def test_ImplicitRegion():
ellipse = ImplicitRegion((x, y), (x**2 / 4 + y**2 / 16 - 1))
assert ellipse.equation == x**2 / 4 + y**2 / 16 - 1
assert ellipse.variables == (x, y)
assert ellipse.degree == 2
r = ImplicitRegion((x, y, z), Eq(x**4 + y**2 - x * y, 6))
assert r.equation == x**4 + y**2 - x * y - 6
assert r.variables == (x, y, z)
assert r.degree == 4
def test_rational_parametrization():
p = ImplicitRegion((x, ), x - 2)
assert p.rational_parametrization() == (x - 2, )
line = ImplicitRegion((x, y), Eq(y, 3 * x + 2))
assert line.rational_parametrization() == (x, 3 * x + 2)
circle1 = ImplicitRegion((x, y), (x - 2)**2 + (y + 3)**2 - 4)
assert circle1.rational_parametrization(
parameters=t) == (4 * t / (t**2 + 1) + 2, 4 * t**2 / (t**2 + 1) - 5)
circle2 = ImplicitRegion((x, y), (x - S.Half)**2 + y**2 - (S(1) / 2)**2)
assert circle2.rational_parametrization(
parameters=t) == (t / (t**2 + 1) + S(1) / 2,
t**2 / (t**2 + 1) - S(1) / 2)
circle3 = ImplicitRegion((x, y), Eq(x**2 + y**2, 2 * x))
assert circle3.rational_parametrization(
parameters=(t, )) == (2 * t / (t**2 + 1) + 1,
2 * t**2 / (t**2 + 1) - 1)
parabola = ImplicitRegion((x, y), (y - 3)**2 - 4 * (x + 6))
assert parabola.rational_parametrization(t) == (-6 + 4 / t**2, 3 + 4 / t)
rect_hyperbola = ImplicitRegion((x, y), x * y - 1)
assert rect_hyperbola.rational_parametrization(t) == (-1 + (t + 1) / t, t)
cubic_curve = ImplicitRegion((x, y), x**3 + x**2 - y**2)
assert cubic_curve.rational_parametrization(parameters=(t)) == (t**2 - 1,
t *
(t**2 - 1))
cuspidal = ImplicitRegion((x, y), (x**3 - y**2))
assert cuspidal.rational_parametrization(t) == (t**2, t**3)
I = ImplicitRegion((x, y), x**3 + x**2 - y**2)
assert I.rational_parametrization(t) == (t**2 - 1, t * (t**2 - 1))
sphere = ImplicitRegion((x, y, z), Eq(x**2 + y**2 + z**2, 2 * x))
assert sphere.rational_parametrization(
parameters=(s,
t)) == (2 / (s**2 + t**2 + 1), 2 * t / (s**2 + t**2 + 1),
2 * s / (s**2 + t**2 + 1))
conic = ImplicitRegion((x, y),
Eq(x**2 + 4 * x * y + 3 * y**2 + x - y + 10, 0))
conic.rational_parametrization(t) == (17 / 2 + 4 / (3 * t**2 + 4 * t + 1),
4 * t / (3 * t**2 + 4 * t + 1) -
11 / 2)
r1 = ImplicitRegion((x, y), y**2 - x**3 + x)
raises(NotImplementedError, lambda: r1.rational_parametrization())
r2 = ImplicitRegion((x, y), y**2 - x**3 - x**2 + 1)
raises(NotImplementedError, lambda: r2.rational_parametrization())
def test_singular_points_and_multiplicty():
r1 = ImplicitRegion((x, y, z), Eq(x + y + z, 0))
assert r1.singular_points() == FiniteSet((-y - z, y, z))
assert r1.multiplicity((0, 0, 0)) == 1
assert r1.multiplicity((-y - z, y, z)) == 1
r2 = ImplicitRegion((x, y, z), x * y * z + y**4 - x**2 * z**2)
assert r2.singular_points() == FiniteSet((0, 0, z), ((-y*sqrt(4*y**2 + 1)/2 + y/2)/z, y, z),\
((y*sqrt(4*y**2 + 1)/2 + y/2)/z, y, z))
assert r2.multiplicity((0, 0, 0)) == 3
assert r2.multiplicity((0, 0, 6)) == 2
r3 = ImplicitRegion((x, y, z), z**2 - x**2 - y**2)
assert r3.singular_points() == FiniteSet((0, 0, 0))
assert r3.multiplicity((0, 0, 0)) == 2
r4 = ImplicitRegion((x, y), x**2 + y**2 - 2 * x)
assert r4.singular_points() == EmptySet
assert r4.multiplicity(Point(1, 3)) == 0
def test_regular_point():
r1 = ImplicitRegion((x, ), x**2 - 16)
r1.regular_point() == (-4, )
c1 = ImplicitRegion((x, y), x**2 + y**2 - 4)
c1.regular_point() == (2, 0)
c2 = ImplicitRegion((x, y), (x - S(5) / 2)**2 + y**2 - (S(1) / 4)**2)
c2.regular_point() == (11 / 4, 0)
c3 = ImplicitRegion((x, y), (y - 5)**2 - 16 * (x - 5))
c3.regular_point() == (5, 5)
r2 = ImplicitRegion((x, y),
x**2 - 4 * x * y - 3 * y**2 + 4 * x + 8 * y - 5)
r2.regular_point == (4 / 7, 13 / 21)
r3 = ImplicitRegion((x, y),
x**2 - 2 * x * y + 3 * y**2 - 2 * x - 5 * y + 3 / 2)
raises(ValueError, lambda: r3.regular_point())
def test_regular_point():
r1 = ImplicitRegion((x,), x**2 - 16)
assert r1.regular_point() == (-4,)
c1 = ImplicitRegion((x, y), x**2 + y**2 - 4)
assert c1.regular_point() == (0, -2)
c2 = ImplicitRegion((x, y), (x - S(5)/2)**2 + y**2 - (S(1)/4)**2)
assert c2.regular_point() == (S(5)/2, -S(1)/4)
c3 = ImplicitRegion((x, y), (y - 5)**2 - 16*(x - 5))
assert c3.regular_point() == (5, 5)
r2 = ImplicitRegion((x, y), x**2 - 4*x*y - 3*y**2 + 4*x + 8*y - 5)
assert r2.regular_point() == (S(4)/7, S(9)/7)
r3 = ImplicitRegion((x, y), x**2 - 2*x*y + 3*y**2 - 2*x - 5*y + 3/2)
raises(ValueError, lambda: r3.regular_point())
def test_rational_parametrization():
p = ImplicitRegion((x, ), x - 2)
assert p.rational_parametrization() == (x - 2, )
line = ImplicitRegion((x, y), Eq(y, 3 * x + 2))
assert line.rational_parametrization() == (x, 3 * x + 2)
circle1 = ImplicitRegion((x, y), ((x - 2)**2 + (y + 3)**2 - 4))
assert circle1.rational_parametrization(parameters=t) == (4 / (t**2 + 1),
4 * t /
(t**2 + 1) - 3)
circle2 = ImplicitRegion((x, y), (x - 1 / 2)**2 + y**2 - (1 / 4)**2)
raises(NotImplementedError, lambda: circle2.rational_parametrization())
circle2.rational_parametrization(t, reg_point=Point(
0.75, 0)) == (3 / 4 - 0.5 / (t**2 + 1), -0.5 * t / (t**2 + 1))
circle3 = ImplicitRegion((x, y), Eq(x**2 + y**2, 2 * x))
assert circle3.rational_parametrization(
parameters=(t, )) == (2 / (t**2 + 1), 2 * t / (t**2 + 1))
parabola = ImplicitRegion((x, y), (y - 3)**2 - 4 * (x + 6))
assert parabola.rational_parametrization(t) == (-6 + 4 / t**2, 3 + 4 / t)
rect_hyperbola = ImplicitRegion((x, y), x * y - 1)
assert rect_hyperbola.rational_parametrization(t) == (
-100 + (100 * t + S(1) / 100) / t, 100 * t)
cubic_curve = ImplicitRegion((x, y), x**3 + x**2 - y**2)
assert cubic_curve.rational_parametrization(parameters=(t)) == (t**2 - 1,
t *
(t**2 - 1))
cuspidal = ImplicitRegion((x, y), (x**3 - y**2))
assert cuspidal.rational_parametrization(t) == (t**2, t**3)
I = ImplicitRegion((x, y), x**3 + x**2 - y**2)
assert I.rational_parametrization(t) == (t**2 - 1, t * (t**2 - 1))
sphere = ImplicitRegion((x, y, z), Eq(x**2 + y**2 + z**2, 2 * x))
assert sphere.rational_parametrization(
parameters=(s,
t)) == (2 / (s**2 + t**2 + 1), 2 * t / (s**2 + t**2 + 1),
2 * s / (s**2 + t**2 + 1))
conic = ImplicitRegion((x, y),
Eq(x**2 + 4 * x * y + 3 * y**2 + x - y + 10, 0))
conic.rational_parametrization(t) == (-100 - 205/(3*(3*t**2 - sqrt(41881)*t + 4*t - 2*sqrt(41881)/3 + 1)),\
-205*t/(3*(3*t**2 - sqrt(41881)*t + 4*t - 2*sqrt(41881)/3 + 1)) - sqrt(41881)/6 + 401/6)
r1 = ImplicitRegion((x, y), y**2 - x**3 + x)
raises(NotImplementedError, lambda: r1.rational_parametrization())
r2 = ImplicitRegion((x, y), y**2 - x**3 - x**2 + 1)
raises(NotImplementedError, lambda: r2.rational_parametrization())
def test_regular_point():
r1 = ImplicitRegion((x, ), x**2 - 16)
r1.regular_point() == (-4, )
c = ImplicitRegion((x, y), x**2 + y**2 - 4)
c.regular_point() == (-2, 0)
r2 = ImplicitRegion((x, y), (x - S(5) / 2)**2 + y**2 - (S(1) / 4)**2)
raises(NotImplementedError, lambda: r2.regular_point())