def Astroid(x, y):
"""Private: solve astroid equation."""
# Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
# This solution is adapted from Geocentric::Reverse.
p = Math.sq(x)
q = Math.sq(y)
r = (p + q - 1) / 6
if not (q == 0 and r <= 0):
# Avoid possible division by zero when r = 0 by multiplying equations
# for s and t by r^3 and r, resp.
S = p * q / 4 # S = r^3 * s
r2 = Math.sq(r)
r3 = r * r2
# The discrimant of the quadratic equation for T3. This is zero on
# the evolute curve p^(1/3)+q^(1/3) = 1
disc = S * (S + 2 * r3)
u = r
if (disc >= 0):
T3 = S + r3
# Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
# of precision due to cancellation. The result is unchanged because
# of the way the T is used in definition of u.
T3 += -math.sqrt(disc) if T3 < 0 else math.sqrt(
disc) # T3 = (r * t)^3
# N.B. cbrt always returns the real root. cbrt(-8) = -2.
T = Math.cbrt(T3) # T = r * t
# T can be zero; but then r2 / T -> 0.
u += T + (r2 / T if T != 0 else 0)
else:
# T is complex, but the way u is defined the result is real.
ang = math.atan2(math.sqrt(-disc), -(S + r3))
# There are three possible cube roots. We choose the root which
# avoids cancellation. Note that disc < 0 implies that r < 0.
u += 2 * r * math.cos(ang / 3)
v = math.sqrt(Math.sq(u) + q) # guaranteed positive
# Avoid loss of accuracy when u < 0.
uv = q / (v - u) if u < 0 else u + v # u+v, guaranteed positive
w = (uv - q) / (2 * v) # positive?
# Rearrange expression for k to avoid loss of accuracy due to
# subtraction. Division by 0 not possible because uv > 0, w >= 0.
k = uv / (math.sqrt(uv + Math.sq(w)) + w) # guaranteed positive
else: # q == 0 && r <= 0
# y = 0 with |x| <= 1. Handle this case directly.
# for y small, positive root is k = abs(y)/sqrt(1-x^2)
k = 0
return k
def Astroid(x, y):
"""Private: solve astroid equation."""
# Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
# This solution is adapted from Geocentric::Reverse.
p = Math.sq(x)
q = Math.sq(y)
r = (p + q - 1) / 6
if not(q == 0 and r <= 0):
# Avoid possible division by zero when r = 0 by multiplying equations
# for s and t by r^3 and r, resp.
S = p * q / 4 # S = r^3 * s
r2 = Math.sq(r)
r3 = r * r2
# The discrimant of the quadratic equation for T3. This is zero on
# the evolute curve p^(1/3)+q^(1/3) = 1
disc = S * (S + 2 * r3)
u = r
if (disc >= 0):
T3 = S + r3
# Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
# of precision due to cancellation. The result is unchanged because
# of the way the T is used in definition of u.
T3 += -math.sqrt(disc) if T3 < 0 else math.sqrt(disc) # T3 = (r * t)^3
# N.B. cbrt always returns the real root. cbrt(-8) = -2.
T = Math.cbrt(T3) # T = r * t
# T can be zero; but then r2 / T -> 0.
u += T + (r2 / T if T != 0 else 0)
else:
# T is complex, but the way u is defined the result is real.
ang = math.atan2(math.sqrt(-disc), -(S + r3))
# There are three possible cube roots. We choose the root which
# avoids cancellation. Note that disc < 0 implies that r < 0.
u += 2 * r * math.cos(ang / 3)
v = math.sqrt(Math.sq(u) + q) # guaranteed positive
# Avoid loss of accuracy when u < 0.
uv = q / (v - u) if u < 0 else u + v # u+v, guaranteed positive
w = (uv - q) / (2 * v) # positive?
# Rearrange expression for k to avoid loss of accuracy due to
# subtraction. Division by 0 not possible because uv > 0, w >= 0.
k = uv / (math.sqrt(uv + Math.sq(w)) + w) # guaranteed positive
else: # q == 0 && r <= 0
# y = 0 with |x| <= 1. Handle this case directly.
# for y small, positive root is k = abs(y)/sqrt(1-x^2)
k = 0
return k
