def test_projection():
p1 = Point(0, 0)
p2 = Point3D(0, 0, 0)
p3 = Point(-x1, x1)
l1 = Line(p1, Point(1, 1))
l2 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
l3 = Line3D(p2, Point3D(1, 1, 1))
r1 = Ray(Point(1, 1), Point(2, 2))
s1 = Segment(Point2D(0, 0), Point2D(0, 1))
s2 = Segment(Point2D(1, 0), Point2D(2, 1/2))
assert Line(Point(x1, x1), Point(y1, y1)).projection(Point(y1, y1)) == Point(y1, y1)
assert Line(Point(x1, x1), Point(x1, 1 + x1)).projection(Point(1, 1)) == Point(x1, 1)
assert Segment(Point(-2, 2), Point(0, 4)).projection(r1) == Segment(Point(-1, 3), Point(0, 4))
assert Segment(Point(0, 4), Point(-2, 2)).projection(r1) == Segment(Point(0, 4), Point(-1, 3))
assert s2.projection(s1) == EmptySet
assert l1.projection(p3) == p1
assert l1.projection(Ray(p1, Point(-1, 5))) == Ray(Point(0, 0), Point(2, 2))
assert l1.projection(Ray(p1, Point(-1, 1))) == p1
assert r1.projection(Ray(Point(1, 1), Point(-1, -1))) == Point(1, 1)
assert r1.projection(Ray(Point(0, 4), Point(-1, -5))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Segment(Point(-1, 5), Point(-5, -10))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Ray(Point(1, 1), Point(-1, -1))) == Point(1, 1)
assert r1.projection(Ray(Point(0, 4), Point(-1, -5))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Segment(Point(-1, 5), Point(-5, -10))) == Segment(Point(1, 1), Point(2, 2))
assert l3.projection(Ray3D(p2, Point3D(-1, 5, 0))) == Ray3D(Point3D(0, 0, 0), Point3D(Rational(4, 3), Rational(4, 3), Rational(4, 3)))
assert l3.projection(Ray3D(p2, Point3D(-1, 1, 1))) == Ray3D(Point3D(0, 0, 0), Point3D(Rational(1, 3), Rational(1, 3), Rational(1, 3)))
assert l2.projection(Point3D(5, 5, 0)) == Point3D(5, 0)
assert l2.projection(Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))).equals(l2)
def test_line():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
p5 = Point(x1, 1 + x1)
p6 = Point(1, 0)
p7 = Point(0, 1)
p8 = Point(2, 0)
p9 = Point(2, 1)
l1 = Line(p1, p2)
l2 = Line(p3, p4)
l3 = Line(p3, p5)
l4 = Line(p1, p6)
l5 = Line(p1, p7)
l6 = Line(p8, p9)
l7 = Line(p2, p9)
raises(ValueError, lambda: Line(Point(0, 0), Point(0, 0)))
# Basic stuff
assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
raises(ValueError, lambda: Line((1, 1), 1))
assert Line(p1, p2) == Line(p2, p1)
assert l1 == l2
assert l1 != l3
assert l1.slope == 1
assert l1.length == oo
assert l3.slope == oo
assert l4.slope == 0
assert l4.coefficients == (0, 1, 0)
assert l4.equation(x=x, y=y) == y
assert l5.slope == oo
assert l5.coefficients == (1, 0, 0)
assert l5.equation() == x
assert l6.equation() == x - 2
assert l7.equation() == y - 1
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert Line((-x, x), (-x + 1, x - 1)).coefficients == (1, 1, 0)
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert Line(p1, p2).scale(2, 1) == Line(p1, p9)
assert l2.arbitrary_point() in l2
for ind in xrange(0, 5):
assert l3.random_point() in l3
# Orthogonality
p1_1 = Point(-x1, x1)
l1_1 = Line(p1, p1_1)
assert l1.perpendicular_line(p1) == l1_1
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1, l2) == False
p = l1.random_point()
assert l1.perpendicular_segment(p) == p
# Parallelity
p2_1 = Point(-2 * x1, 0)
l2_1 = Line(p3, p5)
assert l2.parallel_line(p1_1) == Line(p2_1, p1_1)
assert l2_1.parallel_line(p1) == Line(p1, Point(0, 2))
assert Line.is_parallel(l1, l2)
assert Line.is_parallel(l2, l3) == False
assert Line.is_parallel(l2, l2.parallel_line(p1_1))
assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))
# Intersection
assert intersection(l1, p1) == [p1]
assert intersection(l1, p5) == []
assert intersection(l1, l2) in [[l1], [l2]]
assert intersection(l1, l1.parallel_line(p5)) == []
# Concurrency
l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
assert Line.is_concurrent(l1) == False
assert Line.is_concurrent(l1, l3)
assert Line.is_concurrent(l1, l3, l3_1)
assert Line.is_concurrent(l1, l1_1, l3) == False
# Projection
assert l2.projection(p4) == p4
assert l1.projection(p1_1) == p1
assert l3.projection(p2) == Point(x1, 1)
raises(GeometryError, lambda: Line(Point(0, 0), Point(1, 0)).projection(Circle(Point(0, 0), 1)))
# Finding angles
l1_1 = Line(p1, Point(5, 0))
assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf() / 4)
# Testing Rays and Segments (very similar to Lines)
assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2))
assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0))
assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
# XXX don't know why this fails without str
assert str(Ray((1, 1), angle=4.2 * pi)) == str(Ray(Point(1, 1), Point(2, 1 + C.tan(0.2 * pi))))
assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + C.tan(5)))
raises(ValueError, lambda: Ray((1, 1), 1))
r1 = Ray(p1, Point(-1, 5))
r2 = Ray(p1, Point(-1, 1))
r3 = Ray(p3, p5)
r4 = Ray(p1, p2)
r5 = Ray(p2, p1)
r6 = Ray(Point(0, 1), Point(1, 2))
r7 = Ray(Point(0.5, 0.5), Point(1, 1))
assert l1.projection(r1) == Ray(p1, p2)
assert l1.projection(r2) == p1
assert r3 != r1
t = Symbol("t", real=True)
assert Ray((1, 1), angle=pi / 4).arbitrary_point() == Point(1 / (1 - t), 1 / (1 - t))
s1 = Segment(p1, p2)
s2 = Segment(p1, p1_1)
assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
assert s2.length == sqrt(2 * (x1 ** 2))
assert s1.perpendicular_bisector() == Line(Point(0, 1), Point(1, 0))
assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t)
# intersections
assert s1.intersection(Line(p6, p9)) == []
s3 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
assert s1.intersection(s3) == [s1]
assert s3.intersection(s1) == [s3]
assert r4.intersection(s3) == [s3]
assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
s3 = Segment(Point(1, 1), Point(2, 2))
assert s1.intersection(s3) == [Point(1, 1)]
s3 = Segment(Point(0.5, 0.5), Point(1.5, 1.5))
assert s1.intersection(s3) == [Segment(Point(0.5, 0.5), p2)]
assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
assert r4.intersection(r5) == [s1]
assert r5.intersection(r6) == []
assert r4.intersection(r7) == r7.intersection(r4) == [r7]
# Segment contains
a, b = symbols("a,b")
s = Segment((0, a), (0, b))
assert Point(0, (a + b) / 2) in s
s = Segment((a, 0), (b, 0))
assert Point((a + b) / 2, 0) in s
raises(Undecidable, lambda: Point(2 * a, 0) in s)
# Testing distance from a Segment to an object
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
pt1 = Point(0, 0)
pt2 = Point(Rational(3) / 2, Rational(3) / 2)
assert s1.distance(pt1) == 0
assert s2.distance(pt1) == 2 ** (half) / 2
assert s2.distance(pt2) == 2 ** (half)
# Special cases of projection and intersection
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(2, 2), Point(0, 0))
r3 = Ray(Point(1, 1), Point(-1, -1))
r4 = Ray(Point(0, 4), Point(-1, -5))
r5 = Ray(Point(2, 2), Point(3, 3))
assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, r3) == [Point(1, 1)]
assert r1.projection(r3) == Point(1, 1)
assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))
r5 = Ray(Point(0, 0), Point(0, 1))
r6 = Ray(Point(0, 0), Point(0, 2))
assert r5 in r6
assert r6 in r5
s1 = Segment(Point(0, 0), Point(2, 2))
s2 = Segment(Point(-1, 5), Point(-5, -10))
s3 = Segment(Point(0, 4), Point(-2, 2))
assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))
l1 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(0, 0), Point(3, 4))
s1 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, l1) == [l1]
assert intersection(l1, r1) == [r1]
assert intersection(l1, s1) == [s1]
assert intersection(r1, l1) == [r1]
assert intersection(s1, l1) == [s1]
entity1 = Segment(Point(-10, 10), Point(10, 10))
entity2 = Segment(Point(-5, -5), Point(-5, 5))
assert intersection(entity1, entity2) == []
r1 = Ray(p1, Point(0, 1))
r2 = Ray(Point(0, 1), p1)
r3 = Ray(p1, p2)
r4 = Ray(p2, p1)
s1 = Segment(p1, Point(0, 1))
assert Line(r1.source, r1.random_point()).slope == r1.slope
assert Line(r2.source, r2.random_point()).slope == r2.slope
assert Segment(Point(0, -1), s1.random_point()).slope == s1.slope
p_r3 = r3.random_point()
p_r4 = r4.random_point()
assert p_r3.x >= p1.x and p_r3.y >= p1.y
assert p_r4.x <= p2.x and p_r4.y <= p2.y
p10 = Point(2000, 2000)
s1 = Segment(p1, p10)
p_s1 = s1.random_point()
assert p1.x <= p_s1.x and p_s1.x <= p10.x and p1.y <= p_s1.y and p_s1.y <= p10.y
s2 = Segment(p10, p1)
assert hash(s1) == hash(s2)
p11 = p10.scale(2, 2)
assert s1.is_similar(Segment(p10, p11))
assert s1.is_similar(r1) == False
assert (r1 in s1) == False
assert Segment(p1, p2) in s1
assert s1.plot_interval() == [t, 0, 1]
assert s1 in Line(p1, p10)
assert Line(p1, p10) == Line(p10, p1)
assert Line(p1, p10) != p1
assert Line(p1, p10).plot_interval() == [t, -5, 5]
assert Ray((0, 0), angle=pi / 4).plot_interval() == [t, 0, 5 * sqrt(2) / (1 + 5 * sqrt(2))]
def test_line():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
p5 = Point(x1, 1 + x1)
p6 = Point(1, 0)
p7 = Point(0, 1)
p8 = Point(2, 0)
p9 = Point(2, 1)
l1 = Line(p1, p2)
l2 = Line(p3, p4)
l3 = Line(p3, p5)
l4 = Line(p1, p6)
l5 = Line(p1, p7)
l6 = Line(p8, p9)
l7 = Line(p2, p9)
raises(ValueError, 'Line(Point(0, 0), Point(0, 0))')
# Basic stuff
assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
raises(ValueError, 'Line((1, 1), 1)')
assert Line(p1, p2) == Line(p2, p1)
assert l1 == l2
assert l1 != l3
assert l1.slope == 1
assert l1.length == oo
assert l3.slope == oo
assert l4.slope == 0
assert l4.coefficients == (0, 1, 0)
assert l4.equation(x=x, y=y) == y
assert l5.slope == oo
assert l5.coefficients == (1, 0, 0)
assert l5.equation() == x
assert l6.equation() == x - 2
assert l7.equation() == y - 1
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert Line(p1, p2).scale(2, 1) == Line(p1, p9)
assert l2.arbitrary_point() in l2
for ind in xrange(0, 5):
assert l3.random_point() in l3
# Orthogonality
p1_1 = Point(-x1, x1)
l1_1 = Line(p1, p1_1)
assert l1.perpendicular_line(p1) == l1_1
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1, l2) == False
p = l1.random_point()
assert l1.perpendicular_segment(p) == p
# Parallelity
p2_1 = Point(-2 * x1, 0)
l2_1 = Line(p3, p5)
assert l2.parallel_line(p1_1) == Line(p2_1, p1_1)
assert l2_1.parallel_line(p1) == Line(p1, Point(0, 2))
assert Line.is_parallel(l1, l2)
assert Line.is_parallel(l2, l3) == False
assert Line.is_parallel(l2, l2.parallel_line(p1_1))
assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))
# Intersection
assert intersection(l1, p1) == [p1]
assert intersection(l1, p5) == []
assert intersection(l1, l2) in [[l1], [l2]]
assert intersection(l1, l1.parallel_line(p5)) == []
# Concurrency
l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
assert Line.is_concurrent(l1) == False
assert Line.is_concurrent(l1, l3)
assert Line.is_concurrent(l1, l3, l3_1)
assert Line.is_concurrent(l1, l1_1, l3) == False
# Projection
assert l2.projection(p4) == p4
assert l1.projection(p1_1) == p1
assert l3.projection(p2) == Point(x1, 1)
raises(
GeometryError,
'Line(Point(0, 0), Point(1, 0)).projection(Circle(Point(0, 0), 1))')
# Finding angles
l1_1 = Line(p1, Point(5, 0))
assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf() / 4)
# Testing Rays and Segments (very similar to Lines)
assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2))
assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0))
assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
# XXX don't know why this fails without str
assert str(Ray(
(1, 1),
angle=4.2 * pi)) == str(Ray(Point(1, 1), Point(2,
1 + C.tan(0.2 * pi))))
assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + C.tan(5)))
raises(ValueError, 'Ray((1, 1), 1)')
r1 = Ray(p1, Point(-1, 5))
r2 = Ray(p1, Point(-1, 1))
r3 = Ray(p3, p5)
r4 = Ray(p1, p2)
r5 = Ray(p2, p1)
r6 = Ray(Point(0, 1), Point(1, 2))
r7 = Ray(Point(0.5, 0.5), Point(1, 1))
assert l1.projection(r1) == Ray(p1, p2)
assert l1.projection(r2) == p1
assert r3 != r1
t = Symbol('t', real=True)
assert Ray(
(1, 1),
angle=pi / 4).arbitrary_point() == Point(1 / (1 - t), 1 / (1 - t))
s1 = Segment(p1, p2)
s2 = Segment(p1, p1_1)
assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
assert s2.length == sqrt(2 * (x1**2))
assert s1.perpendicular_bisector() == Line(Point(0, 1), Point(1, 0))
assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t)
# intersections
assert s1.intersection(Line(p6, p9)) == []
s3 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
assert s1.intersection(s3) == [s1]
assert s3.intersection(s1) == [s3]
assert r4.intersection(s3) == [s3]
assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
assert r4.intersection(Segment(Point(-1, -1), Point(
0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
s3 = Segment(Point(1, 1), Point(2, 2))
assert s1.intersection(s3) == [Point(1, 1)]
s3 = Segment(Point(0.5, 0.5), Point(1.5, 1.5))
assert s1.intersection(s3) == [Segment(Point(0.5, 0.5), p2)]
assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
assert s1.intersection(Segment(Point(-1, -1), Point(
0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
assert r4.intersection(r5) == [s1]
assert r5.intersection(r6) == []
assert r4.intersection(r7) == r7.intersection(r4) == [r7]
# Segment contains
a, b = symbols('a,b')
s = Segment((0, a), (0, b))
assert Point(0, (a + b) / 2) in s
s = Segment((a, 0), (b, 0))
assert Point((a + b) / 2, 0) in s
raises(Undecidable, "Point(2*a, 0) in s")
# Testing distance from a Segment to an object
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
pt1 = Point(0, 0)
pt2 = Point(Rational(3) / 2, Rational(3) / 2)
assert s1.distance(pt1) == 0
assert s2.distance(pt1) == 2**(half) / 2
assert s2.distance(pt2) == 2**(half)
# Special cases of projection and intersection
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(2, 2), Point(0, 0))
r3 = Ray(Point(1, 1), Point(-1, -1))
r4 = Ray(Point(0, 4), Point(-1, -5))
r5 = Ray(Point(2, 2), Point(3, 3))
assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, r3) == [Point(1, 1)]
assert r1.projection(r3) == Point(1, 1)
assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))
r5 = Ray(Point(0, 0), Point(0, 1))
r6 = Ray(Point(0, 0), Point(0, 2))
assert r5 in r6
assert r6 in r5
s1 = Segment(Point(0, 0), Point(2, 2))
s2 = Segment(Point(-1, 5), Point(-5, -10))
s3 = Segment(Point(0, 4), Point(-2, 2))
assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))
l1 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(0, 0), Point(3, 4))
s1 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, l1) == [l1]
assert intersection(l1, r1) == [r1]
assert intersection(l1, s1) == [s1]
assert intersection(r1, l1) == [r1]
assert intersection(s1, l1) == [s1]
entity1 = Segment(Point(-10, 10), Point(10, 10))
entity2 = Segment(Point(-5, -5), Point(-5, 5))
assert intersection(entity1, entity2) == []
r1 = Ray(p1, Point(0, 1))
r2 = Ray(Point(0, 1), p1)
r3 = Ray(p1, p2)
r4 = Ray(p2, p1)
s1 = Segment(p1, Point(0, 1))
assert Line(r1.source, r1.random_point()).slope == r1.slope
assert Line(r2.source, r2.random_point()).slope == r2.slope
assert Segment(Point(0, -1), s1.random_point()).slope == s1.slope
p_r3 = r3.random_point()
p_r4 = r4.random_point()
assert p_r3.x >= p1.x and p_r3.y >= p1.y
assert p_r4.x <= p2.x and p_r4.y <= p2.y
p10 = Point(2000, 2000)
s1 = Segment(p1, p10)
p_s1 = s1.random_point()
assert p1.x <= p_s1.x and p_s1.x <= p10.x and p1.y <= p_s1.y and p_s1.y <= p10.y
s2 = Segment(p10, p1)
assert hash(s1) == hash(s2)
p11 = p10.scale(2, 2)
assert s1.is_similar(Segment(p10, p11))
assert s1.is_similar(r1) == False
assert (r1 in s1) == False
assert Segment(p1, p2) in s1
assert s1.plot_interval() == [t, 0, 1]
assert s1 in Line(p1, p10)
assert Line(p1, p10) == Line(p10, p1)
assert Line(p1, p10) != p1
assert Line(p1, p10).plot_interval() == [t, -5, 5]
def test_line():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
p5 = Point(x1, 1 + x1)
l1 = Line(p1, p2)
l2 = Line(p3, p4)
l3 = Line(p3, p5)
# Basic stuff
assert Line(p1, p2) == Line(p2, p1)
assert l1 == l2
assert l1 != l3
assert l1.slope == 1
assert l3.slope == oo
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert simplify(l1.equation()) in (x-y, y-x)
assert simplify(l3.equation()) in (x-x1, x1-x)
assert l2.arbitrary_point() in l2
for ind in xrange(0, 5):
assert l3.random_point() in l3
# Orthogonality
p1_1 = Point(-x1, x1)
l1_1 = Line(p1, p1_1)
assert l1.perpendicular_line(p1) == l1_1
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1 , l2) == False
# Parallelity
p2_1 = Point(-2*x1, 0)
l2_1 = Line(p3, p5)
assert l2.parallel_line(p1_1) == Line(p2_1, p1_1)
assert l2_1.parallel_line(p1) == Line(p1, Point(0, 2))
assert Line.is_parallel(l1, l2)
assert Line.is_parallel(l2, l3) == False
assert Line.is_parallel(l2, l2.parallel_line(p1_1))
assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))
# Intersection
assert intersection(l1, p1) == [p1]
assert intersection(l1, p5) == []
assert intersection(l1, l2) in [[l1], [l2]]
assert intersection(l1, l1.parallel_line(p5)) == []
# Concurrency
l3_1 = Line(Point(5, x1), Point(-Rational(3,5), x1))
assert Line.is_concurrent(l1, l3)
assert Line.is_concurrent(l1, l3, l3_1)
assert Line.is_concurrent(l1, l1_1, l3) == False
# Projection
assert l2.projection(p4) == p4
assert l1.projection(p1_1) == p1
assert l3.projection(p2) == Point(x1, 1)
# Finding angles
l1_1 = Line(p1, Point(5, 0))
assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf()/4)
# Testing Rays and Segments (very similar to Lines)
r1 = Ray(p1, Point(-1, 5))
r2 = Ray(p1, Point(-1, 1))
r3 = Ray(p3, p5)
assert l1.projection(r1) == Ray(p1, p2)
assert l1.projection(r2) == p1
assert r3 != r1
s1 = Segment(p1, p2)
s2 = Segment(p1, p1_1)
assert s1.midpoint == Point(Rational(1,2), Rational(1,2))
assert s2.length == sqrt( 2*(x1**2) )
assert s1.perpendicular_bisector() == Line(Point(0, 1), Point(1, 0))
# Testing distance from a Segment to an object
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
pt1 = Point(0, 0)
pt2 = Point(Rational(3)/2, Rational(3)/2)
assert s1.distance(pt1) == 0
assert s2.distance(pt1) == 2**(half)/2
assert s2.distance(pt2) == 2**(half)
# Special cases of projection and intersection
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(2, 2), Point(0, 0))
r3 = Ray(Point(1, 1), Point(-1, -1))
r4 = Ray(Point(0, 4), Point(-1, -5))
assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, r3) == [Point(1, 1)]
assert r1.projection(r3) == Point(1, 1)
assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))
r5 = Ray(Point(0, 0), Point(0, 1))
r6 = Ray(Point(0, 0), Point(0, 2))
assert r5 in r6
assert r6 in r5
s1 = Segment(Point(0, 0), Point(2, 2))
s2 = Segment(Point(-1, 5), Point(-5, -10))
s3 = Segment(Point(0, 4), Point(-2, 2))
assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))
l1 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(0, 0), Point(3, 4))
s1 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, l1) == [l1]
assert intersection(l1, r1) == [r1]
assert intersection(l1, s1) == [s1]
assert intersection(r1, l1) == [r1]
assert intersection(s1, l1) == [s1]
entity1 = Segment(Point(-10,10), Point(10,10))
entity2 = Segment(Point(-5,-5), Point(-5,5))
assert intersection(entity1, entity2) == []
def test_line():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
p5 = Point(x1, 1 + x1)
p6 = Point(1, 0)
p7 = Point(0, 1)
p8 = Point(2, 0)
p9 = Point(2, 1)
l1 = Line(p1, p2)
l2 = Line(p3, p4)
l3 = Line(p3, p5)
l4 = Line(p1, p6)
l5 = Line(p1, p7)
l6 = Line(p8, p9)
l7 = Line(p2, p9)
# Basic stuff
assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
raises(ValueError, "Line((1, 1), 1)")
assert Line(p1, p2) == Line(p2, p1)
assert l1 == l2
assert l1 != l3
assert l1.slope == 1
assert l3.slope == oo
assert l4.slope == 0
assert l4.coefficients == (0, 1, 0)
assert l4.equation(x=x, y=y) == y
assert l5.slope == oo
assert l5.coefficients == (1, 0, 0)
assert l5.equation() == x
assert l6.equation() == x - 2
assert l7.equation() == y - 1
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert l2.arbitrary_point() in l2
for ind in xrange(0, 5):
assert l3.random_point() in l3
# Orthogonality
p1_1 = Point(-x1, x1)
l1_1 = Line(p1, p1_1)
assert l1.perpendicular_line(p1) == l1_1
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1, l2) == False
# Parallelity
p2_1 = Point(-2 * x1, 0)
l2_1 = Line(p3, p5)
assert l2.parallel_line(p1_1) == Line(p2_1, p1_1)
assert l2_1.parallel_line(p1) == Line(p1, Point(0, 2))
assert Line.is_parallel(l1, l2)
assert Line.is_parallel(l2, l3) == False
assert Line.is_parallel(l2, l2.parallel_line(p1_1))
assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))
# Intersection
assert intersection(l1, p1) == [p1]
assert intersection(l1, p5) == []
assert intersection(l1, l2) in [[l1], [l2]]
assert intersection(l1, l1.parallel_line(p5)) == []
# Concurrency
l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
assert Line.is_concurrent(l1, l3)
assert Line.is_concurrent(l1, l3, l3_1)
assert Line.is_concurrent(l1, l1_1, l3) == False
# Projection
assert l2.projection(p4) == p4
assert l1.projection(p1_1) == p1
assert l3.projection(p2) == Point(x1, 1)
# Finding angles
l1_1 = Line(p1, Point(5, 0))
assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf() / 4)
# Testing Rays and Segments (very similar to Lines)
assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2))
assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0))
assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
# XXX don't know why this fails without str
assert str(Ray((1, 1), angle=4.2 * pi)) == str(Ray(Point(1, 1), Point(2, 1 + C.tan(0.2 * pi))))
assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + C.tan(5)))
raises(ValueError, "Ray((1, 1), 1)")
r1 = Ray(p1, Point(-1, 5))
r2 = Ray(p1, Point(-1, 1))
r3 = Ray(p3, p5)
assert l1.projection(r1) == Ray(p1, p2)
assert l1.projection(r2) == p1
assert r3 != r1
t = Symbol("t", real=True)
assert Ray((1, 1), angle=pi / 4).arbitrary_point() == Point(1 / (1 - t), 1 / (1 - t))
s1 = Segment(p1, p2)
s2 = Segment(p1, p1_1)
assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
assert s2.length == sqrt(2 * (x1 ** 2))
assert s1.perpendicular_bisector() == Line(Point(0, 1), Point(1, 0))
assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t)
# Segment contains
a, b = symbols("a,b")
s = Segment((0, a), (0, b))
assert Point(0, (a + b) / 2) in s
s = Segment((a, 0), (b, 0))
assert Point((a + b) / 2, 0) in s
assert (Point(2 * a, 0) in s) is False # XXX should be None?
# Testing distance from a Segment to an object
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
pt1 = Point(0, 0)
pt2 = Point(Rational(3) / 2, Rational(3) / 2)
assert s1.distance(pt1) == 0
assert s2.distance(pt1) == 2 ** (half) / 2
assert s2.distance(pt2) == 2 ** (half)
# Special cases of projection and intersection
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(2, 2), Point(0, 0))
r3 = Ray(Point(1, 1), Point(-1, -1))
r4 = Ray(Point(0, 4), Point(-1, -5))
assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, r3) == [Point(1, 1)]
assert r1.projection(r3) == Point(1, 1)
assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))
r5 = Ray(Point(0, 0), Point(0, 1))
r6 = Ray(Point(0, 0), Point(0, 2))
assert r5 in r6
assert r6 in r5
s1 = Segment(Point(0, 0), Point(2, 2))
s2 = Segment(Point(-1, 5), Point(-5, -10))
s3 = Segment(Point(0, 4), Point(-2, 2))
assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))
l1 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(0, 0), Point(3, 4))
s1 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, l1) == [l1]
assert intersection(l1, r1) == [r1]
assert intersection(l1, s1) == [s1]
assert intersection(r1, l1) == [r1]
assert intersection(s1, l1) == [s1]
entity1 = Segment(Point(-10, 10), Point(10, 10))
entity2 = Segment(Point(-5, -5), Point(-5, 5))
assert intersection(entity1, entity2) == []
def test_line():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
p5 = Point(x1, 1 + x1)
l1 = Line(p1, p2)
l2 = Line(p3, p4)
l3 = Line(p3, p5)
# Basic stuff
assert Line(p1, p2) == Line(p2, p1)
assert l1 == l2
assert l1 != l3
assert l1.slope == 1
assert l3.slope == oo
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert l2.arbitrary_point() in l2
for ind in xrange(0, 5):
assert l3.random_point() in l3
# Orthogonality
p1_1 = Point(-x1, x1)
l1_1 = Line(p1, p1_1)
assert l1.perpendicular_line(p1) == l1_1
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1, l2) == False
# Parallelity
p2_1 = Point(-2 * x1, 0)
l2_1 = Line(p3, p5)
assert l2.parallel_line(p1_1) == Line(p2_1, p1_1)
assert l2_1.parallel_line(p1) == Line(p1, Point(0, 2))
assert Line.is_parallel(l1, l2)
assert Line.is_parallel(l2, l3) == False
assert Line.is_parallel(l2, l2.parallel_line(p1_1))
assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))
# Intersection
assert intersection(l1, p1) == [p1]
assert intersection(l1, p5) == []
assert intersection(l1, l2) in [[l1], [l2]]
assert intersection(l1, l1.parallel_line(p5)) == []
# Concurrency
l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
assert Line.is_concurrent(l1, l3)
assert Line.is_concurrent(l1, l3, l3_1)
assert Line.is_concurrent(l1, l1_1, l3) == False
# Projection
assert l2.projection(p4) == p4
assert l1.projection(p1_1) == p1
assert l3.projection(p2) == Point(x1, 1)
# Finding angles
l1_1 = Line(p1, Point(5, 0))
assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf() / 4)
# Testing Rays and Segments (very similar to Lines)
r1 = Ray(p1, Point(-1, 5))
r2 = Ray(p1, Point(-1, 1))
r3 = Ray(p3, p5)
assert l1.projection(r1) == Ray(p1, p2)
assert l1.projection(r2) == p1
assert r3 != r1
s1 = Segment(p1, p2)
s2 = Segment(p1, p1_1)
assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
assert s2.length == sqrt(2 * (x1**2))
assert s1.perpendicular_bisector() == Line(Point(0, 1), Point(1, 0))
# Testing distance from a Segment to an object
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
pt1 = Point(0, 0)
pt2 = Point(Rational(3) / 2, Rational(3) / 2)
assert s1.distance(pt1) == 0
assert s2.distance(pt1) == 2**(half) / 2
assert s2.distance(pt2) == 2**(half)
# Special cases of projection and intersection
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(2, 2), Point(0, 0))
r3 = Ray(Point(1, 1), Point(-1, -1))
r4 = Ray(Point(0, 4), Point(-1, -5))
assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, r3) == [Point(1, 1)]
assert r1.projection(r3) == Point(1, 1)
assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))
r5 = Ray(Point(0, 0), Point(0, 1))
r6 = Ray(Point(0, 0), Point(0, 2))
assert r5 in r6
assert r6 in r5
s1 = Segment(Point(0, 0), Point(2, 2))
s2 = Segment(Point(-1, 5), Point(-5, -10))
s3 = Segment(Point(0, 4), Point(-2, 2))
assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))
l1 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(0, 0), Point(3, 4))
s1 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, l1) == [l1]
assert intersection(l1, r1) == [r1]
assert intersection(l1, s1) == [s1]
assert intersection(r1, l1) == [r1]
assert intersection(s1, l1) == [s1]
entity1 = Segment(Point(-10, 10), Point(10, 10))
entity2 = Segment(Point(-5, -5), Point(-5, 5))
assert intersection(entity1, entity2) == []
def test_line_geom():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
x1 = Symbol('x1', real=True)
y1 = Symbol('y1', real=True)
half = Rational(1, 2)
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
p5 = Point(x1, 1 + x1)
p6 = Point(1, 0)
p7 = Point(0, 1)
p8 = Point(2, 0)
p9 = Point(2, 1)
l1 = Line(p1, p2)
l2 = Line(p3, p4)
l3 = Line(p3, p5)
l4 = Line(p1, p6)
l5 = Line(p1, p7)
l6 = Line(p8, p9)
l7 = Line(p2, p9)
raises(ValueError, lambda: Line(Point(0, 0), Point(0, 0)))
# Basic stuff
assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
raises(TypeError, lambda: Line((1, 1), 1))
assert Line(p1, p2) == Line(p1, p2)
assert Line(p1, p2) != Line(p2, p1)
assert l1 != l2
assert l1 != l3
assert l1.slope == 1
assert l1.length == oo
assert l3.slope == oo
assert l4.slope == 0
assert l4.coefficients == (0, 1, 0)
assert l4.equation(x=x, y=y) == y
assert l5.slope == oo
assert l5.coefficients == (1, 0, 0)
assert l5.equation() == x
assert l6.equation() == x - 2
assert l7.equation() == y - 1
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert Line((-x, x), (-x + 1, x - 1)).coefficients == (1, 1, 0)
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert Line(p1, p2).scale(2, 1) == Line(p1, p9)
assert l2.arbitrary_point() in l2
for ind in range(0, 5):
assert l3.random_point() in l3
# Orthogonality
p1_1 = Point(-x1, x1)
l1_1 = Line(p1, p1_1)
assert l1.perpendicular_line(p1.args).equals(
Line(Point(0, 0), Point(1, -1)))
assert l1.perpendicular_line(p1).equals(Line(Point(0, 0), Point(1, -1)))
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1, l2) is False
p = l1.random_point()
assert l1.perpendicular_segment(p) == p
# Parallelity
l2_1 = Line(p3, p5)
assert l2.parallel_line(p1_1).equals(
Line(Point(-x1, x1), Point(-y1, 2 * x1 - y1)))
assert l2_1.parallel_line(p1.args).equals(Line(Point(0, 0), Point(0, -1)))
assert l2_1.parallel_line(p1).equals(Line(Point(0, 0), Point(0, -1)))
assert Line.is_parallel(l1, l2)
assert Line.is_parallel(l2, l3) is False
assert Line.is_parallel(l2, l2.parallel_line(p1_1))
assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))
# Intersection
assert intersection(l1, p1) == [p1]
assert intersection(l1, p5) == []
assert intersection(l1, l2) in [[l1], [l2]]
assert intersection(l1, l1.parallel_line(p5)) == []
# Concurrency
l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
assert Line.are_concurrent(l1) is False
assert Line.are_concurrent(l1, l3)
assert Line.are_concurrent(l1, l1, l1, l3)
assert Line.are_concurrent(l1, l3, l3_1)
assert Line.are_concurrent(l1, l1_1, l3) is False
# Projection
assert l2.projection(p4) == p4
assert l1.projection(p1_1) == p1
assert l3.projection(p2) == Point(x1, 1)
raises(
GeometryError, lambda: Line(Point(0, 0), Point(1, 0)).projection(
Circle(Point(0, 0), 1)))
# Finding angles
l1_1 = Line(p1, Point(5, 0))
assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf() / 4)
a = Point(1, 2, 3, 4)
b = a.orthogonal_direction
o = a.origin
assert Line(a, o).angle_between(Line(b, o)) == pi / 2
# Testing Rays and Segments (very similar to Lines)
assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2))
assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0))
assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=4.05 * pi) == Ray(
Point(1, 1),
Point(
2, -sqrt(5) * sqrt(2 * sqrt(5) + 10) / 4 -
sqrt(2 * sqrt(5) + 10) / 4 + 2 + sqrt(5)))
assert Ray((1, 1), angle=4.02 * pi) == Ray(Point(1, 1),
Point(2, 1 + tan(4.02 * pi)))
assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + tan(5)))
raises(TypeError, lambda: Ray((1, 1), 1))
# issue 7963
r = Ray((0, 0), angle=x)
assert r.subs(x, 3 * pi / 4) == Ray((0, 0), (-1, 1))
assert r.subs(x, 5 * pi / 4) == Ray((0, 0), (-1, -1))
assert r.subs(x, -pi / 4) == Ray((0, 0), (1, -1))
assert r.subs(x, pi / 2) == Ray((0, 0), (0, 1))
assert r.subs(x, -pi / 2) == Ray((0, 0), (0, -1))
r1 = Ray(p1, Point(-1, 5))
r2 = Ray(p1, Point(-1, 1))
r3 = Ray(p3, p5)
r4 = Ray(p1, p2)
r5 = Ray(p2, p1)
r6 = Ray(Point(0, 1), Point(1, 2))
r7 = Ray(Point(0.5, 0.5), Point(1, 1))
assert l1.projection(r1) == Ray(Point(0, 0), Point(2, 2))
assert l1.projection(r2) == p1
assert r3 != r1
t = Symbol('t', real=True)
assert Ray((1, 1), angle=pi/4).arbitrary_point() == \
Point(t + 1, t + 1)
r8 = Ray(Point(0, 0), Point(0, 4))
r9 = Ray(Point(0, 1), Point(0, -1))
assert r8.intersection(r9) == [Segment(Point(0, 0), Point(0, 1))]
s1 = Segment(p1, p2)
s2 = Segment(p1, p1_1)
assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
assert s2.length == sqrt(2 * (x1**2))
assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t)
aline = Line(Point(1 / 2, 1 / 2), Point(3 / 2, -1 / 2))
assert s1.perpendicular_bisector().equals(aline)
on_line = Segment(Point(1 / 2, 1 / 2), Point(3 / 2, -1 / 2)).midpoint
assert s1.perpendicular_bisector(on_line) == Segment(s1.midpoint, on_line)
assert s1.perpendicular_bisector(on_line + (1, 0)).equals(aline)
# intersections
assert s1.intersection(Line(p6, p9)) == []
s3 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
assert s1.intersection(s3) == [s3]
assert s3.intersection(s1) == [s3]
assert r4.intersection(s3) == [s3]
assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == \
[Segment(p1, Point(0.5, 0.5))]
s3 = Segment(Point(1, 1), Point(2, 2))
assert s1.intersection(s3) == [Point(1, 1)]
s3 = Segment(Point(0.5, 0.5), Point(1.5, 1.5))
assert s1.intersection(s3) == [Segment(Point(0.5, 0.5), p2)]
assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == \
[Segment(p1, Point(0.5, 0.5))]
assert r4.intersection(r5) == [s1]
assert r5.intersection(r6) == []
assert r4.intersection(r7) == r7.intersection(r4) == [r7]
# Segment contains
a, b = symbols('a,b', real=True)
s = Segment((0, a), (0, b))
assert Point(0, (a + b) / 2) in s
s = Segment((a, 0), (b, 0))
assert Point((a + b) / 2, 0) in s
raises(Undecidable, lambda: Point(2 * a, 0) in s)
# Testing distance from a Segment to an object
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
pt1 = Point(0, 0)
pt2 = Point(Rational(3) / 2, Rational(3) / 2)
assert s1.distance(pt1) == 0
assert s1.distance((0, 0)) == 0
assert s2.distance(pt1) == 2**(half) / 2
assert s2.distance(pt2) == 2**(half)
# Line to point
p1, p2 = Point(0, 0), Point(1, 1)
s = Line(p1, p2)
assert s.distance(Point(-1, 1)) == sqrt(2)
assert s.distance(Point(1, -1)) == sqrt(2)
assert s.distance(Point(2, 2)) == 0
assert s.distance((-1, 1)) == sqrt(2)
assert Line((0, 0), (0, 1)).distance(p1) == 0
assert Line((0, 0), (0, 1)).distance(p2) == 1
assert Line((0, 0), (1, 0)).distance(p1) == 0
assert Line((0, 0), (1, 0)).distance(p2) == 1
m = symbols('m', real=True)
l = Line((0, 5), slope=m)
p = Point(2, 3)
assert (l.distance(p) - 2 * abs(m + 1) / sqrt(m**2 + 1)).equals(0)
# Ray to point
r = Ray(p1, p2)
assert r.distance(Point(-1, -1)) == sqrt(2)
assert r.distance(Point(1, 1)) == 0
assert r.distance(Point(-1, 1)) == sqrt(2)
assert Ray((1, 1), (2, 2)).distance(Point(1.5, 3)) == 3 * sqrt(2) / 4
assert r.distance((1, 1)) == 0
#Line contains
p1, p2 = Point(0, 1), Point(3, 4)
l = Line(p1, p2)
assert l.contains(p1) is True
assert l.contains((0, 1)) is True
assert l.contains((0, 0)) is False
#Ray contains
p1, p2 = Point(0, 0), Point(4, 4)
r = Ray(p1, p2)
assert r.contains(p1) is True
assert r.contains((1, 1)) is True
assert r.contains((1, 3)) is False
s = Segment((1, 1), (2, 2))
assert r.contains(s) is True
s = Segment((1, 2), (2, 5))
assert r.contains(s) is False
r1 = Ray((2, 2), (3, 3))
assert r.contains(r1) is True
r1 = Ray((2, 2), (3, 5))
assert r.contains(r1) is False
# Special cases of projection and intersection
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(2, 2), Point(0, 0))
r3 = Ray(Point(1, 1), Point(-1, -1))
r4 = Ray(Point(0, 4), Point(-1, -5))
r5 = Ray(Point(2, 2), Point(3, 3))
assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, r3) == [Point(1, 1)]
assert r1.projection(r3) == Point(1, 1)
assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))
r5 = Ray(Point(0, 0), Point(0, 1))
r6 = Ray(Point(0, 0), Point(0, 2))
assert r5 in r6
assert r6 in r5
s1 = Segment(Point(0, 0), Point(2, 2))
s2 = Segment(Point(-1, 5), Point(-5, -10))
s3 = Segment(Point(0, 4), Point(-2, 2))
assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))
l1 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(0, 0), Point(3, 4))
s1 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, l1) == [l1]
assert intersection(l1, r1) == [r1]
assert intersection(l1, s1) == [s1]
assert intersection(r1, l1) == [r1]
assert intersection(s1, l1) == [s1]
entity1 = Segment(Point(-10, 10), Point(10, 10))
entity2 = Segment(Point(-5, -5), Point(-5, 5))
assert intersection(entity1, entity2) == []
r1 = Ray(p1, Point(0, 1))
r2 = Ray(Point(0, 1), p1)
r3 = Ray(p1, p2)
r4 = Ray(p2, p1)
s1 = Segment(p1, Point(0, 1))
assert Line(r1.source, r1.random_point()).slope == r1.slope
assert Line(r2.source, r2.random_point()).slope == r2.slope
assert Segment(Point(0, -1), s1.random_point()).slope == s1.slope
p_r3 = r3.random_point()
p_r4 = r4.random_point()
assert p_r3.x >= p1.x and p_r3.y >= p1.y
assert p_r4.x <= p2.x and p_r4.y <= p2.y
p10 = Point(2000, 2000)
s1 = Segment(p1, p10)
p_s1 = s1.random_point()
assert p1.x <= p_s1.x and p_s1.x <= p10.x and \
p1.y <= p_s1.y and p_s1.y <= p10.y
s2 = Segment(p10, p1)
assert hash(s1) == hash(s2)
p11 = p10.scale(2, 2)
assert s1.is_similar(Segment(p10, p11))
assert s1.is_similar(r1) is False
assert (r1 in s1) is False
assert Segment(p1, p2) in s1
assert s1.plot_interval() == [t, 0, 1]
assert s1 in Line(p1, p10)
assert Line(p1, p10) != Line(p10, p1)
assert Line(p1, p10) != p1
assert Line(p1, p10).plot_interval() == [t, -5, 5]
assert Ray((0, 0), angle=pi/4).plot_interval() == \
[t, 0, 10]
def test_line_geom():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
x1 = Symbol('x1', real=True)
y1 = Symbol('y1', real=True)
half = Rational(1, 2)
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
p5 = Point(x1, 1 + x1)
p6 = Point(1, 0)
p7 = Point(0, 1)
p8 = Point(2, 0)
p9 = Point(2, 1)
l1 = Line(p1, p2)
l2 = Line(p3, p4)
l3 = Line(p3, p5)
l4 = Line(p1, p6)
l5 = Line(p1, p7)
l6 = Line(p8, p9)
l7 = Line(p2, p9)
raises(ValueError, lambda: Line(Point(0, 0), Point(0, 0)))
# Basic stuff
assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
raises(TypeError, lambda: Line((1, 1), 1))
assert Line(p1, p2) == Line(p1, p2)
assert Line(p1, p2) != Line(p2, p1)
assert l1 != l2
assert l1 != l3
assert l1.slope == 1
assert l1.length == oo
assert l3.slope == oo
assert l4.slope == 0
assert l4.coefficients == (0, 1, 0)
assert l4.equation(x=x, y=y) == y
assert l5.slope == oo
assert l5.coefficients == (1, 0, 0)
assert l5.equation() == x
assert l6.equation() == x - 2
assert l7.equation() == y - 1
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert Line((-x, x), (-x + 1, x - 1)).coefficients == (1, 1, 0)
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert Line(p1, p2).scale(2, 1) == Line(p1, p9)
assert l2.arbitrary_point() in l2
for ind in range(0, 5):
assert l3.random_point() in l3
# Orthogonality
p1_1 = Point(-x1, x1)
l1_1 = Line(p1, p1_1)
assert l1.perpendicular_line(p1.args).equals( Line(Point(0, 0), Point(1, -1)) )
assert l1.perpendicular_line(p1).equals( Line(Point(0, 0), Point(1, -1)) )
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1, l2) is False
p = l1.random_point()
assert l1.perpendicular_segment(p) == p
# Parallelity
l2_1 = Line(p3, p5)
assert l2.parallel_line(p1_1).equals( Line(Point(-x1, x1), Point(-y1, 2*x1 - y1)) )
assert l2_1.parallel_line(p1.args).equals( Line(Point(0, 0), Point(0, -1)) )
assert l2_1.parallel_line(p1).equals( Line(Point(0, 0), Point(0, -1)) )
assert Line.is_parallel(l1, l2)
assert Line.is_parallel(l2, l3) is False
assert Line.is_parallel(l2, l2.parallel_line(p1_1))
assert Line.is_parallel(l2_1, l2_1.parallel_line(p1))
# Intersection
assert intersection(l1, p1) == [p1]
assert intersection(l1, p5) == []
assert intersection(l1, l2) in [[l1], [l2]]
assert intersection(l1, l1.parallel_line(p5)) == []
# Concurrency
l3_1 = Line(Point(5, x1), Point(-Rational(3, 5), x1))
assert Line.are_concurrent(l1) is False
assert Line.are_concurrent(l1, l3)
assert Line.are_concurrent(l1, l1, l1, l3)
assert Line.are_concurrent(l1, l3, l3_1)
assert Line.are_concurrent(l1, l1_1, l3) is False
# Projection
assert l2.projection(p4) == p4
assert l1.projection(p1_1) == p1
assert l3.projection(p2) == Point(x1, 1)
raises(GeometryError, lambda: Line(Point(0, 0), Point(1, 0))
.projection(Circle(Point(0, 0), 1)))
# Finding angles
l1_1 = Line(p1, Point(5, 0))
assert feq(Line.angle_between(l1, l1_1).evalf(), pi.evalf()/4)
a = Point(1, 2, 3, 4)
b = a.orthogonal_direction
o = a.origin
assert Line(a, o).angle_between(Line(b, o)) == pi/2
# Testing Rays and Segments (very similar to Lines)
assert Ray((1, 1), angle=pi/4) == Ray((1, 1), (2, 2))
assert Ray((1, 1), angle=pi/2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=-pi/2) == Ray((1, 1), (1, 0))
assert Ray((1, 1), angle=-3*pi/2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5*pi/2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5.0*pi/2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=3.0*pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=4.0*pi) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=4.05*pi) == Ray(Point(1, 1),
Point(2, -sqrt(5)*sqrt(2*sqrt(5) + 10)/4 - sqrt(2*sqrt(5) + 10)/4 + 2 + sqrt(5)))
assert Ray((1, 1), angle=4.02*pi) == Ray(Point(1, 1),
Point(2, 1 + tan(4.02*pi)))
assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + tan(5)))
raises(TypeError, lambda: Ray((1, 1), 1))
# issue 7963
r = Ray((0, 0), angle=x)
assert r.subs(x, 3*pi/4) == Ray((0, 0), (-1, 1))
assert r.subs(x, 5*pi/4) == Ray((0, 0), (-1, -1))
assert r.subs(x, -pi/4) == Ray((0, 0), (1, -1))
assert r.subs(x, pi/2) == Ray((0, 0), (0, 1))
assert r.subs(x, -pi/2) == Ray((0, 0), (0, -1))
r1 = Ray(p1, Point(-1, 5))
r2 = Ray(p1, Point(-1, 1))
r3 = Ray(p3, p5)
r4 = Ray(p1, p2)
r5 = Ray(p2, p1)
r6 = Ray(Point(0, 1), Point(1, 2))
r7 = Ray(Point(0.5, 0.5), Point(1, 1))
assert l1.projection(r1) == Ray(Point(0, 0), Point(2, 2))
assert l1.projection(r2) == p1
assert r3 != r1
t = Symbol('t', real=True)
assert Ray((1, 1), angle=pi/4).arbitrary_point() == \
Point(t + 1, t + 1)
r8 = Ray(Point(0, 0), Point(0, 4))
r9 = Ray(Point(0, 1), Point(0, -1))
assert r8.intersection(r9) == [Segment(Point(0, 0), Point(0, 1))]
s1 = Segment(p1, p2)
s2 = Segment(p1, p1_1)
assert s1.midpoint == Point(Rational(1, 2), Rational(1, 2))
assert s2.length == sqrt( 2*(x1**2) )
assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2*t)
aline = Line(Point(1/2, 1/2), Point(3/2, -1/2))
assert s1.perpendicular_bisector().equals(aline)
on_line = Segment(Point(1/2, 1/2), Point(3/2, -1/2)).midpoint
assert s1.perpendicular_bisector(on_line) == Segment(s1.midpoint, on_line)
assert s1.perpendicular_bisector(on_line + (1, 0)).equals(aline)
# intersections
assert s1.intersection(Line(p6, p9)) == []
s3 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
assert s1.intersection(s3) == [s3]
assert s3.intersection(s1) == [s3]
assert r4.intersection(s3) == [s3]
assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == \
[Segment(p1, Point(0.5, 0.5))]
s3 = Segment(Point(1, 1), Point(2, 2))
assert s1.intersection(s3) == [Point(1, 1)]
s3 = Segment(Point(0.5, 0.5), Point(1.5, 1.5))
assert s1.intersection(s3) == [Segment(Point(0.5, 0.5), p2)]
assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == \
[Segment(p1, Point(0.5, 0.5))]
assert r4.intersection(r5) == [s1]
assert r5.intersection(r6) == []
assert r4.intersection(r7) == r7.intersection(r4) == [r7]
# Segment contains
a, b = symbols('a,b', real=True)
s = Segment((0, a), (0, b))
assert Point(0, (a + b)/2) in s
s = Segment((a, 0), (b, 0))
assert Point((a + b)/2, 0) in s
raises(Undecidable, lambda: Point(2*a, 0) in s)
# Testing distance from a Segment to an object
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
pt1 = Point(0, 0)
pt2 = Point(Rational(3)/2, Rational(3)/2)
assert s1.distance(pt1) == 0
assert s1.distance((0, 0)) == 0
assert s2.distance(pt1) == 2**(half)/2
assert s2.distance(pt2) == 2**(half)
# Line to point
p1, p2 = Point(0, 0), Point(1, 1)
s = Line(p1, p2)
assert s.distance(Point(-1, 1)) == sqrt(2)
assert s.distance(Point(1, -1)) == sqrt(2)
assert s.distance(Point(2, 2)) == 0
assert s.distance((-1, 1)) == sqrt(2)
assert Line((0, 0), (0, 1)).distance(p1) == 0
assert Line((0, 0), (0, 1)).distance(p2) == 1
assert Line((0, 0), (1, 0)).distance(p1) == 0
assert Line((0, 0), (1, 0)).distance(p2) == 1
m = symbols('m', real=True)
l = Line((0, 5), slope=m)
p = Point(2, 3)
assert (l.distance(p) - 2*abs(m + 1)/sqrt(m**2 + 1)).equals(0)
# Ray to point
r = Ray(p1, p2)
assert r.distance(Point(-1, -1)) == sqrt(2)
assert r.distance(Point(1, 1)) == 0
assert r.distance(Point(-1, 1)) == sqrt(2)
assert Ray((1, 1), (2, 2)).distance(Point(1.5, 3)) == 3*sqrt(2)/4
assert r.distance((1, 1)) == 0
#Line contains
p1, p2 = Point(0, 1), Point(3, 4)
l = Line(p1, p2)
assert l.contains(p1) is True
assert l.contains((0, 1)) is True
assert l.contains((0, 0)) is False
#Ray contains
p1, p2 = Point(0, 0), Point(4, 4)
r = Ray(p1, p2)
assert r.contains(p1) is True
assert r.contains((1, 1)) is True
assert r.contains((1, 3)) is False
s = Segment((1, 1), (2, 2))
assert r.contains(s) is True
s = Segment((1, 2), (2, 5))
assert r.contains(s) is False
r1 = Ray((2, 2), (3, 3))
assert r.contains(r1) is True
r1 = Ray((2, 2), (3, 5))
assert r.contains(r1) is False
# Special cases of projection and intersection
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(2, 2), Point(0, 0))
r3 = Ray(Point(1, 1), Point(-1, -1))
r4 = Ray(Point(0, 4), Point(-1, -5))
r5 = Ray(Point(2, 2), Point(3, 3))
assert intersection(r1, r2) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, r3) == [Point(1, 1)]
assert r1.projection(r3) == Point(1, 1)
assert r1.projection(r4) == Segment(Point(1, 1), Point(2, 2))
r5 = Ray(Point(0, 0), Point(0, 1))
r6 = Ray(Point(0, 0), Point(0, 2))
assert r5 in r6
assert r6 in r5
s1 = Segment(Point(0, 0), Point(2, 2))
s2 = Segment(Point(-1, 5), Point(-5, -10))
s3 = Segment(Point(0, 4), Point(-2, 2))
assert intersection(r1, s1) == [Segment(Point(1, 1), Point(2, 2))]
assert r1.projection(s2) == Segment(Point(1, 1), Point(2, 2))
assert s3.projection(r1) == Segment(Point(0, 4), Point(-1, 3))
l1 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(0, 0), Point(3, 4))
s1 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, l1) == [l1]
assert intersection(l1, r1) == [r1]
assert intersection(l1, s1) == [s1]
assert intersection(r1, l1) == [r1]
assert intersection(s1, l1) == [s1]
entity1 = Segment(Point(-10, 10), Point(10, 10))
entity2 = Segment(Point(-5, -5), Point(-5, 5))
assert intersection(entity1, entity2) == []
r1 = Ray(p1, Point(0, 1))
r2 = Ray(Point(0, 1), p1)
r3 = Ray(p1, p2)
r4 = Ray(p2, p1)
s1 = Segment(p1, Point(0, 1))
assert Line(r1.source, r1.random_point()).slope == r1.slope
assert Line(r2.source, r2.random_point()).slope == r2.slope
assert Segment(Point(0, -1), s1.random_point()).slope == s1.slope
p_r3 = r3.random_point()
p_r4 = r4.random_point()
assert p_r3.x >= p1.x and p_r3.y >= p1.y
assert p_r4.x <= p2.x and p_r4.y <= p2.y
p10 = Point(2000, 2000)
s1 = Segment(p1, p10)
p_s1 = s1.random_point()
assert p1.x <= p_s1.x and p_s1.x <= p10.x and \
p1.y <= p_s1.y and p_s1.y <= p10.y
s2 = Segment(p10, p1)
assert hash(s1) == hash(s2)
p11 = p10.scale(2, 2)
assert s1.is_similar(Segment(p10, p11))
assert s1.is_similar(r1) is False
assert (r1 in s1) is False
assert Segment(p1, p2) in s1
assert s1.plot_interval() == [t, 0, 1]
assert s1 in Line(p1, p10)
assert Line(p1, p10) != Line(p10, p1)
assert Line(p1, p10) != p1
assert Line(p1, p10).plot_interval() == [t, -5, 5]
assert Ray((0, 0), angle=pi/4).plot_interval() == \
[t, 0, 10]