ellipsoid.py 5.27 KB
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from sympy import *
import numpy as np
import collections

x, y, z = symbols("x y z")
a, b, c = symbols("a b c")

# simplify entries of an array
def simplify_all(A):
  if isinstance(A, collections.Iterable):
    return list(map(lambda a: simplify_all(a), A))
  else:
    return simplify(A)

def add(A,B):
  if isinstance(A, collections.Iterable):
    return list(map(lambda a,b: add(a,b), A,B))
  else:
    return A + B

def negate(A):
  if isinstance(A, collections.Iterable):
    return list(map(lambda a: negate(a), A))
  else:
    return -A


X = [x,y,z]

# normal vector
div = sqrt(b**4*c**4*x**2 + a**4*c**4*y**2 + a**4*b**4**z**2)
N = [b**2*c**2*x/div, a**2*c**2*y/div, a**2*b**2*z/div]

print("N = ")
print("return {")
for i in range(3):
  print("  ",ccode(N[i]),",")
print("};")
print("")

# projection
P = [[ (1 if i==j else 0) - N[i]*N[j] for j in range(3)] for i in range(3)]

# parametrization
nrm_X = sqrt(x**2 + y**2 + z**2)
X0 = [x/nrm_X, y/nrm_X, z/nrm_X]
u = atan(X0[1]/X0[0])
v = acos(X0[2])
X1 = [a*cos(u)*sin(v), b*sin(u)*sin(v), c*cos(v)]

# jacobian of parametrization
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J = [[diff(X1[i],X[j]) for i in range(3)] for j in range(3)]
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print("J = ")
print("return {")
for i in range(3):
  print("  {")
  for j in range(3):
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    print("    ",ccode(simplify(J[i][j])),",")
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  print("  },")
print("};")
print("")

# P(F)_j
def project1(F):
  return simplify_all([
    np.sum([
      P[i][j] * F[i]
    for i in range(3)])
    for j in range(3)])

# P(F)_kl
def project2(F):
  return simplify_all([[
    np.sum([np.sum([
      P[i][k] * P[j][l] * F[i][j]
    for j in range(3)]) for i in range(3)])
    for l in range(3)]  for k in range(3)])

# P(F)_lmn
def project3(F):
  return simplify_all([[[
    np.sum([np.sum([np.sum([
      P[i][l] * P[j][m] * P[k][n] * F[i][j][k]
    for k in range(3)]) for j in range(3)]) for i in range(3)])
    for n in range(3)]  for m in range(3)]  for l in range(3)])

# P(F)_mnop
def project4(F):
  return simplify_all([[[[
    np.sum([np.sum([np.sum([np.sum([
      P[i][m] * P[j][n] * P[k][o] * P[l][p] * F[i][j][k][l]
    for l in range(3)]) for k in range(3)]) for j in range(3)]) for i in range(3)])
    for p in range(3)]  for o in range(3)]  for n in range(3)]  for m in range(3)])


# Euclidean gradient
def Grad0(f):
  return simplify_all([
    diff(f, X[i])
  for i in range(3)])

# surface gradient (covariant derivative of scalars)
def grad0(f):
  return project1(Grad0(f))

def Grad1(T):
  return simplify_all([[
    diff(T[i], X[j])
  for j in range(3)] for i in range(3)])

# surface shape operator
#B = negate(project2(Grad1(N)))

# covariant derivative of vector field
def grad1(T):
  return simplify_all(add(project2(Grad1(T)), [[
    np.sum([
      B[i][j] * T[k] * N[k]
    for k in range(3)])
    for j in range(3)] for i in range(3)]))

def Grad2(T):
  return simplify_all([[[
    diff(T[i][j], X[k])
  for k in range(3)] for j in range(3)] for i in range(3)])

def grad2(T):
  return simplify_all([[[
    np.sum([np.sum([np.sum([
      P[L1][I1] * P[L2][I2] * P[L3][K] * diff(T[L1][L2], X[L3])
    for L3 in range(3)]) for L2 in range(3)]) for L1 in range(3)]) +
    np.sum([np.sum([
      B[K][I1] * P[J2][I2] * T[L][J2] * N[L]
    for J2 in range(3)]) for L in range(3)]) +
    np.sum([np.sum([
      B[K][I2] * P[J1][I1] * T[J1][L] * N[L]
    for J1 in range(3)]) for L in range(3)])
    for K in range(3)]  for I2 in range(3)] for I1 in range(3)])

def Grad3(T):
  return simplify_all([[[[
    diff(T[i][j][k], X[l])
  for l in range(3)] for k in range(3)] for j in range(3)] for i in range(3)])

def grad3(T):
  return simplify_all(add(project4(Grad3(T)), [[[[
    np.sum([np.sum([np.sum([
      B[K][I1] * P[J2][I2] * P[J3][I3] * T[L][J2][J3] * N[L]
    for J2 in range(3)]) for J3 in range(3)]) for L in range(3)]) +
    np.sum([np.sum([np.sum([
      B[K][I2] * P[J1][I1] * P[J3][I3] * T[J1][L][J3] * N[L]
    for J1 in range(3)]) for J3 in range(3)]) for L in range(3)]) +
    np.sum([np.sum([np.sum([
      B[K][I3] * P[J1][I1] * P[J2][I2] * T[J1][J2][L] * N[L]
    for J1 in range(3)]) for J2 in range(3)]) for L in range(3)])
  for K in range(3)] for I3 in range(3)] for I2 in range(3)] for I1 in range(3)]))



# normal-rotation of scalar field
def rot0(f):
  return [diff(f*N[2],y) - diff(f*N[1],z),
          diff(f*N[0],z) - diff(f*N[2],x),
          diff(f*N[1],x) - diff(f*N[0],y)]

def Div1(F):
  return diff(F[0],x) + diff(F[1],y) + diff(F[2],z)

def div1(F):
  return Div1(project1(F))

# def div2(t):
#   F = Matrix([div1(t.row(0).T), div1(t.row(1).T), div1(t.row(2).T)])
#   return P*F

# div(T)_I1,I2
def div3(T):
  return simplify_all([[
    np.sum([np.sum([np.sum([np.sum([np.sum([
      P[L1][I1] * P[L2][I2] * P[L3][K] * P[L4][K] * diff(T[L1][L2][L3],X[L4])
    for K in range(3)]) for L4 in range(3)]) for L3 in range(3)]) for L2 in range(3)]) for L1 in range(3)]) +
    np.sum([np.sum([
      B[I3][I1] * T[L][I2][I3] * N[L] +
      B[I3][I2] * T[I1][L][I3] * N[L] +
      B[I3][I3] * T[I1][I2][L] * N[L]
    for I3 in range(3)]) for L in range(3)])
    for I2 in range(3)]  for I1 in range(3)])

#p0 = simplify_all( rot0(x*y*z) ) # => vector
#print("p0 = ", p0)

# p1 = simplify_all( grad1(p0) ) # => 2-tensor

# print("p1 = ", p1)

# f = simplify_all( add(negate(div3(grad2(p1))), p1) )
# print("f = ", f)

#F = simplify( -div2(grad1(p0)) + p0 )
#print("F = ", F)
#print("F*N = ", simplify(F.dot(N)))