def test_solve_biquadratic():
x0, y0, x1, y1, r = symbols('x0 y0 x1 y1 r')
f_1 = (x - 1)**2 + (y - 1)**2 - r**2
f_2 = (x - 2)**2 + (y - 2)**2 - r**2
s = sqrt(2*r**2 - 1)
a = (3 - s)/2
b = (3 + s)/2
assert solve_poly_system([f_1, f_2], x, y) == [(a, b), (b, a)]
f_1 = (x - 1)**2 + (y - 2)**2 - r**2
f_2 = (x - 1)**2 + (y - 1)**2 - r**2
assert solve_poly_system([f_1, f_2], x, y) == \
[(1 - sqrt(((2*r - 1)*(2*r + 1)))/2, S(3)/2),
(1 + sqrt(((2*r - 1)*(2*r + 1)))/2, S(3)/2)]
query = lambda expr: expr.is_Pow and expr.exp is S.Half
f_1 = (x - 1 )**2 + (y - 2)**2 - r**2
f_2 = (x - x1)**2 + (y - 1)**2 - r**2
result = solve_poly_system([f_1, f_2], x, y)
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(r.count(query) == 1 for r in flatten(result))
f_1 = (x - x0)**2 + (y - y0)**2 - r**2
f_2 = (x - x1)**2 + (y - y1)**2 - r**2
result = solve_poly_system([f_1, f_2], x, y)
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(len(r.find(query)) == 1 for r in flatten(result))
s1 = (x*y - y, x**2 - x)
assert solve(s1) == [{x: 1}, {x: 0, y: 0}]
s2 = (x*y - x, y**2 - y)
assert solve(s2) == [{y: 1}, {x: 0, y: 0}]
gens = (x, y)
for seq in (s1, s2):
(f, g), opt = parallel_poly_from_expr(seq, *gens)
raises(SolveFailed, lambda: solve_biquadratic(f, g, opt))
seq = (x**2 + y**2 - 2, y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == [
(-1, -1), (-1, 1), (1, -1), (1, 1)]
ans = [(0, -1), (0, 1)]
seq = (x**2 + y**2 - 1, y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == ans
seq = (x**2 + y**2 - 1, x**2 - x + y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == ans
def test_solve_biquadratic():
x0, y0, x1, y1, r = symbols('x0 y0 x1 y1 r')
f_1 = (x - 1)**2 + (y - 1)**2 - r**2
f_2 = (x - 2)**2 + (y - 2)**2 - r**2
s = sqrt(2*r**2 - 1)
a = (3 - s)/2
b = (3 + s)/2
assert solve_poly_system([f_1, f_2], x, y) == [(a, b), (b, a)]
f_1 = (x - 1)**2 + (y - 2)**2 - r**2
f_2 = (x - 1)**2 + (y - 1)**2 - r**2
assert solve_poly_system([f_1, f_2], x, y) == \
[(1 - sqrt(((2*r - 1)*(2*r + 1)))/2, Rational(3, 2)),
(1 + sqrt(((2*r - 1)*(2*r + 1)))/2, Rational(3, 2))]
query = lambda expr: expr.is_Pow and expr.exp is S.Half
f_1 = (x - 1 )**2 + (y - 2)**2 - r**2
f_2 = (x - x1)**2 + (y - 1)**2 - r**2
result = solve_poly_system([f_1, f_2], x, y)
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(r.count(query) == 1 for r in flatten(result))
f_1 = (x - x0)**2 + (y - y0)**2 - r**2
f_2 = (x - x1)**2 + (y - y1)**2 - r**2
result = solve_poly_system([f_1, f_2], x, y)
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(len(r.find(query)) == 1 for r in flatten(result))
s1 = (x*y - y, x**2 - x)
assert solve(s1) == [{x: 1}, {x: 0, y: 0}]
s2 = (x*y - x, y**2 - y)
assert solve(s2) == [{y: 1}, {x: 0, y: 0}]
gens = (x, y)
for seq in (s1, s2):
(f, g), opt = parallel_poly_from_expr(seq, *gens)
raises(SolveFailed, lambda: solve_biquadratic(f, g, opt))
seq = (x**2 + y**2 - 2, y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == [
(-1, -1), (-1, 1), (1, -1), (1, 1)]
ans = [(0, -1), (0, 1)]
seq = (x**2 + y**2 - 1, y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == ans
seq = (x**2 + y**2 - 1, x**2 - x + y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == ans