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