def test_point3D():
p1 = Point3D(x1, x2, x3)
p2 = Point3D(y1, y2, y3)
p3 = Point3D(0, 0, 0)
p4 = Point3D(1, 1, 1)
p5 = Point3D(0, 1, 2)
assert p1 in p1
assert p1 not in p2
assert p2.y == y2
assert (p3 + p4) == p4
assert (p2 - p1) == Point3D(y1 - x1, y2 - x2, y3 - x3)
assert p4*5 == Point3D(5, 5, 5)
assert -p2 == Point3D(-y1, -y2, -y3)
assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
assert Point3D.midpoint(p3, p4) == Point3D(half, half, half)
assert Point3D.midpoint(p1, p4) == Point3D(half + half*x1, half + half*x2,
half + half*x3)
assert Point3D.midpoint(p2, p2) == p2
assert p2.midpoint(p2) == p2
assert Point3D.distance(p3, p4) == sqrt(3)
assert Point3D.distance(p1, p1) == 0
assert Point3D.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2 + p2.z**2)
p1_1 = Point3D(x1, x1, x1)
p1_2 = Point3D(y2, y2, y2)
p1_3 = Point3D(x1 + 1, x1, x1)
# according to the description in the docs, points are collinear
# if they like on a single line. Thus a single point should always
# be collinear
assert Point3D.are_collinear(p3)
assert Point3D.are_collinear(p3, p4)
assert Point3D.are_collinear(p3, p4, p1_1, p1_2)
assert Point3D.are_collinear(p3, p4, p1_1, p1_3) is False
assert Point3D.are_collinear(p3, p3, p4, p5) is False
assert p3.intersection(Point3D(0, 0, 0)) == [p3]
assert p3.intersection(p4) == []
assert p4 * 5 == Point3D(5, 5, 5)
assert p4 / 5 == Point3D(0.2, 0.2, 0.2)
pytest.raises(ValueError, lambda: Point3D(0, 0, 0) + 10)
# Point differences should be simplified
assert Point3D(x*(x - 1), y, 2) - Point3D(x**2 - x, y + 1, 1) == \
Point3D(0, -1, 1)
a, b = Rational(1, 2), Rational(1, 3)
assert Point(a, b).evalf(2) == \
Point(a.evalf(2), b.evalf(2))
pytest.raises(ValueError, lambda: Point(1, 2) + 1)
# test transformations
p = Point3D(1, 1, 1)
assert p.scale(2, 3) == Point3D(2, 3, 1)
assert p.translate(1, 2) == Point3D(2, 3, 1)
assert p.translate(1) == Point3D(2, 1, 1)
assert p.translate(z=1) == Point3D(1, 1, 2)
assert p.translate(*p.args) == Point3D(2, 2, 2)
# Test __new__
assert Point3D(Point3D(1, 2, 3), 4, 5, evaluate=False) == Point3D(1, 2, 3)
# Test length property returns correctly
assert p.length == 0
assert p1_1.length == 0
assert p1_2.length == 0
# Test are_colinear type error
pytest.raises(TypeError, lambda: Point3D.are_collinear(p, x))
# Test are_coplanar
planar2 = Point3D(1, -1, 1)
planar3 = Point3D(-1, 1, 1)
assert Point3D.are_coplanar(p, planar2, planar3) is True
assert Point3D.are_coplanar(p, planar2, planar3, p3) is False
pytest.raises(ValueError, lambda: Point3D.are_coplanar(p, planar2))
planar2 = Point3D(1, 1, 2)
planar3 = Point3D(1, 1, 3)
pytest.raises(ValueError, lambda: Point3D.are_coplanar(p, planar2, planar3))
# Test Intersection
assert planar2.intersection(Line3D(p, planar3)) == [Point3D(1, 1, 2)]
# Test Scale
assert planar2.scale(1, 1, 1) == planar2
assert planar2.scale(2, 2, 2, planar3) == Point3D(1, 1, 1)
assert planar2.scale(1, 1, 1, p3) == planar2
# Test Transform
identity = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])
assert p.transform(identity) == p
trans = Matrix([[1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 0, 0, 1]])
assert p.transform(trans) == Point3D(2, 2, 2)
pytest.raises(ValueError, lambda: p.transform(p))
pytest.raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]])))
# Test Equals
assert p.equals(x1) is False
# Test __sub__
p_2d = Point(0, 0)
pytest.raises(ValueError, lambda: (p - p_2d))
def test_direction_cosine():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
assert p1.direction_cosine(Point3D(1, 0, 0)) == [1, 0, 0]
assert p1.direction_cosine(Point3D(0, 1, 0)) == [0, 1, 0]
assert p1.direction_cosine(Point3D(0, 0, pi)) == [0, 0, 1]
assert p1.direction_cosine(Point3D(5, 0, 0)) == [1, 0, 0]
assert p1.direction_cosine(Point3D(0, sqrt(3), 0)) == [0, 1, 0]
assert p1.direction_cosine(Point3D(0, 0, 5)) == [0, 0, 1]
assert p1.direction_cosine(Point3D(2.4, 2.4,
0)) == [sqrt(2) / 2,
sqrt(2) / 2, 0]
assert p1.direction_cosine(Point3D(
1, 1, 1)) == [sqrt(3) / 3, sqrt(3) / 3,
sqrt(3) / 3]
assert p1.direction_cosine(Point3D(
-12, 0, -15)) == [-4 * sqrt(41) / 41, 0, -5 * sqrt(41) / 41]
assert p2.direction_cosine(Point3D(
0, 0, 0)) == [-sqrt(3) / 3, -sqrt(3) / 3, -sqrt(3) / 3]
assert p2.direction_cosine(Point3D(1, 1, 12)) == [0, 0, 1]
assert p2.direction_cosine(Point3D(12, 1,
12)) == [sqrt(2) / 2, 0,
sqrt(2) / 2]
def test_sympyissue_9214():
p1 = Point3D(4, -2, 6)
p2 = Point3D(1, 2, 3)
p3 = Point3D(7, 2, 3)
assert Point3D.are_collinear(p1, p2, p3) is False
def test_point3D():
p1 = Point3D(x1, x2, x3)
p2 = Point3D(y1, y2, y3)
p3 = Point3D(0, 0, 0)
p4 = Point3D(1, 1, 1)
p5 = Point3D(0, 1, 2)
assert p1 in p1
assert p1 not in p2
assert p2.y == y2
assert (p3 + p4) == p4
assert (p2 - p1) == Point3D(y1 - x1, y2 - x2, y3 - x3)
assert p4 * 5 == Point3D(5, 5, 5)
assert -p2 == Point3D(-y1, -y2, -y3)
assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
assert Point3D.midpoint(p3, p4) == Point3D(half, half, half)
assert Point3D.midpoint(p1,
p4) == Point3D(half + half * x1, half + half * x2,
half + half * x3)
assert Point3D.midpoint(p2, p2) == p2
assert p2.midpoint(p2) == p2
assert Point3D.distance(p3, p4) == sqrt(3)
assert Point3D.distance(p1, p1) == 0
assert Point3D.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2 + p2.z**2)
p1_1 = Point3D(x1, x1, x1)
p1_2 = Point3D(y2, y2, y2)
p1_3 = Point3D(x1 + 1, x1, x1)
# according to the description in the docs, points are collinear
# if they like on a single line. Thus a single point should always
# be collinear
assert Point3D.are_collinear(p3)
assert Point3D.are_collinear(p3, p4)
assert Point3D.are_collinear(p3, p4, p1_1, p1_2)
assert Point3D.are_collinear(p3, p4, p1_1, p1_3) is False
assert Point3D.are_collinear(p3, p3, p4, p5) is False
assert p3.intersection(Point3D(0, 0, 0)) == [p3]
assert p3.intersection(p4) == []
assert p4 * 5 == Point3D(5, 5, 5)
assert p4 / 5 == Point3D(0.2, 0.2, 0.2)
pytest.raises(ValueError, lambda: Point3D(0, 0, 0) + 10)
# Point differences should be simplified
assert Point3D(x*(x - 1), y, 2) - Point3D(x**2 - x, y + 1, 1) == \
Point3D(0, -1, 1)
a, b = Rational(1, 2), Rational(1, 3)
assert Point(a, b).evalf(2) == \
Point(a.evalf(2), b.evalf(2))
pytest.raises(ValueError, lambda: Point(1, 2) + 1)
# test transformations
p = Point3D(1, 1, 1)
assert p.scale(2, 3) == Point3D(2, 3, 1)
assert p.translate(1, 2) == Point3D(2, 3, 1)
assert p.translate(1) == Point3D(2, 1, 1)
assert p.translate(z=1) == Point3D(1, 1, 2)
assert p.translate(*p.args) == Point3D(2, 2, 2)
# Test __new__
assert Point3D(Point3D(1, 2, 3), 4, 5, evaluate=False) == Point3D(1, 2, 3)
# Test length property returns correctly
assert p.length == 0
assert p1_1.length == 0
assert p1_2.length == 0
# Test are_colinear type error
pytest.raises(TypeError, lambda: Point3D.are_collinear(p, x))
# Test are_coplanar
planar2 = Point3D(1, -1, 1)
planar3 = Point3D(-1, 1, 1)
assert Point3D.are_coplanar(p, planar2, planar3) is True
assert Point3D.are_coplanar(p, planar2, planar3, p3) is False
pytest.raises(ValueError, lambda: Point3D.are_coplanar(p, planar2))
planar2 = Point3D(1, 1, 2)
planar3 = Point3D(1, 1, 3)
pytest.raises(ValueError,
lambda: Point3D.are_coplanar(p, planar2, planar3))
# Test Intersection
assert planar2.intersection(Line3D(p, planar3)) == [Point3D(1, 1, 2)]
# Test Scale
assert planar2.scale(1, 1, 1) == planar2
assert planar2.scale(2, 2, 2, planar3) == Point3D(1, 1, 1)
assert planar2.scale(1, 1, 1, p3) == planar2
# Test Transform
identity = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])
assert p.transform(identity) == p
trans = Matrix([[1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 0, 0, 1]])
assert p.transform(trans) == Point3D(2, 2, 2)
pytest.raises(ValueError, lambda: p.transform(p))
pytest.raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]])))
# Test Equals
assert p.equals(x1) is False
# Test __sub__
p_2d = Point(0, 0)
pytest.raises(ValueError, lambda: (p - p_2d))
def test_plane():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(1, 2, 3)
p4 = Point3D(x, x, x)
p5 = Point3D(y, y, y)
pl3 = Plane(p1, p2, p3)
pl4 = Plane(p1, normal_vector=(1, 1, 1))
pl4b = Plane(p1, p2)
pl5 = Plane(p3, normal_vector=(1, 2, 3))
pl6 = Plane(Point3D(2, 3, 7), normal_vector=(2, 2, 2))
pl7 = Plane(Point3D(1, -5, -6), normal_vector=(1, -2, 1))
l1 = Line3D(Point3D(5, 0, 0), Point3D(1, -1, 1))
l2 = Line3D(Point3D(0, -2, 0), Point3D(3, 1, 1))
l3 = Line3D(Point3D(0, -1, 0), Point3D(5, -1, 9))
assert Plane(p1, p2, p3) != Plane(p1, p3, p2)
assert Plane(p1, p2, p3).is_coplanar(Plane(p1, p3, p2))
assert pl3 == Plane(Point3D(0, 0, 0), normal_vector=(1, -2, 1))
assert pl3 != pl4
assert pl4 == pl4b
assert pl5 == Plane(Point3D(1, 2, 3), normal_vector=(1, 2, 3))
assert pl5.equation(x, y, z) == x + 2 * y + 3 * z - 14
assert pl3.equation(x, y, z) == x - 2 * y + z
assert pl3.p1 == p1
assert pl4.p1 == p1
assert pl5.p1 == p3
assert pl4.normal_vector == (1, 1, 1)
assert pl5.normal_vector == (1, 2, 3)
assert p1 in pl3
assert p1 in pl4
assert p3 in pl5
assert pl3.projection(Point(0, 0)) == p1
p = pl3.projection(Point3D(1, 1, 0))
assert p == Point3D(7 / 6, 2 / 3, 1 / 6)
assert p in pl3
l = pl3.projection_line(Line(Point(0, 0), Point(1, 1)))
assert l == Line3D(Point3D(0, 0, 0), Point3D(7 / 6, 2 / 3, 1 / 6))
assert l in pl3
# get a segment that does not intersect the plane which is also
# parallel to pl3's normal veector
t = Dummy()
r = pl3.random_point()
a = pl3.perpendicular_line(r).arbitrary_point(t)
s = Segment3D(a.subs(t, 1), a.subs(t, 2))
assert s.p1 not in pl3 and s.p2 not in pl3
assert pl3.projection_line(s).equals(r)
assert pl3.projection_line(Segment(Point(1, 0), Point(1, 1))) == \
Segment3D(Point3D(5/6, 1/3, -1/6), Point3D(7/6, 2/3, 1/6))
assert pl6.projection_line(Ray(Point(1, 0), Point(1, 1))) == \
Ray3D(Point3D(14/3, 11/3, 11/3), Point3D(13/3, 13/3, 10/3))
assert pl3.perpendicular_line(r.args) == pl3.perpendicular_line(r)
assert pl3.is_parallel(pl6) is False
assert pl4.is_parallel(pl6)
assert pl6.is_parallel(l1) is False
assert pl3.is_perpendicular(pl6)
assert pl4.is_perpendicular(pl7)
assert pl6.is_perpendicular(pl7)
assert pl6.is_perpendicular(l1) is False
assert pl7.distance(Point3D(1, 3, 5)) == 5 * sqrt(6) / 6
assert pl6.distance(Point3D(0, 0, 0)) == 4 * sqrt(3)
assert pl6.distance(pl6.p1) == 0
assert pl7.distance(pl6) == 0
assert pl7.distance(l1) == 0
assert pl6.distance(Segment3D(Point3D(2, 3, 1), Point3D(1, 3, 4))) == 0
pl6.distance(Plane(Point3D(5, 5, 5), normal_vector=(8, 8, 8))) == sqrt(3)
assert pl6.angle_between(pl3) == pi / 2
assert pl6.angle_between(pl6) == 0
assert pl6.angle_between(pl4) == 0
assert pl7.angle_between(Line3D(Point3D(2, 3, 5), Point3D(2, 4, 6))) == \
-asin(sqrt(3)/6)
assert pl6.angle_between(Ray3D(Point3D(2, 4, 1), Point3D(6, 5, 3))) == \
asin(sqrt(7)/3)
assert pl7.angle_between(Segment3D(Point3D(5, 6, 1), Point3D(1, 2, 4))) == \
-asin(7*sqrt(246)/246)
assert are_coplanar(l1, l2, l3) is False
assert are_coplanar(l1) is False
assert are_coplanar(Point3D(2, 7, 2), Point3D(0, 0, 2), Point3D(1, 1, 2),
Point3D(1, 2, 2))
assert are_coplanar(Plane(p1, p2, p3), Plane(p1, p3, p2))
assert Plane.are_concurrent(pl3, pl4, pl5) is False
assert Plane.are_concurrent(pl6) is False
pytest.raises(ValueError, lambda: Plane.are_concurrent(Point3D(0, 0, 0)))
assert pl3.parallel_plane(Point3D(1, 2,
5)) == Plane(Point3D(1, 2, 5),
normal_vector=(1, -2, 1))
# perpendicular_plane
p = Plane((0, 0, 0), (1, 0, 0))
# default
assert p.perpendicular_plane() == Plane(Point3D(0, 0, 0), (0, 1, 0))
# 1 pt
assert p.perpendicular_plane(Point3D(1, 0, 1)) == \
Plane(Point3D(1, 0, 1), (0, 1, 0))
# pts as tuples
assert p.perpendicular_plane((1, 0, 1), (1, 1, 1)) == \
Plane(Point3D(1, 0, 1), (0, 0, -1))
a, b = Point3D(0, 0, 0), Point3D(0, 1, 0)
Z = (0, 0, 1)
p = Plane(a, normal_vector=Z)
# case 4
assert p.perpendicular_plane(a, b) == Plane(a, (1, 0, 0))
n = Point3D(*Z)
# case 1
assert p.perpendicular_plane(a, n) == Plane(a, (-1, 0, 0))
# case 2
assert Plane(a, normal_vector=b.args).perpendicular_plane(a, a + b) == \
Plane(Point3D(0, 0, 0), (1, 0, 0))
# case 1&3
assert Plane(b, normal_vector=Z).perpendicular_plane(b, b + n) == \
Plane(Point3D(0, 1, 0), (-1, 0, 0))
# case 2&3
assert Plane(b, normal_vector=b.args).perpendicular_plane(n, n + b) == \
Plane(Point3D(0, 0, 1), (1, 0, 0))
assert pl6.intersection(pl6) == [pl6]
assert pl4.intersection(pl4.p1) == [pl4.p1]
assert pl3.intersection(pl6) == [
Line3D(Point3D(8, 4, 0), Point3D(2, 4, 6))
]
assert pl3.intersection(Line3D(Point3D(1, 2, 4),
Point3D(4, 4,
2))) == [Point3D(2, 8 / 3, 10 / 3)]
assert pl3.intersection(Plane(Point3D(6, 0, 0),
normal_vector=(2, -5, 3))) == [
Line3D(Point3D(-24, -12, 0),
Point3D(-25, -13, -1))
]
assert pl6.intersection(Ray3D(Point3D(2, 3, 1),
Point3D(1, 3, 4))) == [Point3D(-1, 3, 10)]
assert pl6.intersection(Segment3D(Point3D(2, 3, 1),
Point3D(1, 3,
4))) == [Point3D(-1, 3, 10)]
assert pl7.intersection(Line(Point(2, 3),
Point(4, 2))) == [Point3D(13 / 2, 3 / 4, 0)]
r = Ray(Point(2, 3), Point(4, 2))
assert Plane((1, 2, 0), normal_vector=(0, 0, 1)).intersection(r) == [
Ray3D(Point(2, 3), Point(4, 2))
]
assert pl3.random_point() in pl3
# issue 8570
l2 = Line3D(
Point3D(Rational(50000004459633, 5000000000000),
Rational(-891926590718643, 1000000000000000),
Rational(231800966893633, 100000000000000)),
Point3D(Rational(50000004459633, 50000000000000),
Rational(-222981647679771, 250000000000000),
Rational(231800966893633, 100000000000000)))
p2 = Plane(
Point3D(Rational(402775636372767, 100000000000000),
Rational(-97224357654973, 100000000000000),
Rational(216793600814789, 100000000000000)),
(-Float(9.00000087501922), Float(-4.81170658872543e-13), Float(0.0)))
assert sstr([i.n(2) for i in p2.intersection(l2)]) == \
'[Point3D(4.0, -0.89, 2.3)]'
def test_line3d():
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)
l1 = Line3D(p1, p2)
l2 = Line3D(p3, p4)
l3 = Line3D(p3, p5)
pytest.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))
pytest.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()
pytest.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 sympy/sympy#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 sympy/sympy#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) is 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)
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', extended_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))
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]
pytest.raises(GeometryError,
lambda: parallel_1.projection(Plane(p1, p2, p6)))
# Test __new__
assert Line3D(perp_1) == perp_1
pytest.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))
pytest.raises(ValueError, lambda: Ray3D(pt2d))
# Test zdirection
negz = Ray3D(p1, Point3D(0, 0, -1))
assert negz.zdirection == -oo
# 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))
pytest.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
pytest.raises(NotImplementedError, lambda: ray1.is_similar(ray1))
# Begin Segment
seg1 = Segment3D(p1, Point3D(1, 0, 0))
pytest.raises(TypeError, lambda: seg1.contains([]))
seg2 = Segment3D(Point3D(2, 2, 2), Point3D(3, 2, 2))
assert seg1.contains(seg2) is False
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)
l1 = Line(p1, p2)
pytest.raises(ValueError, lambda: Ellipse(Point3D(0, 0, 0), 1, 1))
pytest.raises(ValueError, lambda: e3.arbitrary_point(y1))
assert e1.ambient_dimension == 2
# 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)
pytest.raises(GeometryError,
lambda: Circle(Point(0, 0), Point(1, 1), Point(2, 2)))
pytest.raises(ValueError, lambda: Ellipse(None, None, None, 1))
pytest.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 # issue sympy/sympy#12303
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
# 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), Rational(1, 2))
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))]
pytest.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)
pytest.raises(NotImplementedError, lambda: e.rotate(pi / 3))
# Circle rotation tests (issue sympy/sympy#11743)
cir = Circle(Point(1, 0), 1)
assert cir.rotate(pi / 2) == Circle(Point(0, 1), 1)
assert cir.rotate(pi / 3) == Circle(Point(Rational(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(Rational(1, 2) + sqrt(3) / 2,
Rational(1, 2) + sqrt(3) / 2), 1)
# 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)
e1 = Ellipse(Point(1, 0), 3, 2)
assert (e1.evolute() == root(4, 3) * y**Rational(2, 3) +
(3 * x - 3)**Rational(2, 3) - root(25, 3))
e1 = Ellipse(Point(0, 0), 3, 2)
p1 = e1.random_point(seed=0)
assert p1.evalf(2) == Point(2.0664, 1.4492)
assert Ellipse((1, 0), 2, 1).rotate(pi / 2) == Ellipse(Point(0, 1), 1, 2)
def test_plane2():
p1 = Plane(Point3D(0, 0, 0), normal_vector=(1, 0, 0))
p2 = Plane(Point3D(0, 0, 0), normal_vector=(1, 0, -1))
assert p1.intersection(p2) == [Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))]
p1 = Plane(Point3D(0, 0, 1), normal_vector=(0, -5, 3))
p2 = Plane(Point3D(0, 0, 2), normal_vector=(0, -5, -3))
assert p1.intersection(p2) == [
Line3D(Point3D(0, 3 / 10, 3 / 2), Point3D(30, 3 / 10, 3 / 2))
]
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(1, 2, 3)
pl3 = Plane(p1, p2, p3)
# get a segment that does not intersect the plane which is also
# parallel to pl3's normal veector
t = Dummy()
r = pl3.random_point()
a = pl3.perpendicular_line(r).arbitrary_point(t)
s = Segment3D(a.subs(t, 1), a.subs(t, 2))
assert s.p1 not in pl3 and s.p2 not in pl3
def test_plane2():
p1 = Plane(Point3D(0, 0, 0), normal_vector=(1, 0, 0))
p2 = Plane(Point3D(0, 0, 0), normal_vector=(1, 0, -1))
assert p1.intersection(p2) == [Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))]
p1 = Plane(Point3D(0, 0, 1), normal_vector=(0, -5, 3))
p2 = Plane(Point3D(0, 0, 2), normal_vector=(0, -5, -3))
assert p1.intersection(p2) == [
Line3D(Point3D(0, 3 / 10, 3 / 2), Point3D(30, 3 / 10, 3 / 2))
]
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(1, 2, 3)
pl3 = Plane(p1, p2, p3)
# get a segment that does not intersect the plane which is also
# parallel to pl3's normal veector
t = Dummy()
r = pl3.random_point()
a = pl3.perpendicular_line(r).arbitrary_point(t)
s = Segment3D(a.subs({t: 1}), a.subs({t: 2}))
assert s.p1 not in pl3 and s.p2 not in pl3
a = Plane(Point3D(5, 0, 0), normal_vector=(1, -1, 1))
b = Plane(Point3D(0, -2, 0), normal_vector=(3, 1, 1))
c = Plane(Point3D(0, -1, 0), normal_vector=(5, -1, 9))
assert Plane.are_concurrent(a, b) is True
assert Plane.are_concurrent(a, b, c) is False
p = Plane((0, 0, 0), (0, 0, 1), (0, 1, 0))
assert p.arbitrary_point(t) == Point3D(0, cos(t), sin(t))
def test_sympyissue_9214():
p1 = Point3D(4, -2, 6)
p2 = Point3D(1, 2, 3)
p3 = Point3D(7, 2, 3)
assert Point3D.are_collinear(p1, p2, p3) is False