 * 5 == Point(5, 5)
    assert p4 / 5 == Point(0.2, 0.2)

    raises(ValueError, lambda: Point(0, 0) + 10)

    # Point differences should be simplified
    assert Point(x*(x - 1), y) - Point(x**2 - x, y + 1) == Point(0, -1)

    a, b = Rational(1, 2), Rational(1, 3)
    assert Point(a, b).evalf(2) == \
        Point(a.n(2), b.n(2))
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test transformations
    p = Point(1, 0)
    assert p.rotate(pi/2) == Point(0, 1)
    assert p.rotate(pi/2, p) == p
    p = Point(1, 1)
    assert p.scale(2, 3) == Point(2, 3)
    assert p.translate(1, 2) == Point(2, 3)
    assert p.translate(1) == Point(2, 1)
    assert p.translate(y=1) == Point(1, 2)
    assert p.translate(*p.args) == Point(2, 2)

    # Check invalid input for transform
    raises(ValueError, lambda: p3.transform(p3))
    raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]])))
Пример #7
0
def test_point():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    x1 = Symbol('x1', real=True)
    x2 = Symbol('x2', real=True)
    y1 = Symbol('y1', real=True)
    y2 = Symbol('y2', real=True)
    half = S.Half
    p1 = Point(x1, x2)
    p2 = Point(y1, y2)
    p3 = Point(0, 0)
    p4 = Point(1, 1)
    p5 = Point(0, 1)
    line = Line(Point(1, 0), slope=1)

    assert p1 in p1
    assert p1 not in p2
    assert p2.y == y2
    assert (p3 + p4) == p4
    assert (p2 - p1) == Point(y1 - x1, y2 - x2)
    assert -p2 == Point(-y1, -y2)
    raises(TypeError, lambda: Point(1))
    raises(ValueError, lambda: Point([1]))
    raises(ValueError, lambda: Point(3, I))
    raises(ValueError, lambda: Point(2*I, I))
    raises(ValueError, lambda: Point(3 + I, I))

    assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
    assert Point.midpoint(p3, p4) == Point(half, half)
    assert Point.midpoint(p1, p4) == Point(half + half*x1, half + half*x2)
    assert Point.midpoint(p2, p2) == p2
    assert p2.midpoint(p2) == p2
    assert p1.origin == Point(0, 0)

    assert Point.distance(p3, p4) == sqrt(2)
    assert Point.distance(p1, p1) == 0
    assert Point.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2)
    raises(TypeError, lambda: Point.distance(p1, 0))
    raises(TypeError, lambda: Point.distance(p1, GeometryEntity()))

    # distance should be symmetric
    assert p1.distance(line) == line.distance(p1)
    assert p4.distance(line) == line.distance(p4)

    assert Point.taxicab_distance(p4, p3) == 2

    assert Point.canberra_distance(p4, p5) == 1
    raises(ValueError, lambda: Point.canberra_distance(p3, p3))

    p1_1 = Point(x1, x1)
    p1_2 = Point(y2, y2)
    p1_3 = Point(x1 + 1, x1)
    assert Point.is_collinear(p3)

    with warns(UserWarning, test_stacklevel=False):
        assert Point.is_collinear(p3, Point(p3, dim=4))
    assert p3.is_collinear()
    assert Point.is_collinear(p3, p4)
    assert Point.is_collinear(p3, p4, p1_1, p1_2)
    assert Point.is_collinear(p3, p4, p1_1, p1_3) is False
    assert Point.is_collinear(p3, p3, p4, p5) is False

    raises(TypeError, lambda: Point.is_collinear(line))
    raises(TypeError, lambda: p1_1.is_collinear(line))

    assert p3.intersection(Point(0, 0)) == [p3]
    assert p3.intersection(p4) == []
    assert p3.intersection(line) == []
    with warns(UserWarning, test_stacklevel=False):
        assert Point.intersection(Point(0, 0, 0), Point(0, 0)) == [Point(0, 0, 0)]

    x_pos = Symbol('x', positive=True)
    p2_1 = Point(x_pos, 0)
    p2_2 = Point(0, x_pos)
    p2_3 = Point(-x_pos, 0)
    p2_4 = Point(0, -x_pos)
    p2_5 = Point(x_pos, 5)
    assert Point.is_concyclic(p2_1)
    assert Point.is_concyclic(p2_1, p2_2)
    assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
    for pts in permutations((p2_1, p2_2, p2_3, p2_5)):
        assert Point.is_concyclic(*pts) is False
    assert Point.is_concyclic(p4, p4 * 2, p4 * 3) is False
    assert Point(0, 0).is_concyclic((1, 1), (2, 2), (2, 1)) is False
    assert Point.is_concyclic(Point(0, 0, 0, 0), Point(1, 0, 0, 0), Point(1, 1, 0, 0), Point(1, 1, 1, 0)) is False

    assert p1.is_scalar_multiple(p1)
    assert p1.is_scalar_multiple(2*p1)
    assert not p1.is_scalar_multiple(p2)
    assert Point.is_scalar_multiple(Point(1, 1), (-1, -1))
    assert Point.is_scalar_multiple(Point(0, 0), (0, -1))
    # test when is_scalar_multiple can't be determined
    raises(Undecidable, lambda: Point.is_scalar_multiple(Point(sympify("x1%y1"), sympify("x2%y2")), Point(0, 1)))

    assert Point(0, 1).orthogonal_direction == Point(1, 0)
    assert Point(1, 0).orthogonal_direction == Point(0, 1)

    assert p1.is_zero is None
    assert p3.is_zero
    assert p4.is_zero is False
    assert p1.is_nonzero is None
    assert p3.is_nonzero is False
    assert p4.is_nonzero

    assert p4.scale(2, 3) == Point(2, 3)
    assert p3.scale(2, 3) == p3

    assert p4.rotate(pi, Point(0.5, 0.5)) == p3
    assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
    assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)

    assert p4 * 5 == Point(5, 5)
    assert p4 / 5 == Point(0.2, 0.2)
    assert 5 * p4 == Point(5, 5)

    raises(ValueError, lambda: Point(0, 0) + 10)

    # Point differences should be simplified
    assert Point(x*(x - 1), y) - Point(x**2 - x, y + 1) == Point(0, -1)

    a, b = S.Half, Rational(1, 3)
    assert Point(a, b).evalf(2) == \
        Point(a.n(2), b.n(2), evaluate=False)
    raises(ValueError, lambda: Point(1, 2) + 1)

    # test project
    assert Point.project((0, 1), (1, 0)) == Point(0, 0)
    assert Point.project((1, 1), (1, 0)) == Point(1, 0)
    raises(ValueError, lambda: Point.project(p1, Point(0, 0)))

    # test transformations
    p = Point(1, 0)
    assert p.rotate(pi/2) == Point(0, 1)
    assert p.rotate(pi/2, p) == p
    p = Point(1, 1)
    assert p.scale(2, 3) == Point(2, 3)
    assert p.translate(1, 2) == Point(2, 3)
    assert p.translate(1) == Point(2, 1)
    assert p.translate(y=1) == Point(1, 2)
    assert p.translate(*p.args) == Point(2, 2)

    # Check invalid input for transform
    raises(ValueError, lambda: p3.transform(p3))
    raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]])))

    # test __contains__
    assert 0 in Point(0, 0, 0, 0)
    assert 1 not in Point(0, 0, 0, 0)

    # test affine_rank
    assert Point.affine_rank() == -1