def test_ellipse_geom():
p1 = Point(0, 0)
p2 = Point(1, 1)
p4 = Point(0, 1)
e1 = Ellipse(p1, 1, 1)
e2 = Ellipse(p2, 0.5, 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: e3.arbitrary_point(y1))
pytest.raises(ValueError, lambda: e3.arbitrary_point(object()))
assert e1.ambient_dimension == 2
# Test creation with three points
cen, rad = Point(1.5, 2), Rational(5, 2)
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(0.5, 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 not c1.tangent_lines(p1)
assert e2.is_tangent(Line(p1_2, p2 + Point(0.5, 1)))
assert e2.is_tangent(Line(p1_3, p2 + Point(0.5, 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)))
assert (c1.normal_lines(Point(1, 1)) == [
Line(Point(-sqrt(2) / 2, -sqrt(2) / 2),
Point(-sqrt(2) / 2 + 1, -sqrt(2) / 2 + 1)),
Line(Point(sqrt(2) / 2, -sqrt(2) / 2),
Point(sqrt(2) / 2 + 1, -1 - sqrt(2) / 2))
])
# Properties
major = 3
minor = 1
e4 = Ellipse(p2, minor, major)
assert e4.focus_distance == sqrt(major**2 - minor**2)
ecc = e4.focus_distance / major
assert e4.eccentricity == ecc
assert e4.periapsis == major * (1 - ecc)
assert e4.apoapsis == major * (1 + ecc)
# independent of orientation
e4 = Ellipse(p2, major, minor)
assert e4.focus_distance == sqrt(major**2 - minor**2)
ecc = e4.focus_distance / major
assert e4.eccentricity == ecc
assert e4.periapsis == major * (1 - ecc)
assert e4.apoapsis == major * (1 + ecc)
# Intersection
l1 = Line(Point(1, -5), Point(1, 5))
l2 = Line(Point(-5, -1), Point(5, -1))
l3 = Line(Point(-1, -1), Point(1, 1))
l4 = Line(Point(-10, 0), Point(0, 10))
pts_c1_l3 = [
Point(sqrt(2) / 2,
sqrt(2) / 2),
Point(-sqrt(2) / 2, -sqrt(2) / 2)
]
assert intersection(e2, l4) == []
assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
assert intersection(c1, l1) == [Point(1, 0)]
assert intersection(c1, l2) == [Point(0, -1)]
assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
assert intersection(c1, c2) == [Point(0, 1), Point(1, 0)]
assert intersection(c1, c3) == [Point(sqrt(2) / 2, sqrt(2) / 2)]
assert e1.intersection(l1) == [Point(1, 0)]
assert e2.intersection(l4) == []
assert e1.intersection(Circle(Point(0, 2), 1)) == [Point(0, 1)]
assert e1.intersection(Circle(Point(5, 0), 1)) == []
assert e1.intersection(Ellipse(Point(2, 0), 1, 1)) == [Point(1, 0)]
assert e1.intersection(Ellipse(
Point(5, 0),
1,
1,
)) == []
assert e1.intersection(Point(2, 0)) == []
assert e1.intersection(e1) == e1
assert e2.intersection(e2) == e2
assert e2.intersection(Circle(Point(0, 0), 10)) == []
pytest.raises(NotImplementedError,
lambda: e2.intersection(Curve((t**2, t), (t, 0, 1))))
# 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 not 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_line_geom():
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)
pytest.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))
pytest.raises(ValueError, 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(5):
assert l3.random_point() in l3
pytest.raises(ValueError, lambda: l3.arbitrary_point('x1'))
assert Line(Point(0, 0), Point(1, 0)).is_similar(
Line(Point(1, 0), Point(2, 0))) is True
assert l1.equal(l1) is True
assert l1.equal(l2) is True
assert l1.equal(l3) is False
assert l1.equal(object()) is False
# Orthogonality
p1_1 = Point(-x1, x1)
l1_1 = Line(p1, p1_1)
assert l1.perpendicular_line(p1.args) == Line(Point(0, 0), Point(1, -1))
assert l1.perpendicular_line(p1) == 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
assert l4.perpendicular_line(p2) == Line(Point(1, 1), Point(1, 0))
# Parallelity
l2_1 = Line(p3, p5)
assert l2.parallel_line(p1_1) == Line(Point(-x1, x1),
Point(-y1, 2 * x1 - y1))
assert l2_1.parallel_line(p1.args) == Line(Point(0, 0), Point(0, -1))
assert l2_1.parallel_line(p1) == 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, 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)
pytest.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)
pytest.raises(ValueError, lambda: Ray((1, 1), I))
pytest.raises(ValueError, lambda: Ray(p1, p1))
pytest.raises(ValueError, lambda: Ray(p1))
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)))
pytest.raises(ValueError, lambda: Ray((1, 1), 1))
# issue sympy/sympy#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', extended_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)
assert s1.perpendicular_bisector() == \
Line(Point(1/2, 1/2), Point(3/2, -1/2))
# 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
pytest.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')
l = Line((0, 5), slope=m)
p = Point(2, 3)
assert l.distance(p) == 2 * abs(m + 1) / sqrt(m**2 + 1)
# 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
assert r.distance((-1, Rational(1, 2))) == sqrt(5) / 2
# 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
assert l.contains(Circle(p1, 1)) is False
assert l.contains(Ray(p1, p1 + p2)) 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
assert r.contains(object()) 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
r1 = Ray(p1, angle=-pi)
assert r1.contains(Point(1, 0)) is False
r1 = Ray(p1, angle=-pi / 2)
assert r1.contains(Point(0, 1)) is False
pytest.raises(Undecidable, lambda: r1.contains(Point(0, x)))
# 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]
p1, p2 = Point(0, 0), Point(4, 1)
r1 = Ray(p1, p2)
assert r1.direction == p2
p1, p2, p3 = Point(0, 0), Point(-1, -1), Point(-1, 0)
r1, r2 = Ray(p1, p2), Ray(p1, p3)
assert r1.ydirection == -oo
assert r2.ydirection == 0
p1, p2, p3 = Point(0, 0), Point(6, 6), Point(5, 1)
s1 = Segment(p1, p2)
assert s1.perpendicular_bisector() == Line(Point(3, 3), Point(9, -3))
assert s1.perpendicular_bisector(p3) == Segment(Point(3, 3), Point(5, 1))
def test_util():
# coverage for some leftover functions in diofant.geometry.util
assert intersection(Point(0, 0)) == []
pytest.raises(ValueError, lambda: intersection(Point(0, 0), 3))
pytest.raises(ValueError, lambda: convex_hull(Point(0, 0), 3))
def test_polygon():
a, b, c = Point(0, 0), Point(2, 0), Point(3, 3)
t = Triangle(a, b, c)
assert Polygon(a) == a
assert Polygon(a, a) == a
assert Polygon(a, b, b, c) == Polygon(a, b, c)
assert Polygon(a, 1, 1, n=4) == RegularPolygon(a, 1, 4, 1)
assert Polygon(a, Point(1, 0), b, c) == t
assert Polygon(Point(1, 0), b, c, a) == t
assert Polygon(b, c, a, Point(1, 0)) == t
# 2 "remove folded" tests
assert Polygon(a, Point(3, 0), b, c) == t
assert Polygon(a, b, Point(3, -1), b, c) == t
pytest.raises(GeometryError, lambda: Polygon((0, 0), (1, 0), (0, 1),
(1, 1)))
# remove multiple collinear points
assert Polygon(Point(-4, 15), Point(-11, 15), Point(-15, 15),
Point(-15, 33/5), Point(-15, -87/10), Point(-15, -15),
Point(-42/5, -15), Point(-2, -15), Point(7, -15), Point(15, -15),
Point(15, -3), Point(15, 10), Point(15, 15)) == \
Polygon(Point(-15, -15), Point(15, -15), Point(15, 15), Point(-15, 15))
p1 = Polygon(Point(0, 0), Point(3, -1), Point(6, 0), Point(4, 5),
Point(2, 3), Point(0, 3))
p2 = Polygon(Point(6, 0), Point(3, -1), Point(0, 0), Point(0, 3),
Point(2, 3), Point(4, 5))
p3 = Polygon(Point(0, 0), Point(3, 0), Point(5, 2), Point(4, 4))
p4 = Polygon(Point(0, 0), Point(4, 4), Point(5, 2), Point(3, 0))
p5 = Polygon(Point(0, 0), Point(4, 4), Point(0, 4))
p6 = Polygon(Point(-11, 1), Point(-9, 6.6), Point(-4, -3),
Point(-8.4, -8.7))
r = Ray(Point(-9, 6.6), Point(-9, 5.5))
#
# General polygon
#
assert p1 == p2
assert len(p1.args) == 6
assert len(p1.sides) == 6
assert p1.perimeter == 5 + 2 * sqrt(10) + sqrt(29) + sqrt(8)
assert p1.area == 22
assert not p1.is_convex()
assert p1.contains(Segment((0, 0), (1, 2))) is False
assert p1.contains(Ray((0, 0), angle=pi / 3)) is False
# ensure convex for both CW and CCW point specification
assert p3.is_convex()
assert p4.is_convex()
dict5 = p5.angles
assert dict5[Point(0, 0)] == pi / 4
assert dict5[Point(0, 4)] == pi / 2
assert p5.encloses_point(Point(x, y)) is None
assert p5.encloses_point(Point(1, 3))
assert p5.encloses_point(Point(0, 0)) is False
assert p5.encloses_point(Point(4, 0)) is False
assert p1.encloses(Circle(Point(2.5, 2.5), 5)) is False
assert p1.encloses(Ellipse(Point(2.5, 2), 5, 6)) is False
assert p5.plot_interval('x') == [x, 0, 1]
assert p5.distance(Polygon(Point(10, 10), Point(14, 14),
Point(10, 14))) == 6 * sqrt(2)
assert p5.distance(
Polygon(Point(1, 8), Point(5, 8), Point(8, 12), Point(1, 12))) == 4
p7 = Polygon(Point(1, 2), Point(3, 7), Point(0, 1))
assert p5.distance(p7) == 9 * sqrt(29) / 29
l1 = Line(Point(0, 0), Point(1, 0))
assert p5.reflect(l1).distance(p7.reflect(l1)) == 9 * sqrt(29) / 29
warnings.filterwarnings(
'error', message='Polygons may intersect producing erroneous output')
pytest.raises(
UserWarning,
lambda: Polygon(Point(0, 0), Point(1, 0), Point(1, 1)).distance(
Polygon(Point(0, 0), Point(0, 1), Point(1, 1))))
warnings.filterwarnings(
'ignore', message='Polygons may intersect producing erroneous output')
assert hash(p5) == hash(Polygon(Point(0, 0), Point(4, 4), Point(0, 4)))
assert p5 == Polygon(Point(4, 4), Point(0, 4), Point(0, 0))
assert Polygon(Point(4, 4), Point(0, 4), Point(0, 0)) in p5
assert p5 != Point(0, 4)
assert Point(0, 1) in p5
assert p5.arbitrary_point('t').subs({Symbol('t', extended_real=True): 0}) == \
Point(0, 0)
pytest.raises(
ValueError, lambda: Polygon(Point(x, 0), Point(0, y), Point(x, y)).
arbitrary_point('x'))
assert p6.intersection(r) == [Point(-9, 33 / 5), Point(-9, -84 / 13)]
#
# Regular polygon
#
p1 = RegularPolygon(Point(0, 0), 10, 5)
p2 = RegularPolygon(Point(0, 0), 5, 5)
pytest.raises(
GeometryError,
lambda: RegularPolygon(Point(0, 0), Point(0, 1), Point(1, 1)))
pytest.raises(GeometryError, lambda: RegularPolygon(Point(0, 0), 1, 2))
pytest.raises(ValueError, lambda: RegularPolygon(Point(0, 0), 1, 2.5))
assert Polygon(Point(0, 0), 10, 5, pi,
n=5) == RegularPolygon(Point(0, 0), 10, 5, pi)
assert p1 != p2
assert p1.interior_angle == 3 * pi / 5
assert p1.exterior_angle == 2 * pi / 5
assert p2.apothem == 5 * cos(pi / 5)
assert p2.circumcenter == p1.circumcenter == Point(0, 0)
assert p1.circumradius == p1.radius == 10
assert p2.circumcircle == Circle(Point(0, 0), 5)
assert p2.incircle == Circle(Point(0, 0), p2.apothem)
assert p2.inradius == p2.apothem == (5 * (1 + sqrt(5)) / 4)
p2.spin(pi / 10)
dict1 = p2.angles
assert dict1[Point(0, 5)] == 3 * pi / 5
assert p1.is_convex()
assert p1.rotation == 0
assert p1.encloses_point(Point(0, 0))
assert p1.encloses_point(Point(11, 0)) is False
assert p2.encloses_point(Point(0, 4.9))
p1.spin(pi / 3)
assert p1.rotation == pi / 3
assert p1.vertices[0] == Point(5, 5 * sqrt(3))
for var in p1.args:
if isinstance(var, Point):
assert var == Point(0, 0)
else:
assert var in (5, 10, pi / 3)
assert p1 != Point(0, 0)
assert p1 != p5
# while spin works in place (notice that rotation is 2pi/3 below)
# rotate returns a new object
p1_old = p1
assert p1.rotate(pi / 3) == RegularPolygon(Point(0, 0), 10, 5, 2 * pi / 3)
assert p1 == p1_old
assert p1.area == (-250 * sqrt(5) + 1250) / (4 * tan(pi / 5))
assert p1.length == 20 * sqrt(-sqrt(5) / 8 + 5 / 8)
assert p1.scale(2, 2) == \
RegularPolygon(p1.center, p1.radius*2, p1._n, p1.rotation)
assert RegularPolygon((0, 0), 1, 4).scale(2, 3) == \
Polygon(Point(2, 0), Point(0, 3), Point(-2, 0), Point(0, -3))
assert repr(p1) == str(p1)
#
# Angles
#
angles = p4.angles
assert feq(angles[Point(0, 0)].evalf(), Float('0.7853981633974483'))
assert feq(angles[Point(4, 4)].evalf(), Float('1.2490457723982544'))
assert feq(angles[Point(5, 2)].evalf(), Float('1.8925468811915388'))
assert feq(angles[Point(3, 0)].evalf(), Float('2.3561944901923449'))
angles = p3.angles
assert feq(angles[Point(0, 0)].evalf(), Float('0.7853981633974483'))
assert feq(angles[Point(4, 4)].evalf(), Float('1.2490457723982544'))
assert feq(angles[Point(5, 2)].evalf(), Float('1.8925468811915388'))
assert feq(angles[Point(3, 0)].evalf(), Float('2.3561944901923449'))
assert (Polygon((0, 0), (10, 0), (2, 1), (0, 3)).angles == {
Point(0, 0): pi / 2,
Point(0, 3): pi / 4,
Point(2, 1): -acos(-9 * sqrt(130) / 130) + 2 * pi,
Point(10, 0): acos(8 * sqrt(65) / 65)
})
#
# Triangle
#
p1 = Point(0, 0)
p2 = Point(5, 0)
p3 = Point(0, 5)
t1 = Triangle(p1, p2, p3)
t2 = Triangle(p1, p2, Point(Rational(5, 2), sqrt(Rational(75, 4))))
t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
s1 = t1.sides
assert Triangle(p1, p2, p1) == Polygon(p1, p2, p1) == Segment(p1, p2)
pytest.raises(GeometryError, lambda: Triangle(Point(0, 0)))
# Basic stuff
assert Triangle(p1, p1, p1) == p1
assert Triangle(p2, p2 * 2, p2 * 3) == Segment(p2, p2 * 3)
assert t1.area == Rational(25, 2)
assert t1.is_right()
assert t2.is_right() is False
assert t3.is_right()
assert p1 in t1
assert t1.sides[0] in t1
assert Segment((0, 0), (1, 0)) in t1
assert Point(5, 5) not in t2
assert t1.is_convex()
assert feq(t1.angles[p1].evalf(), pi.evalf() / 2)
assert t1.is_equilateral() is False
assert t2.is_equilateral()
assert t3.is_equilateral() is False
assert are_similar(t1, t2) is False
assert are_similar(t1, t3)
assert are_similar(t2, t3) is False
assert t1.is_similar(Point(0, 0)) is False
# Bisectors
bisectors = t1.bisectors()
assert bisectors[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
ic = (250 - 125 * sqrt(2)) / 50
assert t1.incenter == Point(ic, ic)
# Inradius
assert t1.inradius == t1.incircle.radius == 5 - 5 * sqrt(2) / 2
assert t2.inradius == t2.incircle.radius == 5 * sqrt(3) / 6
assert t3.inradius == t3.incircle.radius == x1**2 / (
(2 + sqrt(2)) * abs(x1))
# Circumcircle
assert t1.circumcircle.center == Point(2.5, 2.5)
# Medians + Centroid
m = t1.medians
assert t1.centroid == Point(Rational(5, 3), Rational(5, 3))
assert m[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
assert t3.medians[p1] == Segment(p1, Point(x1 / 2, x1 / 2))
assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]
assert t1.medial == Triangle(Point(2.5, 0), Point(0, 2.5), Point(2.5, 2.5))
# Perpendicular
altitudes = t1.altitudes
assert altitudes[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
assert altitudes[p2] == s1[0]
assert altitudes[p3] == s1[2]
assert t1.orthocenter == p1
t = Triangle(
Point(Rational(100080156402737, 5000000000000),
Rational(79782624633431, 500000000000)),
Point(Rational(39223884078253, 2000000000000),
Rational(156345163124289, 1000000000000)),
Point(Rational(31241359188437, 1250000000000),
Rational(338338270939941, 1000000000000000)))
assert t.orthocenter == \
Point(Rational(-78066086905059984021699779471538701955848721853,
80368430960602242240789074233100000000000000),
Rational(20151573611150265741278060334545897615974257,
160736861921204484481578148466200000000000))
# Ensure
assert len(intersection(*bisectors.values())) == 1
assert len(intersection(*altitudes.values())) == 1
assert len(intersection(*m.values())) == 1
# Distance
p1 = Polygon(Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1))
p2 = Polygon(Point(0, Rational(5, 4)), Point(1, Rational(5, 4)),
Point(1, Rational(9, 4)), Point(0, Rational(9, 4)))
p3 = Polygon(Point(1, 2), Point(2, 2), Point(2, 1))
p4 = Polygon(Point(1, 1), Point(Rational(6, 5), 1),
Point(1, Rational(6, 5)))
pt1 = Point(0.5, 0.5)
pt2 = Point(1, 1)
# Polygon to Point
assert p1.distance(pt1) == Rational(1, 2)
assert p1.distance(pt2) == 0
assert p2.distance(pt1) == Rational(3, 4)
assert p3.distance(pt2) == sqrt(2) / 2
# Polygon to Polygon
# p1.distance(p2) emits a warning
# First, test the warning
warnings.filterwarnings(
'error', message='Polygons may intersect producing erroneous output')
pytest.raises(UserWarning, lambda: p1.distance(p2))
# now test the actual output
warnings.filterwarnings(
'ignore', message='Polygons may intersect producing erroneous output')
assert p1.distance(p2) == Rational(1, 4)
assert p1.distance(p3) == sqrt(2) / 2
assert p3.distance(p4) == 2 * sqrt(2) / 5
r = Polygon(Point(0, 0), 1, n=3)
assert r.vertices[0] == Point(1, 0)
mid = Point(1, 1)
assert Polygon((0, 2), (2, 2), mid, (0, 0), (2, 0), mid).area == 0
t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4))
assert t1.is_isosceles() is True
t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4))
assert t1.is_scalene() is True
assert t1.is_isosceles() is False
p1 = Polygon((1, 0), (2, 0), (2, 2), (-4, 3))
p2 = Polygon((1, 0), (2, 0), (3, 2), (-4, 3))
assert (p1 == p2) is False
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_line_geom():
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)
pytest.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))
pytest.raises(ValueError, 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(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) == Line(Point(0, 0), Point(1, -1))
assert l1.perpendicular_line(p1) == 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) == Line(Point(-x1, x1), Point(-y1, 2*x1 - y1))
assert l2_1.parallel_line(p1.args) == Line(Point(0, 0), Point(0, -1))
assert l2_1.parallel_line(p1) == 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, 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)
pytest.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)
pytest.raises(ValueError, lambda: Ray((1, 1), I))
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)))
pytest.raises(ValueError, lambda: Ray((1, 1), 1))
# issue sympy/sympy#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', extended_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)
assert s1.perpendicular_bisector() == \
Line(Point(1/2, 1/2), Point(3/2, -1/2))
# 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
pytest.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')
l = Line((0, 5), slope=m)
p = Point(2, 3)
assert l.distance(p) == 2*abs(m + 1)/sqrt(m**2 + 1)
# 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_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
