def test_eulerline():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).eulerline \
== Line(Point2D(0, 0), Point2D(S(1)/2, S(1)/2))
assert Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3))).eulerline \
== Point2D(5, 5*sqrt(3)/3)
assert Triangle(Point(4, -6), Point(4, -1), Point(-3, 3)).eulerline \
== Line(Point2D(S(64)/7, 3), Point2D(-S(29)/14, -S(7)/2))
def test_intersection():
poly1 = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
poly2 = Polygon(Point(0, 1), Point(-5, 0),
Point(0, -4), Point(0, S(1)/5),
Point(S(1)/2, -0.1), Point(1,0), Point(0, 1))
assert poly1.intersection(poly2) == [Point2D(S(1)/3, 0),
Segment(Point(0, S(1)/5), Point(0, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly1) == [Point(S(1)/3, 0),
Segment(Point(0, 0), Point(0, S(1)/5)),
Segment(Point(1, 0), Point(0, 1))]
assert poly1.intersection(Point(0, 0)) == [Point(0, 0)]
assert poly1.intersection(Point(-12, -43)) == []
assert poly2.intersection(Line((-12, 0), (12, 0))) == [Point(-5, 0),
Point(0, 0),Point(S(1)/3, 0), Point(1, 0)]
assert poly2.intersection(Line((-12, 12), (12, 12))) == []
assert poly2.intersection(Ray((-3,4), (1,0))) == [Segment(Point(1, 0),
Point(0, 1))]
assert poly2.intersection(Circle((0, -1), 1)) == [Point(0, -2),
Point(0, 0)]
assert poly1.intersection(poly1) == [Segment(Point(0, 0), Point(1, 0)),
Segment(Point(0, 1), Point(0, 0)), Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly2) == [Segment(Point(-5, 0), Point(0, -4)),
Segment(Point(0, -4), Point(0, S(1)/5)),
Segment(Point(0, S(1)/5), Point(S(1)/2, -S(1)/10)),
Segment(Point(0, 1), Point(-5, 0)),
Segment(Point(S(1)/2, -S(1)/10), Point(1, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(Triangle(Point(0, 1), Point(1, 0), Point(-1, 1))) \
== [Point(-S(5)/7, S(6)/7), Segment(Point2D(0, 1), Point(1, 0))]
assert poly1.intersection(RegularPolygon((-12, -15), 3, 3)) == []
def test_intersection():
poly1 = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
poly2 = Polygon(Point(0, 1), Point(-5, 0),
Point(0, -4), Point(0, Rational(1, 5)),
Point(S.Half, -0.1), Point(1, 0), Point(0, 1))
assert poly1.intersection(poly2) == [Point2D(Rational(1, 3), 0),
Segment(Point(0, Rational(1, 5)), Point(0, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly1) == [Point(Rational(1, 3), 0),
Segment(Point(0, 0), Point(0, Rational(1, 5))),
Segment(Point(1, 0), Point(0, 1))]
assert poly1.intersection(Point(0, 0)) == [Point(0, 0)]
assert poly1.intersection(Point(-12, -43)) == []
assert poly2.intersection(Line((-12, 0), (12, 0))) == [Point(-5, 0),
Point(0, 0), Point(Rational(1, 3), 0), Point(1, 0)]
assert poly2.intersection(Line((-12, 12), (12, 12))) == []
assert poly2.intersection(Ray((-3, 4), (1, 0))) == [Segment(Point(1, 0),
Point(0, 1))]
assert poly2.intersection(Circle((0, -1), 1)) == [Point(0, -2),
Point(0, 0)]
assert poly1.intersection(poly1) == [Segment(Point(0, 0), Point(1, 0)),
Segment(Point(0, 1), Point(0, 0)), Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly2) == [Segment(Point(-5, 0), Point(0, -4)),
Segment(Point(0, -4), Point(0, Rational(1, 5))),
Segment(Point(0, Rational(1, 5)), Point(S.Half, Rational(-1, 10))),
Segment(Point(0, 1), Point(-5, 0)),
Segment(Point(S.Half, Rational(-1, 10)), Point(1, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(Triangle(Point(0, 1), Point(1, 0), Point(-1, 1))) \
== [Point(Rational(-5, 7), Rational(6, 7)), Segment(Point2D(0, 1), Point(1, 0))]
assert poly1.intersection(RegularPolygon((-12, -15), 3, 3)) == []
def test_eulerline():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).eulerline \
== Line(Point2D(0, 0), Point2D(1/2, 1/2))
assert Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3))).eulerline \
== Point2D(5, 5*sqrt(3)/3)
assert Triangle(Point(4, -6), Point(4, -1), Point(-3, 3)).eulerline \
== Line(Point2D(64/7, 3), Point2D(-29/14, -7/2))
def test_eulerline():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).eulerline \
== Line(Point2D(0, 0), Point2D(S.Half, S.Half))
assert Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3))).eulerline \
== Point2D(5, 5*sqrt(3)/3)
assert Triangle(Point(4, -6), Point(4, -1), Point(-3, 3)).eulerline \
== Line(Point2D(Rational(64, 7), 3), Point2D(Rational(-29, 14), Rational(-7, 2)))
def test_transform():
pts = [Point(0, 0), Point(S.Half, Rational(1, 4)), Point(1, 1)]
pts_out = [Point(-4, -10), Point(-3, Rational(-37, 4)), Point(-2, -7)]
assert Triangle(*pts).scale(2, 3, (4, 5)) == Triangle(*pts_out)
assert RegularPolygon((0, 0), 1, 4).scale(2, 3, (4, 5)) == \
Polygon(Point(-2, -10), Point(-4, -7), Point(-6, -10), Point(-4, -13))
# Checks for symmetric scaling
assert RegularPolygon((0, 0), 1, 4).scale(2, 2) == \
RegularPolygon(Point2D(0, 0), 2, 4, 0)
def test_Geometry():
assert sstr(Point(0, 0)) == 'Point2D(0, 0)'
assert sstr(Circle(Point(0, 0), 3)) == 'Circle(Point2D(0, 0), 3)'
assert sstr(Ellipse(Point(1, 2), 3, 4)) == 'Ellipse(Point2D(1, 2), 3, 4)'
assert sstr(Triangle(Point(1, 1), Point(7, 8), Point(0, -1))) == \
'Triangle(Point2D(1, 1), Point2D(7, 8), Point2D(0, -1))'
assert sstr(Polygon(Point(5, 6), Point(-2, -3), Point(0, 0), Point(4, 7))) == \
'Polygon(Point2D(5, 6), Point2D(-2, -3), Point2D(0, 0), Point2D(4, 7))'
assert sstr(Triangle(Point(0, 0), Point(1, 0), Point(0, 1)), sympy_integers=True) == \
'Triangle(Point2D(S(0), S(0)), Point2D(S(1), S(0)), Point2D(S(0), S(1)))'
assert sstr(Ellipse(Point(1, 2), 3, 4), sympy_integers=True) == \
'Ellipse(Point2D(S(1), S(2)), S(3), S(4))'
def test_subs():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
p = Point(x, 2)
q = Point(1, 1)
r = Point(3, 4)
for o in [
p,
Segment(p, q),
Ray(p, q),
Line(p, q),
Triangle(p, q, r),
RegularPolygon(p, 3, 6),
Polygon(p, q, r, Point(5, 4)),
Circle(p, 3),
Ellipse(p, 3, 4)
]:
assert 'y' in str(o.subs(x, y))
assert p.subs({x: 1}) == Point(1, 2)
assert Point(1, 2).subs(Point(1, 2), Point(3, 4)) == Point(3, 4)
assert Point(1, 2).subs((1, 2), Point(3, 4)) == Point(3, 4)
assert Point(1, 2).subs(Point(1, 2), Point(3, 4)) == Point(3, 4)
assert Point(1, 2).subs(set([(1, 2)])) == Point(2, 2)
raises(ValueError, lambda: Point(1, 2).subs(1))
raises(ValueError, lambda: Point(1, 1).subs(
(Point(1, 1), Point(1, 2)), 1, 2))
def test_reflect():
b = Symbol('b')
m = Symbol('m')
l = Line((0, b), slope=m)
p = Point(x, y)
r = p.reflect(l)
dp = l.perpendicular_segment(p).length
dr = l.perpendicular_segment(r).length
assert test_numerically(dp, dr)
t = Triangle((0, 0), (1, 0), (2, 3))
assert t.area == -t.reflect(l).area
e = Ellipse((1, 0), 1, 2)
assert e.area == -e.reflect(Line((1, 0), slope=0)).area
assert e.area == -e.reflect(Line((1, 0), slope=oo)).area
raises(NotImplementedError, lambda: e.reflect(Line((1, 0), slope=m)))
# test entity overrides
c = Circle((x, y), 3)
cr = c.reflect(l)
assert cr == Circle(r, -3)
assert c.area == -cr.area
pent = RegularPolygon((1, 2), 1, 5)
l = Line((0, pi), slope=sqrt(2))
rpent = pent.reflect(l)
poly_pent = Polygon(*pent.vertices)
assert rpent.center == pent.center.reflect(l)
assert str([w.n(3) for w in rpent.vertices
]) == ('[Point(-0.586, 4.27), Point(-1.69, 4.66), '
'Point(-2.41, 3.73), Point(-1.74, 2.76), '
'Point(-0.616, 3.10)]')
assert pent.area.equals(-rpent.area)
def test_reflect():
b = Symbol('b')
m = Symbol('m')
l = Line((0, b), slope=m)
t1 = Triangle((0, 0), (1, 0), (2, 3))
assert t1.area == -t1.reflect(l).area
e = Ellipse((1, 0), 1, 2)
assert e.area == -e.reflect(Line((1, 0), slope=0)).area
assert e.area == -e.reflect(Line((1, 0), slope=oo)).area
raises(NotImplementedError, lambda: e.reflect(Line((1, 0), slope=m)))
def test_triangle_kwargs():
assert Triangle(sss=(3, 4, 5)) == \
Triangle(Point(0, 0), Point(3, 0), Point(3, 4))
assert Triangle(asa=(30, 2, 30)) == \
Triangle(Point(0, 0), Point(2, 0), Point(1, sqrt(3)/3))
assert Triangle(sas=(1, 45, 2)) == \
Triangle(Point(0, 0), Point(2, 0), Point(sqrt(2)/2, sqrt(2)/2))
assert Triangle(sss=(1, 2, 5)) is None
assert deg(rad(180)) == 180
def test_reflect():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
b = Symbol('b')
m = Symbol('m')
l = Line((0, b), slope=m)
p = Point(x, y)
r = p.reflect(l)
dp = l.perpendicular_segment(p).length
dr = l.perpendicular_segment(r).length
assert verify_numerically(dp, dr)
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=oo)) \
== Triangle(Point(5, 0), Point(4, 0), Point(4, 2))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=oo)) \
== Triangle(Point(-1, 0), Point(-2, 0), Point(-2, 2))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=0)) \
== Triangle(Point(1, 6), Point(2, 6), Point(2, 4))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=0)) \
== Triangle(Point(1, 0), Point(2, 0), Point(2, -2))
def intri(p, a, b, c):
p = points[p]
a = points[a]
b = points[b]
c = points[c]
t = Triangle(a, b, c)
if hasattr(t, "encloses_point") and t.encloses_point(p):
return 1
if onbounds(p, a, b, c):
return 0
return -1
def test_subs():
p = Point(x, 2)
q = Point(1, 1)
r = Point(3, 4)
for o in [p,
Segment(p, q),
Ray(p, q),
Line(p, q),
Triangle(p, q, r),
RegularPolygon(p, 3, 6),
Polygon(p, q, r, Point(5,4)),
Circle(p, 3),
Ellipse(p, 3, 4)]:
assert 'y' in str(o.subs(x, y))
def incircle(p, a, b, c):
p = points[p]
a = points[a]
b = points[b]
c = points[c]
t = Triangle(a, b, c)
if hasattr(t, "circumcircle") and t.circumcircle.encloses_point(p):
return 1
if hasattr(t, "circumcircle") and intersection(
t.circumcenter,
p): #and t.circumcenter.distance(p) == t.circumradius:
return 0
return -1
def test_geometry_transforms():
from sympy import Tuple
c = Curve((x, x**2), (x, 0, 1))
pts = [Point(0, 0), Point(S(1) / 2, S(1) / 4), Point(1, 1)]
cout = Curve((2 * x - 4, 3 * x**2 - 10), (x, 0, 1))
pts_out = [Point(-4, -10), Point(-3, -S(37) / 4), Point(-2, -7)]
assert c.scale(2, 3, (4, 5)) == cout
assert [c.subs(x, xi / 2) for xi in Tuple(0, 1, 2)] == pts
assert [cout.subs(x, xi / 2) for xi in Tuple(0, 1, 2)] == pts_out
assert Triangle(*pts).scale(2, 3, (4, 5)) == Triangle(*pts_out)
assert Ellipse((0, 0), 2, 3).scale(2, 3, (4, 5)) == \
Ellipse(Point(-4, -10), 4, 9)
assert Circle((0, 0), 2).scale(2, 3, (4, 5)) == \
Ellipse(Point(-4, -10), 4, 6)
assert Ellipse((0, 0), 2, 3).scale(3, 3, (4, 5)) == \
Ellipse(Point(-8, -10), 6, 9)
assert Circle((0, 0), 2).scale(3, 3, (4, 5)) == \
Circle(Point(-8, -10), 6)
assert Circle(Point(-8, -10), 6).scale(S(1)/3, S(1)/3, (4, 5)) == \
Circle((0, 0), 2)
assert Curve((x + y, 3*x), (x, 0, 1)).subs(y, S.Half) == \
Curve((x + S(1)/2, 3*x), (x, 0, 1))
assert Curve((x, 3*x), (x, 0, 1)).translate(4, 5) == \
Curve((x + 4, 3*x + 5), (x, 0, 1))
assert Circle((0, 0), 2).translate(4, 5) == \
Circle((4, 5), 2)
assert Circle((0, 0), 2).scale(3, 3) == \
Circle((0, 0), 6)
assert Point(1, 1).scale(2, 3, (4, 5)) == \
Point(-2, -7)
assert Point(1, 1).translate(4, 5) == \
Point(5, 6)
assert scale(1, 2, (3, 4)).tolist() == \
[[1, 0, 0], [0, 2, 0], [0, -4, 1]]
assert RegularPolygon((0, 0), 1, 4).scale(2, 3, (4, 5)) == \
Polygon(Point(-2, -10), Point(-4, -7), Point(-6, -10), Point(-4, -13))
def test_bisectors():
p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
p = Polygon(Point(0, 0), Point(2, 0), Point(1, 1), Point(0, 3))
q = Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(-1, 5))
poly = Polygon(Point(3, 4), Point(0, 0), Point(8, 7), Point(-1, 1), Point(19, -19))
t = Triangle(p1, p2, p3)
assert t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1))
assert p.bisectors()[Point2D(0, 3)] == Ray2D(Point2D(0, 3), \
Point2D(sin(acos(2*sqrt(5)/5)/2), 3 - cos(acos(2*sqrt(5)/5)/2)))
assert q.bisectors()[Point2D(-1, 5)] == \
Ray2D(Point2D(-1, 5), Point2D(-1 + sqrt(29)*(5*sin(acos(9*sqrt(145)/145)/2) + \
2*cos(acos(9*sqrt(145)/145)/2))/29, sqrt(29)*(-5*cos(acos(9*sqrt(145)/145)/2) + \
2*sin(acos(9*sqrt(145)/145)/2))/29 + 5))
assert poly.bisectors()[Point2D(-1, 1)] == Ray2D(Point2D(-1, 1), \
Point2D(-1 + sin(acos(sqrt(26)/26)/2 + pi/4), 1 - sin(-acos(sqrt(26)/26)/2 + pi/4)))
def test_line_intersection():
assert asa(120, 8, 52) == \
Triangle(
Point(0, 0),
Point(8, 0),
Point(-4 * cos(19 * pi / 90) / sin(2 * pi / 45),
4 * sqrt(3) * cos(19 * pi / 90) / sin(2 * pi / 45)))
assert Line((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)]
assert Line((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (10, 10)).contains(Segment((1, 1), (2, 2))) is True
assert Segment((1, 1), (2, 2)) in Line((0, 0), (10, 10))
x = 8 * tan(13 * pi / 45) / (tan(13 * pi / 45) + sqrt(3))
y = (-8 * sqrt(3) * tan(13 * pi / 45) ** 2 + 24 * tan(13 * pi / 45)) / (-3 + tan(13 * pi / 45) ** 2)
assert Line(Point(0, 0), Point(1, -sqrt(3))).contains(Point(x, y)) is True
def test_geometry_EvalfMixin():
x = pi
t = Symbol('t')
for g in [
Point(x, x),
Plane(Point(0, x, 0), (0, 0, x)),
Curve((x * t, x), (t, 0, x)),
Ellipse((x, x), x, -x),
Circle((x, x), x),
Line((0, x), (x, 0)),
Segment((0, x), (x, 0)),
Ray((0, x), (x, 0)),
Parabola((0, x), Line((-x, 0), (x, 0))),
Polygon((0, 0), (0, x), (x, 0), (x, x)),
RegularPolygon((0, x), x, 4, x),
Triangle((0, 0), (x, 0), (x, x)),
]:
assert str(g).replace('pi', '3.1') == str(g.n(2))
def test_ellipse():
p1 = Point(0, 0)
p2 = Point(1, 1)
p4 = Point(0, 1)
e1 = Ellipse(p1, 1, 1)
e2 = Ellipse(p2, half, 1)
e3 = Ellipse(p1, y1, y1)
c1 = Circle(p1, 1)
c2 = Circle(p2, 1)
c3 = Circle(Point(sqrt(2), sqrt(2)), 1)
# Test creation with three points
cen, rad = Point(3 * half, 2), 5 * half
assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
raises(GeometryError, "Circle(Point(0,0), Point(1,1), Point(2,2))")
raises(ValueError, "Ellipse(None, None, None, 1)")
raises(GeometryError, "Circle(Point(0,0))")
# Basic Stuff
assert Ellipse(None, 1, 1).center == Point(0, 0)
assert e1 == c1
assert e1 != e2
assert p4 in e1
assert p2 not in e2
assert e1.area == pi
assert e2.area == pi / 2
assert e3.area == pi * (y1**2)
assert c1.area == e1.area
assert c1.circumference == e1.circumference
assert e3.circumference == 2 * pi * y1
assert e1.plot_interval() == e2.plot_interval() == [t, -pi, pi]
assert e1.plot_interval(x) == e2.plot_interval(x) == [x, -pi, pi]
assert Ellipse(None, 1, None, 1).circumference == 2 * pi
assert c1.minor == 1
# Private Functions
assert hash(c1) == hash(Circle(Point(1, 0), Point(0, 1), Point(0, -1)))
assert c1 in e1
assert (Line(p1, p2) in e1) == False
assert e1.__cmp__(e1) == 0
assert e1.__cmp__(Point(0, 0)) > 0
# Encloses
assert e1.encloses(Segment(Point(-0.5, -0.5), Point(0.5, 0.5))) == True
assert e1.encloses(Line(p1, p2)) == False
assert e1.encloses(Ray(p1, p2)) == False
assert e1.encloses(e1) == False
assert e1.encloses(
Polygon(Point(-0.5, -0.5), Point(-0.5, 0.5), Point(0.5, 0.5))) == True
assert e1.encloses(RegularPolygon(p1, 0.5, 3)) == True
assert e1.encloses(RegularPolygon(p1, 5, 3)) == False
assert e1.encloses(RegularPolygon(p2, 5, 3)) == False
# with generic symbols, the hradius is assumed to contain the major radius
M = Symbol('M')
m = Symbol('m')
c = Ellipse(p1, M, m).circumference
_x = c.atoms(Dummy).pop()
assert c == \
4*M*C.Integral(sqrt((1 - _x**2*(M**2 - m**2)/M**2)/(1 - _x**2)), (_x, 0, 1))
assert e2.arbitrary_point() in e2
# Foci
f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
ef = Ellipse(Point(0, 0), 4, 2)
assert ef.foci in [(f1, f2), (f2, f1)]
# Tangents
v = sqrt(2) / 2
p1_1 = Point(v, v)
p1_2 = p2 + Point(half, 0)
p1_3 = p2 + Point(0, 1)
assert e1.tangent_lines(p4) == c1.tangent_lines(p4)
assert e2.tangent_lines(p1_2) == [Line(p1_2, p2 + Point(half, 1))]
assert e2.tangent_lines(p1_3) == [Line(p1_3, p2 + Point(half, 1))]
assert c1.tangent_lines(p1_1) == [Line(p1_1, Point(0, sqrt(2)))]
assert c1.tangent_lines(p1) == []
assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) == False
assert c1.is_tangent(e1) == False
assert c1.is_tangent(Ellipse(Point(2, 0), 1, 1)) == True
assert c1.is_tangent(Polygon(Point(1, 1), Point(1, -1), Point(2,
0))) == True
assert c1.is_tangent(Polygon(Point(1, 1), Point(1, 0), Point(2,
0))) == False
assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(0, 0)) == \
[Line(Point(0, 0), Point(S(77)/25, S(132)/25)),
Line(Point(0, 0), Point(S(33)/5, S(22)/5))]
assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(3, 4)) == \
[Line(Point(3, 4), Point(4, 4)), Line(Point(3, 4), Point(3, 5))]
assert Circle(Point(5, 5), 2).tangent_lines(Point(3, 3)) == \
[Line(Point(3, 3), Point(4, 3)), Line(Point(3, 3), Point(3, 4))]
assert Circle(Point(5, 5), 2).tangent_lines(Point(5 - 2*sqrt(2), 5)) == \
[Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 - sqrt(2))),
Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 + sqrt(2))),]
# Properties
major = 3
minor = 1
e4 = Ellipse(p2, minor, major)
assert e4.focus_distance == sqrt(major**2 - minor**2)
ecc = e4.focus_distance / major
assert e4.eccentricity == ecc
assert e4.periapsis == major * (1 - ecc)
assert e4.apoapsis == major * (1 + ecc)
# independent of orientation
e4 = Ellipse(p2, major, minor)
assert e4.focus_distance == sqrt(major**2 - minor**2)
ecc = e4.focus_distance / major
assert e4.eccentricity == ecc
assert e4.periapsis == major * (1 - ecc)
assert e4.apoapsis == major * (1 + ecc)
# Intersection
l1 = Line(Point(1, -5), Point(1, 5))
l2 = Line(Point(-5, -1), Point(5, -1))
l3 = Line(Point(-1, -1), Point(1, 1))
l4 = Line(Point(-10, 0), Point(0, 10))
pts_c1_l3 = [
Point(sqrt(2) / 2,
sqrt(2) / 2),
Point(-sqrt(2) / 2, -sqrt(2) / 2)
]
assert intersection(e2, l4) == []
assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
assert intersection(c1, l1) == [Point(1, 0)]
assert intersection(c1, l2) == [Point(0, -1)]
assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
assert intersection(c1, c2) in [[(1, 0), (0, 1)], [(0, 1), (1, 0)]]
assert intersection(c1, c3) == [(sqrt(2) / 2, sqrt(2) / 2)]
assert e1.intersection(l1) == [Point(1, 0)]
assert e2.intersection(l4) == []
assert e1.intersection(Circle(Point(0, 2), 1)) == [Point(0, 1)]
assert e1.intersection(Circle(Point(5, 0), 1)) == []
assert e1.intersection(Ellipse(Point(2, 0), 1, 1)) == [Point(1, 0)]
assert e1.intersection(Ellipse(
Point(5, 0),
1,
1,
)) == []
assert e1.intersection(Point(2, 0)) == []
assert e1.intersection(e1) == e1
# some special case intersections
csmall = Circle(p1, 3)
cbig = Circle(p1, 5)
cout = Circle(Point(5, 5), 1)
# one circle inside of another
assert csmall.intersection(cbig) == []
# separate circles
assert csmall.intersection(cout) == []
# coincident circles
assert csmall.intersection(csmall) == csmall
v = sqrt(2)
t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
points = intersection(t1, c1)
assert len(points) == 4
assert Point(0, 1) in points
assert Point(0, -1) in points
assert Point(v / 2, v / 2) in points
assert Point(v / 2, -v / 2) in points
circ = Circle(Point(0, 0), 5)
elip = Ellipse(Point(0, 0), 5, 20)
assert intersection(circ, elip) in \
[[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
assert elip.tangent_lines(Point(0, 0)) == []
elip = Ellipse(Point(0, 0), 3, 2)
assert elip.tangent_lines(Point(3,
0)) == [Line(Point(3, 0), Point(3, -12))]
e1 = Ellipse(Point(0, 0), 5, 10)
e2 = Ellipse(Point(2, 1), 4, 8)
a = S(53) / 17
c = 2 * sqrt(3991) / 17
ans = [Point(a - c / 8, a / 2 + c), Point(a + c / 8, a / 2 - c)]
assert e1.intersection(e2) == ans
e2 = Ellipse(Point(x, y), 4, 8)
ans = list(reversed(ans))
assert [p.subs({x: 2, y: 1}) for p in e1.intersection(e2)] == ans
# Combinations of above
assert e3.is_tangent(e3.tangent_lines(p1 + Point(y1, 0))[0])
e = Ellipse((1, 2), 3, 2)
assert e.tangent_lines(Point(10, 0)) == \
[Line(Point(10, 0), Point(1, 0)),
Line(Point(10, 0), Point(S(14)/5, S(18)/5))]
# encloses_point
e = Ellipse((0, 0), 1, 2)
assert e.encloses_point(e.center)
assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
assert e.encloses_point(e.center +
Point(e.hradius + Rational(1, 10), 0)) is False
e = Ellipse((0, 0), 2, 1)
assert e.encloses_point(e.center)
assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
assert e.encloses_point(e.center +
Point(e.hradius + Rational(1, 10), 0)) is False
assert c1.encloses_point(Point(1, 0)) is False
assert c1.encloses_point(Point(0.3, 0.4)) is True
def test_polygon():
t = Triangle(Point(0, 0), Point(2, 0), Point(3, 3))
assert Polygon(Point(0, 0), Point(1, 0), Point(2, 0), Point(3, 3)) == t
assert Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(0, 0)) == t
assert Polygon(Point(2, 0), Point(3, 3), Point(0, 0), Point(1, 0)) == t
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))
#
# General polygon
#
assert p1 == p2
assert len(p1) == 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 p3.is_convex()
assert p4.is_convex(
) # ensure convex for both CW and CCW point specification
dict5 = p5.angles
assert dict5[Point(0, 0)] == pi / 4
assert dict5[Point(0, 4)] == pi / 2
assert p5.encloses_point(Point(x, y)) == None
assert p5.encloses_point(Point(1, 3))
assert p5.encloses_point(Point(0, 0)) == False
assert p5.encloses_point(Point(4, 0)) == False
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
raises(
UserWarning,
'Polygon(Point(0, 0), Point(1, 0), Point(1,1)).distance(Polygon(Point(0, 0), Point(0, 1), Point(1, 1)))'
)
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', real=True),
0) == Point(0, 0)
raises(
ValueError,
"Polygon(Point(x, 0), Point(0, y), Point(x, y)).arbitrary_point('x')")
#
# Regular polygon
#
p1 = RegularPolygon(Point(0, 0), 10, 5)
p2 = RegularPolygon(Point(0, 0), 5, 5)
raises(GeometryError,
'RegularPolygon(Point(0, 0), Point(0, 1), Point(1, 1))')
raises(GeometryError, 'RegularPolygon(Point(0, 0), 1, 2)')
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)) == False
assert p2.encloses_point(Point(0, 4.9))
p1.spin(pi / 3)
assert p1.rotation == pi / 3
assert p1[0] == Point(5, 5 * sqrt(3))
for var in p1:
if isinstance(var, Point):
assert var == Point(0, 0)
else:
assert var == 5 or var == 10 or var == pi / 3
assert p1 != Point(0, 0)
assert p1 != p5
raises(IndexError, 'RegularPolygon(Point(0, 0), 1, 3)[4]')
# 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 ` == 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"))
#
# 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)
raises(GeometryError, '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() == 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() == False
assert t2.is_equilateral()
assert t3.is_equilateral() == False
assert are_similar(t1, t2) == False
assert are_similar(t1, t3)
assert are_similar(t2, t3) == False
assert t1.is_similar(Point(0, 0)) == 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
# 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(half, half)
pt2 = Point(1, 1)
'''Polygon to Point'''
assert p1.distance(pt1) == half
assert p1.distance(pt2) == 0
assert p2.distance(pt1) == Rational(3) / 4
assert p3.distance(pt2) == sqrt(2) / 2
def test_intersection_2d():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
l1 = Line(p1, p2)
l3 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(0, 0), Point(3, 4))
r4 = Ray(p1, p2)
r6 = Ray(Point(0, 1), Point(1, 2))
r7 = Ray(Point(0.5, 0.5), Point(1, 1))
s1 = Segment(p1, p2)
s2 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
s3 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, p1) == [p1]
assert intersection(l1, Point(x1, 1 + x1)) == []
assert intersection(l1, Line(p3, p4)) in [[l1], [Line(p3, p4)]]
assert intersection(l1, l1.parallel_line(Point(x1, 1 + x1))) == []
assert intersection(l3, l3) == [l3]
assert intersection(l3, r2) == [r2]
assert intersection(l3, s3) == [s3]
assert intersection(s3, l3) == [s3]
assert (intersection(Segment(Point(-10, 10), Point(10, 10)),
Segment(Point(-5, -5), Point(-5, 5))) == [])
assert intersection(r2, l3) == [r2]
assert intersection(r1,
Ray(Point(2, 2),
Point(0,
0))) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, Ray(Point(1, 1), Point(-1, -1))) == [Point(1, 1)]
assert intersection(r1, Segment(Point(0, 0), Point(
2, 2))) == [Segment(Point(1, 1), Point(2, 2))]
assert r4.intersection(s2) == [s2]
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))]
assert r4.intersection(Ray(p2, p1)) == [s1]
assert Ray(p2, p1).intersection(r6) == []
assert r4.intersection(r7) == r7.intersection(r4) == [r7]
assert Ray3D((0, 0), (3, 0)).intersection(Ray3D(
(1, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
assert Ray3D((1, 0), (3, 0)).intersection(Ray3D(
(0, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
assert Ray(Point(0, 0), Point(0, 4)).intersection(
Ray(Point(0, 1), Point(0, -1))) == [Segment(Point(0, 0), Point(0, 1))]
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((2, 0), (3, 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)]
assert s1.intersection(Segment(Point(1, 1), Point(2, 2))) == [Point(1, 1)]
assert s1.intersection(Segment(Point(0.5, 0.5), Point(
1.5, 1.5))) == [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 s1.intersection(Line(Point(1, 0), Point(2, 1))) == []
assert s1.intersection(s2) == [s2]
assert s2.intersection(s1) == [s2]
assert asa(120, 8, 52) == Triangle(
Point(0, 0),
Point(8, 0),
Point(
-4 * cos(19 * pi / 90) / sin(2 * pi / 45),
4 * sqrt(3) * cos(19 * pi / 90) / sin(2 * pi / 45),
),
)
assert Line((0, 0), (1, 1)).intersection(Ray((1, 0),
(1, 2))) == [Point(1, 1)]
assert Line((0, 0), (1, 1)).intersection(Segment((1, 0),
(1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (1, 1)).intersection(Ray((1, 0),
(1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (1, 1)).intersection(Segment((1, 0),
(1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (10, 10)).contains(Segment((1, 1), (2, 2))) is True
assert Segment((1, 1), (2, 2)) in Line((0, 0), (10, 10))
assert s1.intersection(Ray((1, 1), (4, 4))) == [Point(1, 1)]
def test_cross_section():
I = Symbol('I')
l = Symbol('l')
E = Symbol('E')
C3, C4 = symbols('C3, C4')
a, c, g, h, r, n = symbols('a, c, g, h, r, n')
# test for second_moment and cross_section setter
b0 = Beam(l, E, I)
assert b0.second_moment == I
assert b0.cross_section == None
b0.cross_section = Circle((0, 0), 5)
assert b0.second_moment == pi*Rational(625, 4)
assert b0.cross_section == Circle((0, 0), 5)
b0.second_moment = 2*n - 6
assert b0.second_moment == 2*n-6
assert b0.cross_section == None
with raises(ValueError):
b0.second_moment = Circle((0, 0), 5)
# beam with a circular cross-section
b1 = Beam(50, E, Circle((0, 0), r))
assert b1.cross_section == Circle((0, 0), r)
assert b1.second_moment == pi*r*Abs(r)**3/4
b1.apply_load(-10, 0, -1)
b1.apply_load(R1, 5, -1)
b1.apply_load(R2, 50, -1)
b1.apply_load(90, 45, -2)
b1.solve_for_reaction_loads(R1, R2)
assert b1.load == (-10*SingularityFunction(x, 0, -1) + 82*SingularityFunction(x, 5, -1)/S(9)
+ 90*SingularityFunction(x, 45, -2) + 8*SingularityFunction(x, 50, -1)/9)
assert b1.bending_moment() == (-10*SingularityFunction(x, 0, 1) + 82*SingularityFunction(x, 5, 1)/9
+ 90*SingularityFunction(x, 45, 0) + 8*SingularityFunction(x, 50, 1)/9)
q = (-5*SingularityFunction(x, 0, 2) + 41*SingularityFunction(x, 5, 2)/S(9)
+ 90*SingularityFunction(x, 45, 1) + 4*SingularityFunction(x, 50, 2)/S(9))/(pi*E*r*Abs(r)**3)
assert b1.slope() == C3 + 4*q
q = (-5*SingularityFunction(x, 0, 3)/3 + 41*SingularityFunction(x, 5, 3)/27 + 45*SingularityFunction(x, 45, 2)
+ 4*SingularityFunction(x, 50, 3)/27)/(pi*E*r*Abs(r)**3)
assert b1.deflection() == C3*x + C4 + 4*q
# beam with a recatangular cross-section
b2 = Beam(20, E, Polygon((0, 0), (a, 0), (a, c), (0, c)))
assert b2.cross_section == Polygon((0, 0), (a, 0), (a, c), (0, c))
assert b2.second_moment == a*c**3/12
# beam with a triangular cross-section
b3 = Beam(15, E, Triangle((0, 0), (g, 0), (g/2, h)))
assert b3.cross_section == Triangle(Point2D(0, 0), Point2D(g, 0), Point2D(g/2, h))
assert b3.second_moment == g*h**3/36
# composite beam
b = b2.join(b3, "fixed")
b.apply_load(-30, 0, -1)
b.apply_load(65, 0, -2)
b.apply_load(40, 0, -1)
b.bc_slope = [(0, 0)]
b.bc_deflection = [(0, 0)]
assert b.second_moment == Piecewise((a*c**3/12, x <= 20), (g*h**3/36, x <= 35))
assert b.cross_section == None
assert b.length == 35
assert b.slope().subs(x, 7) == 8400/(E*a*c**3)
assert b.slope().subs(x, 25) == 52200/(E*g*h**3) + 39600/(E*a*c**3)
assert b.deflection().subs(x, 30) == 537000/(E*g*h**3) + 712000/(E*a*c**3)
def test_polygon():
t = Triangle(Point(0, 0), Point(2, 0), Point(3, 3))
assert Polygon(Point(0, 0), Point(1, 0), Point(2, 0), Point(3, 3)) == t
assert Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(0, 0)) == t
assert Polygon(Point(2, 0), Point(3, 3), Point(0, 0), Point(1, 0)) == t
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))
#
# General polygon
#
assert p1 == p2
assert len(p1) == 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 p3.is_convex()
assert p4.is_convex(
) # ensure convex for both CW and CCW point specification
#
# Regular polygon
#
p1 = RegularPolygon(Point(0, 0), 10, 5)
p2 = RegularPolygon(Point(0, 0), 5, 5)
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.circumcircle == Circle(Point(0, 0), 5)
assert p2.incircle == Circle(Point(0, 0), p2.apothem)
assert p1.is_convex()
assert p1.rotation == 0
p1.spin(pi / 3)
assert p1.rotation == pi / 3
assert p1[0] == Point(5, 5 * sqrt(3))
# 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
#
# 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"))
#
# 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
# 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() == 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() == False
assert t2.is_equilateral()
assert t3.is_equilateral() == False
assert are_similar(t1, t2) == False
assert are_similar(t1, t3)
assert are_similar(t2, t3) == 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 == 5 - 5 * sqrt(2) / 2
assert t2.inradius == 5 * sqrt(3) / 6
assert t3.inradius == x1**2 / ((2 + sqrt(2)) * Abs(x1))
# 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]
# 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]
# 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(half, half)
pt2 = Point(1, 1)
'''Polygon to Point'''
assert p1.distance(pt1) == half
assert p1.distance(pt2) == 0
assert p2.distance(pt1) == Rational(3) / 4
assert p3.distance(pt2) == sqrt(2) / 2
def test_polygon():
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))
#
# General polygon
#
assert p1 == p2
assert len(p1) == Rational(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 p3.is_convex()
assert p4.is_convex(
) # ensure convex for both CW and CCW point specification
#
# Regular polygon
#
p1 = RegularPolygon(Point(0, 0), 10, 5)
p2 = RegularPolygon(Point(0, 0), 5, 5)
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.circumcircle == Circle(Point(0, 0), 5)
assert p2.incircle == Circle(Point(0, 0), p2.apothem)
assert p1.is_convex()
#
# Angles
#
angles = p4.angles
assert feq(angles[Point(0, 0)].evalf(), Real("0.7853981633974483"))
assert feq(angles[Point(4, 4)].evalf(), Real("1.2490457723982544"))
assert feq(angles[Point(5, 2)].evalf(), Real("1.8925468811915388"))
assert feq(angles[Point(3, 0)].evalf(), Real("2.3561944901923449"))
angles = p3.angles
assert feq(angles[Point(0, 0)].evalf(), Real("0.7853981633974483"))
assert feq(angles[Point(4, 4)].evalf(), Real("1.2490457723982544"))
assert feq(angles[Point(5, 2)].evalf(), Real("1.8925468811915388"))
assert feq(angles[Point(3, 0)].evalf(), Real("2.3561944901923449"))
#
# 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
s2 = t2.sides
s3 = t3.sides
# Basic stuff
assert t1.area == Rational(25, 2)
assert t1.is_right()
assert t2.is_right() == False
assert t3.is_right()
assert p1 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() == False
assert t2.is_equilateral()
assert t3.is_equilateral() == False
assert are_similar(t1, t2) == False
assert are_similar(t1, t3)
assert are_similar(t2, t3) == 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 == 5 - 5 * 2**(S(1) / 2) / 2
assert t2.inradius == 5 * 3**(S(1) / 2) / 6
assert t3.inradius == (2 * x1**2 * Abs(x1) -
2**(S(1) / 2) * x1**2 * Abs(x1)) / (2 * x1**2)
# 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]
# 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]
# 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))
p5 = Polygon(Point(half, 3**(half) / 2), Point(-half, 3**(half) / 2),
Point(-1, 0), Point(-half, -(3)**(half) / 2),
Point(half, -(3)**(half) / 2), Point(1, 0))
p6 = Polygon(Point(2,
Rational(3) / 10), Point(Rational(17) / 10, 0),
Point(2, -Rational(3) / 10), Point(Rational(23) / 10, 0))
pt1 = Point(half, half)
pt2 = Point(1, 1)
'''Polygon to Point'''
assert p1.distance(pt1) == half
assert p1.distance(pt2) == 0
assert p2.distance(pt1) == Rational(3) / 4
assert p3.distance(pt2) == sqrt(2) / 2
'''Polygon to Polygon'''
assert p1.distance(p2) == half / 2
assert p1.distance(p3) == sqrt(2) / 2
assert p3.distance(p4) == (sqrt(2) / 2 - sqrt(Rational(2) / 25) / 2)
assert p5.distance(p6) == Rational(7) / 10
def test_ellipse():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x2)
p4 = Point(0, 1)
p5 = Point(-1, 0)
e1 = Ellipse(p1, 1, 1)
e2 = Ellipse(p2, half, 1)
e3 = Ellipse(p1, y1, y1)
c1 = Circle(p1, 1)
c2 = Circle(p2, 1)
c3 = Circle(Point(sqrt(2), sqrt(2)), 1)
# Test creation with three points
cen, rad = Point(3 * half, 2), 5 * half
assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
raises(GeometryError, "Circle(Point(0,0), Point(1,1), Point(2,2))")
# Basic Stuff
assert e1 == c1
assert e1 != e2
assert p4 in e1
assert p2 not in e2
assert e1.area == pi
assert e2.area == pi / 2
assert e3.area == pi * (y1**2)
assert c1.area == e1.area
assert c1.circumference == e1.circumference
assert e3.circumference == 2 * pi * y1
a = Symbol('a')
b = Symbol('b')
e5 = Ellipse(p1, a, b)
assert e5.circumference == 4*a*C.Integral(((1 - x**2*Abs(b**2 - a**2)/a**2)/(1 - x**2))**(S(1)/2),\
(x, 0, 1))
assert e2.arbitrary_point() in e2
# Foci
f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
ef = Ellipse(Point(0, 0), 4, 2)
assert ef.foci in [(f1, f2), (f2, f1)]
# Tangents
v = sqrt(2) / 2
p1_1 = Point(v, v)
p1_2 = p2 + Point(half, 0)
p1_3 = p2 + Point(0, 1)
assert e1.tangent_line(p4) == c1.tangent_line(p4)
assert e2.tangent_line(p1_2) == Line(p1_2, p2 + Point(half, 1))
assert e2.tangent_line(p1_3) == Line(p1_3, p2 + Point(half, 1))
assert c1.tangent_line(p1_1) == Line(p1_1, Point(0, sqrt(2)))
assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) == False
# Intersection
l1 = Line(Point(1, -5), Point(1, 5))
l2 = Line(Point(-5, -1), Point(5, -1))
l3 = Line(Point(-1, -1), Point(1, 1))
l4 = Line(Point(-10, 0), Point(0, 10))
pts_c1_l3 = [
Point(sqrt(2) / 2,
sqrt(2) / 2),
Point(-sqrt(2) / 2, -sqrt(2) / 2)
]
assert intersection(e2, l4) == []
assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
assert intersection(c1, l1) == [Point(1, 0)]
assert intersection(c1, l2) == [Point(0, -1)]
assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
assert intersection(c1, c2) in [[(1, 0), (0, 1)], [(0, 1), (1, 0)]]
assert intersection(c1, c3) == [(sqrt(2) / 2, sqrt(2) / 2)]
# 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
e1 = Circle(Point(0, 0), 5)
e2 = Ellipse(Point(0, 0), 5, 20)
assert intersection(e1, e2) in \
[[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
# FAILING ELLIPSE INTERSECTION GOES HERE
# Combinations of above
assert e3.is_tangent(e3.tangent_line(p1 + Point(y1, 0)))
major = 3
minor = 1
e4 = Ellipse(p2, major, minor)
assert e4.focus_distance == sqrt(abs(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)
def test_polygon():
t = Triangle(Point(0, 0), Point(2, 0), Point(3, 3))
assert Polygon(Point(0, 0), Point(1, 0), Point(2, 0), Point(3, 3)) == t
assert Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(0, 0)) == t
assert Polygon(Point(2, 0), Point(3, 3), Point(0, 0), Point(1, 0)) == t
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))
#
# 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()
# 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
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
warnings.filterwarnings(
"error", message="Polygons may intersect producing erroneous output")
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', real=True), 0) == \
Point(0, 0)
raises(
ValueError, lambda: Polygon(Point(x, 0), Point(0, y), Point(x, y)).
arbitrary_point('x'))
#
# Regular polygon
#
p1 = RegularPolygon(Point(0, 0), 10, 5)
p2 = RegularPolygon(Point(0, 0), 5, 5)
raises(GeometryError,
lambda: RegularPolygon(Point(0, 0), Point(0, 1), Point(1, 1)))
raises(GeometryError, lambda: RegularPolygon(Point(0, 0), 1, 2))
raises(ValueError, lambda: RegularPolygon(Point(0, 0), 1, 2.5))
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 == 5 or var == 10 or var == 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 + S(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"))
#
# 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)
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 = S('''Triangle(
Point(100080156402737/5000000000000, 79782624633431/500000000000),
Point(39223884078253/2000000000000, 156345163124289/1000000000000),
Point(31241359188437/1250000000000, 338338270939941/1000000000000000))''')
assert t.orthocenter == S(
'''Point(-780660869050599840216997'''
'''79471538701955848721853/80368430960602242240789074233100000000000000,'''
'''20151573611150265741278060334545897615974257/16073686192120448448157'''
'''8148466200000000000)''')
# 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(half, half)
pt2 = Point(1, 1)
'''Polygon to Point'''
assert p1.distance(pt1) == half
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")
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) == half / 2
assert p1.distance(p3) == sqrt(2) / 2
assert p3.distance(p4) == (sqrt(2) / 2 - sqrt(Rational(2) / 25) / 2)
def test_ellipse():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x2)
p4 = Point(0, 1)
p5 = Point(-1, 0)
e1 = Ellipse(p1, 1, 1)
e2 = Ellipse(p2, half, 1)
e3 = Ellipse(p1, y1, y1)
c1 = Circle(p1, 1)
c2 = Circle(p2, 1)
c3 = Circle(Point(sqrt(2), sqrt(2)), 1)
# Test creation with three points
cen, rad = Point(3 * half, 2), 5 * half
assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
raises(GeometryError, "Circle(Point(0,0), Point(1,1), Point(2,2))")
# Basic Stuff
assert e1 == c1
assert e1 != e2
assert p4 in e1
assert p2 not in e2
assert e1.area == pi
assert e2.area == pi / 2
assert e3.area == pi * (y1**2)
assert c1.area == e1.area
assert c1.circumference == 2 * pi
assert e2.arbitrary_point() in e2
for ind in xrange(0, 5):
assert e3.random_point() in e3
# Foci
f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
ef = Ellipse(Point(0, 0), 4, 2)
assert ef.foci in [(f1, f2), (f2, f1)]
# Tangents
v = sqrt(2) / 2
p1_1 = Point(v, v)
p1_2 = p2 + Point(half, 0)
p1_3 = p2 + Point(0, 1)
assert e1.tangent_line(p4) == c1.tangent_line(p4)
assert e2.tangent_line(p1_2) == Line(p1_2, p2 + Point(half, 1))
assert e2.tangent_line(p1_3) == Line(p1_3, p2 + Point(half, 1))
assert c1.tangent_line(p1_1) == Line(p1_1, Point(0, sqrt(2)))
assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) == False
# Intersection
l1 = Line(Point(1, -5), Point(1, 5))
l2 = Line(Point(-5, -1), Point(5, -1))
l3 = Line(Point(-1, -1), Point(1, 1))
l4 = Line(Point(-10, 0), Point(0, 10))
pts_c1_l3 = [
Point(sqrt(2) / 2,
sqrt(2) / 2),
Point(-sqrt(2) / 2, -sqrt(2) / 2)
]
assert intersection(e2, l4) == []
assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
assert intersection(c1, l1) == [Point(1, 0)]
assert intersection(c1, l2) == [Point(0, -1)]
assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
assert intersection(c1, c2) in [[(1, 0), (0, 1)], [(0, 1), (1, 0)]]
assert intersection(c1, c3) == [(sqrt(2) / 2, sqrt(2) / 2)]
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
e1 = Circle(Point(0, 0), 5)
e2 = Ellipse(Point(0, 0), 5, 20)
assert intersection(e1, e2) in \
[[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
# Combinations of above
assert e3.is_tangent(e3.tangent_line(p1 + Point(y1, 0)))
def test_ellipse_geom():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
t = Symbol('t', real=True)
y1 = Symbol('y1', real=True)
half = Rational(1, 2)
p1 = Point(0, 0)
p2 = Point(1, 1)
p4 = Point(0, 1)
e1 = Ellipse(p1, 1, 1)
e2 = Ellipse(p2, half, 1)
e3 = Ellipse(p1, y1, y1)
c1 = Circle(p1, 1)
c2 = Circle(p2, 1)
c3 = Circle(Point(sqrt(2), sqrt(2)), 1)
l1 = Line(p1, p2)
# Test creation with three points
cen, rad = Point(3*half, 2), 5*half
assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
assert Circle(Point(0, 0), Point(1, 1), Point(2, 2)) == Segment2D(Point2D(0, 0), Point2D(2, 2))
raises(ValueError, lambda: Ellipse(None, None, None, 1))
raises(GeometryError, lambda: Circle(Point(0, 0)))
# Basic Stuff
assert Ellipse(None, 1, 1).center == Point(0, 0)
assert e1 == c1
assert e1 != e2
assert e1 != l1
assert p4 in e1
assert p2 not in e2
assert e1.area == pi
assert e2.area == pi/2
assert e3.area == pi*y1*abs(y1)
assert c1.area == e1.area
assert c1.circumference == e1.circumference
assert e3.circumference == 2*pi*y1
assert e1.plot_interval() == e2.plot_interval() == [t, -pi, pi]
assert e1.plot_interval(x) == e2.plot_interval(x) == [x, -pi, pi]
assert c1.minor == 1
assert c1.major == 1
assert c1.hradius == 1
assert c1.vradius == 1
assert Ellipse((1, 1), 0, 0) == Point(1, 1)
assert Ellipse((1, 1), 1, 0) == Segment(Point(0, 1), Point(2, 1))
assert Ellipse((1, 1), 0, 1) == Segment(Point(1, 0), Point(1, 2))
# Private Functions
assert hash(c1) == hash(Circle(Point(1, 0), Point(0, 1), Point(0, -1)))
assert c1 in e1
assert (Line(p1, p2) in e1) is False
assert e1.__cmp__(e1) == 0
assert e1.__cmp__(Point(0, 0)) > 0
# Encloses
assert e1.encloses(Segment(Point(-0.5, -0.5), Point(0.5, 0.5))) is True
assert e1.encloses(Line(p1, p2)) is False
assert e1.encloses(Ray(p1, p2)) is False
assert e1.encloses(e1) is False
assert e1.encloses(
Polygon(Point(-0.5, -0.5), Point(-0.5, 0.5), Point(0.5, 0.5))) is True
assert e1.encloses(RegularPolygon(p1, 0.5, 3)) is True
assert e1.encloses(RegularPolygon(p1, 5, 3)) is False
assert e1.encloses(RegularPolygon(p2, 5, 3)) is False
assert e2.arbitrary_point() in e2
# Foci
f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
ef = Ellipse(Point(0, 0), 4, 2)
assert ef.foci in [(f1, f2), (f2, f1)]
# Tangents
v = sqrt(2) / 2
p1_1 = Point(v, v)
p1_2 = p2 + Point(half, 0)
p1_3 = p2 + Point(0, 1)
assert e1.tangent_lines(p4) == c1.tangent_lines(p4)
assert e2.tangent_lines(p1_2) == [Line(Point(S(3)/2, 1), Point(S(3)/2, S(1)/2))]
assert e2.tangent_lines(p1_3) == [Line(Point(1, 2), Point(S(5)/4, 2))]
assert c1.tangent_lines(p1_1) != [Line(p1_1, Point(0, sqrt(2)))]
assert c1.tangent_lines(p1) == []
assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) is False
assert c1.is_tangent(e1) is True
assert c1.is_tangent(Ellipse(Point(2, 0), 1, 1)) is True
assert c1.is_tangent(
Polygon(Point(1, 1), Point(1, -1), Point(2, 0))) is True
assert c1.is_tangent(
Polygon(Point(1, 1), Point(1, 0), Point(2, 0))) is False
assert Circle(Point(5, 5), 3).is_tangent(Circle(Point(0, 5), 1)) is False
assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(0, 0)) == \
[Line(Point(0, 0), Point(S(77)/25, S(132)/25)),
Line(Point(0, 0), Point(S(33)/5, S(22)/5))]
assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(3, 4)) == \
[Line(Point(3, 4), Point(4, 4)), Line(Point(3, 4), Point(3, 5))]
assert Circle(Point(5, 5), 2).tangent_lines(Point(3, 3)) == \
[Line(Point(3, 3), Point(4, 3)), Line(Point(3, 3), Point(3, 4))]
assert Circle(Point(5, 5), 2).tangent_lines(Point(5 - 2*sqrt(2), 5)) == \
[Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 - sqrt(2))),
Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 + sqrt(2))), ]
# for numerical calculations, we shouldn't demand exact equality,
# so only test up to the desired precision
def lines_close(l1, l2, prec):
""" tests whether l1 and 12 are within 10**(-prec)
of each other """
return abs(l1.p1 - l2.p1) < 10**(-prec) and abs(l1.p2 - l2.p2) < 10**(-prec)
def line_list_close(ll1, ll2, prec):
return all(lines_close(l1, l2, prec) for l1, l2 in zip(ll1, ll2))
e = Ellipse(Point(0, 0), 2, 1)
assert e.normal_lines(Point(0, 0)) == \
[Line(Point(0, 0), Point(0, 1)), Line(Point(0, 0), Point(1, 0))]
assert e.normal_lines(Point(1, 0)) == \
[Line(Point(0, 0), Point(1, 0))]
assert e.normal_lines((0, 1)) == \
[Line(Point(0, 0), Point(0, 1))]
assert line_list_close(e.normal_lines(Point(1, 1), 2), [
Line(Point(-S(51)/26, -S(1)/5), Point(-S(25)/26, S(17)/83)),
Line(Point(S(28)/29, -S(7)/8), Point(S(57)/29, -S(9)/2))], 2)
# test the failure of Poly.intervals and checks a point on the boundary
p = Point(sqrt(3), S.Half)
assert p in e
assert line_list_close(e.normal_lines(p, 2), [
Line(Point(-S(341)/171, -S(1)/13), Point(-S(170)/171, S(5)/64)),
Line(Point(S(26)/15, -S(1)/2), Point(S(41)/15, -S(43)/26))], 2)
# be sure to use the slope that isn't undefined on boundary
e = Ellipse((0, 0), 2, 2*sqrt(3)/3)
assert line_list_close(e.normal_lines((1, 1), 2), [
Line(Point(-S(64)/33, -S(20)/71), Point(-S(31)/33, S(2)/13)),
Line(Point(1, -1), Point(2, -4))], 2)
# general ellipse fails except under certain conditions
e = Ellipse((0, 0), x, 1)
assert e.normal_lines((x + 1, 0)) == [Line(Point(0, 0), Point(1, 0))]
raises(NotImplementedError, lambda: e.normal_lines((x + 1, 1)))
# Properties
major = 3
minor = 1
e4 = Ellipse(p2, minor, major)
assert e4.focus_distance == sqrt(major**2 - minor**2)
ecc = e4.focus_distance / major
assert e4.eccentricity == ecc
assert e4.periapsis == major*(1 - ecc)
assert e4.apoapsis == major*(1 + ecc)
assert e4.semilatus_rectum == major*(1 - ecc ** 2)
# independent of orientation
e4 = Ellipse(p2, major, minor)
assert e4.focus_distance == sqrt(major**2 - minor**2)
ecc = e4.focus_distance / major
assert e4.eccentricity == ecc
assert e4.periapsis == major*(1 - ecc)
assert e4.apoapsis == major*(1 + ecc)
# Intersection
l1 = Line(Point(1, -5), Point(1, 5))
l2 = Line(Point(-5, -1), Point(5, -1))
l3 = Line(Point(-1, -1), Point(1, 1))
l4 = Line(Point(-10, 0), Point(0, 10))
pts_c1_l3 = [Point(sqrt(2)/2, sqrt(2)/2), Point(-sqrt(2)/2, -sqrt(2)/2)]
assert intersection(e2, l4) == []
assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
assert intersection(c1, l1) == [Point(1, 0)]
assert intersection(c1, l2) == [Point(0, -1)]
assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
assert intersection(c1, c2) == [Point(0, 1), Point(1, 0)]
assert intersection(c1, c3) == [Point(sqrt(2)/2, sqrt(2)/2)]
assert e1.intersection(l1) == [Point(1, 0)]
assert e2.intersection(l4) == []
assert e1.intersection(Circle(Point(0, 2), 1)) == [Point(0, 1)]
assert e1.intersection(Circle(Point(5, 0), 1)) == []
assert e1.intersection(Ellipse(Point(2, 0), 1, 1)) == [Point(1, 0)]
assert e1.intersection(Ellipse(Point(5, 0), 1, 1)) == []
assert e1.intersection(Point(2, 0)) == []
assert e1.intersection(e1) == e1
assert intersection(Ellipse(Point(0, 0), 2, 1), Ellipse(Point(3, 0), 1, 2)) == [Point(2, 0)]
assert intersection(Circle(Point(0, 0), 2), Circle(Point(3, 0), 1)) == [Point(2, 0)]
assert intersection(Circle(Point(0, 0), 2), Circle(Point(7, 0), 1)) == []
assert intersection(Ellipse(Point(0, 0), 5, 17), Ellipse(Point(4, 0), 1, 0.2)) == [Point(5, 0)]
assert intersection(Ellipse(Point(0, 0), 5, 17), Ellipse(Point(4, 0), 0.999, 0.2)) == []
assert Circle((0, 0), S(1)/2).intersection(
Triangle((-1, 0), (1, 0), (0, 1))) == [
Point(-S(1)/2, 0), Point(S(1)/2, 0)]
raises(TypeError, lambda: intersection(e2, Line((0, 0, 0), (0, 0, 1))))
raises(TypeError, lambda: intersection(e2, Rational(12)))
# some special case intersections
csmall = Circle(p1, 3)
cbig = Circle(p1, 5)
cout = Circle(Point(5, 5), 1)
# one circle inside of another
assert csmall.intersection(cbig) == []
# separate circles
assert csmall.intersection(cout) == []
# coincident circles
assert csmall.intersection(csmall) == csmall
v = sqrt(2)
t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
points = intersection(t1, c1)
assert len(points) == 4
assert Point(0, 1) in points
assert Point(0, -1) in points
assert Point(v/2, v/2) in points
assert Point(v/2, -v/2) in points
circ = Circle(Point(0, 0), 5)
elip = Ellipse(Point(0, 0), 5, 20)
assert intersection(circ, elip) in \
[[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
assert elip.tangent_lines(Point(0, 0)) == []
elip = Ellipse(Point(0, 0), 3, 2)
assert elip.tangent_lines(Point(3, 0)) == \
[Line(Point(3, 0), Point(3, -12))]
e1 = Ellipse(Point(0, 0), 5, 10)
e2 = Ellipse(Point(2, 1), 4, 8)
a = S(53)/17
c = 2*sqrt(3991)/17
ans = [Point(a - c/8, a/2 + c), Point(a + c/8, a/2 - c)]
assert e1.intersection(e2) == ans
e2 = Ellipse(Point(x, y), 4, 8)
c = sqrt(3991)
ans = [Point(-c/68 + a, 2*c/17 + a/2), Point(c/68 + a, -2*c/17 + a/2)]
assert [p.subs({x: 2, y:1}) for p in e1.intersection(e2)] == ans
# Combinations of above
assert e3.is_tangent(e3.tangent_lines(p1 + Point(y1, 0))[0])
e = Ellipse((1, 2), 3, 2)
assert e.tangent_lines(Point(10, 0)) == \
[Line(Point(10, 0), Point(1, 0)),
Line(Point(10, 0), Point(S(14)/5, S(18)/5))]
# encloses_point
e = Ellipse((0, 0), 1, 2)
assert e.encloses_point(e.center)
assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
assert e.encloses_point(
e.center + Point(e.hradius + Rational(1, 10), 0)) is False
e = Ellipse((0, 0), 2, 1)
assert e.encloses_point(e.center)
assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
assert e.encloses_point(
e.center + Point(e.hradius + Rational(1, 10), 0)) is False
assert c1.encloses_point(Point(1, 0)) is False
assert c1.encloses_point(Point(0.3, 0.4)) is True
assert e.scale(2, 3) == Ellipse((0, 0), 4, 3)
assert e.scale(3, 6) == Ellipse((0, 0), 6, 6)
assert e.rotate(pi) == e
assert e.rotate(pi, (1, 2)) == Ellipse(Point(2, 4), 2, 1)
raises(NotImplementedError, lambda: e.rotate(pi/3))
# Circle rotation tests (Issue #11743)
# Link - https://github.com/sympy/sympy/issues/11743
cir = Circle(Point(1, 0), 1)
assert cir.rotate(pi/2) == Circle(Point(0, 1), 1)
assert cir.rotate(pi/3) == Circle(Point(S(1)/2, sqrt(3)/2), 1)
assert cir.rotate(pi/3, Point(1, 0)) == Circle(Point(1, 0), 1)
assert cir.rotate(pi/3, Point(0, 1)) == Circle(Point(S(1)/2 + sqrt(3)/2, S(1)/2 + sqrt(3)/2), 1)
Run code block in SymPy Live from sympy import * from sympy.geometry import * x = Point(0, 0) y = Point(1, 1) z = Point(2, 2) zp = Point(1, 0) Point.is_collinear(x, y, z) True Point.is_collinear(x, y, zp) False t = Triangle(zp, y, x) t.area 1/2 t.medians[x] Segment2D(Point2D(0, 0), Point2D(1, 1/2)) m = t.medians intersection(m[x], m[y], m[zp]) [Point2D(2/3, 1/3)] c = Circle(x, 5) l = Line(Point(5, -5), Point(5, 5)) c.is_tangent(l) # is l tangent to c? True l = Line(x, y) c.is_tangent(l) # is l tangent to c? False intersection(c, l) [Point2D(-5*sqrt(2)/2, -5*sqrt(2)/2), Point2D(5*sqrt(2)/2, 5*sqrt(2)/2)] Run code block in SymPy Live >>> from sympy import symbols
