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""" | |
in trace_a s d=[0,0] n=1 | |
in trace_b s d=[0,0] n=1 | |
in colorize s d=1 n=2 | |
in reverse s d=1 n=2 | |
out centers s | |
out radii s | |
out hue s | |
""" | |
# source: http://bugman123.com/Fractals/index.html Kleinian Double 1/15 Cusp Group Pov-Ray Code | |
import cmath | |
ta = trace_a[0] + trace_a[1]*1.0j | |
tb = trace_b[0] + trace_b[1]*1.0j | |
# ta = 1.958591030 - 0.011278560 * 1.0j | |
# tb = 2.0 + 0.0j | |
# tab = (ta*tb - cmath.sqrt(ta**2*tb**2 - 4*(ta**2 + tb**2))) / 2.0 | |
tab = (ta*tb + cmath.sqrt((ta**2)*(tb**2) - 4*(ta**2 + tb**2))) / 2.0 | |
z0 = (tab-2.0)*tb / (tb*tab - 2*ta + 2*tab*1.0j) | |
b = np.array([[(tb - 2.0j)/2.0, tb/2.0], [tb/2.0, (tb + 2.0j)/2.0]]) | |
B = np.linalg.inv(b) | |
a = np.array([[tab, (tab-2.0)/z0], [(tab+2.0)*z0, tab]]).dot(B) | |
# a = np.array([[ta/2.0, (ta*tab - 2*tb + 4.0j)/(2*tab + 4)/z0], [(ta*tab - 2*tb - 4.0j)*z0/(2*tab - 4), ta/2.0]]) | |
A = np.linalg.inv(a) | |
def Fix(X): | |
return (X[0][0] - X[1][1] \ | |
- cmath.sqrt(4.0*X[0][1]*X[1][0] + (X[0][0] - X[1][1])**2)) / (2.0*X[1][0]) | |
def ToMatrix(z, r): | |
Z = np.array([[z, r**2 - z*z.conjugate()], [1.0, -z.conjugate()]]) | |
return (1.0j/r) * Z | |
def MotherCircle(M1, M2, M3): | |
z1, z2, z3 = Fix(M1), Fix(M2), Fix(M3) | |
x1, x2, x3 = z1.real, z2.real, z3.real | |
y1, y2, y3 = z1.imag, z2.imag, z3.imag | |
zx = (x3**2)*(y1 - y2) + ((x1**2) + (y1 - y2)*(y1-y3))*(y2-y3) + (x2**2)*(y3-y1) | |
zy = -(x2**2)*x3 + (x1**2)*(x3 - x2) + x3*(y1 - y2)*(y1 + y2) \ | |
+ x1*(x2**2 - x3**2 + y2**2 - y3**2) + x2*(x3**2-y1**2+y3**2) \ | |
/ (2*(x3*(y1 - y2) + x1*(y2 - y3) + x2*(y3 - y1))) | |
z = zx + (zy * 1.0j) | |
return ToMatrix(z, abs(z1 - z)) | |
# C1 = MotherCircle(b, np.dot(a, np.dot(b, A)), np.dot(a, np.dot(b, np.dot(A, B)))) | |
# C2 = MotherCircle(np.dot(b, np.linalg.matrix_power(a,15)), np.dot(a, np.dot(b, np.linalg.matrix_power(a,14))), np.dot(a, np.dot(b, np.dot(A,B)))); | |
C1 = MotherCircle(b, a.dot(b.dot(A)), a.dot(b.dot(A.dot(B)))) | |
C2 = MotherCircle(b.dot(np.linalg.matrix_power(a,15)), a.dot(b.dot(np.linalg.matrix_power(a,14))), a.dot(b.dot(A.dot(B)))); | |
def Reflect(C, M): | |
return M.dot(C.dot(np.linalg.inv( \ | |
np.array([[M[0][0].conjugate(), M[0][1].conjugate()], \ | |
[M[1][0].conjugate(), M[1][1].conjugate()]])))) | |
def zcen(Z): | |
return (Z[0][0] / Z[1][0]) | |
def rad(Z): | |
return (1.0j/Z[1][0]).real | |
def spiral(C0, M, n, colorize_, reverse_): | |
c = [] | |
r = [] | |
h = [] | |
C = C0 | |
i = 0 | |
while i < (n - 1): | |
if colorize_: | |
hue = (18 + (1 if reverse else -1)*i)/15 % 1 | |
h.append(hue) | |
z = zcen(C) | |
c.append([float(z.real), float(z.imag), 0.0]) | |
r.append(float(rad(C))) | |
i = i + 1 | |
C = Reflect(C, M) | |
return c, r, h | |
centers_np = np.array([]) | |
radii_np = np.array([]) | |
hue_np = np.array([]) | |
c, r, h = spiral(C1, a, 83, colorize, reverse) | |
centers_np = np.append(centers_np, c) | |
radii_np = np.append(radii_np, r) | |
hue_np = np.append(hue_np, h) | |
c, r, h = spiral(C1, A, 83, colorize, reverse) | |
centers_np = np.append(centers_np, c) | |
radii_np = np.append(radii_np, r) | |
hue_np = np.append(hue_np, h) | |
c, r, h = spiral(C2, a, 91, colorize, reverse) | |
centers_np = np.append(centers_np, c) | |
radii_np = np.append(radii_np, r) | |
hue_np = np.append(hue_np, h) | |
c, r, h = spiral(C2, A, 76, colorize, reverse) | |
centers_np = np.append(centers_np, c) | |
radii_np = np.append(radii_np, r) | |
hue_np = np.append(hue_np, h) | |
centers = [centers_np.reshape(-1,3).tolist()] | |
radii = [radii_np.tolist()] | |
hue = [hue_np.tolist()] | |
# spiral(C1, a, 83, 1, 0) | |
# spiral(C1, A, 83, 1, 1) | |
# spiral(C2, a, 91, 0, 0) | |
# spiral(C2, A, 76, 0, 1) | |
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