def VectorComponents(X, basis):
(coefs, bases) = linear_expand(X.obj)
cdict = {}
for (coef, base) in zip(coefs, bases):
cdict[str(base)] = coef
comp = []
for base in basis:
if base in cdict:
comp.append(cdict[base])
else:
comp.append(0)
return comp
def VectorComponents(X, basis):
(coefs, bases) = linear_expand(X.obj)
cdict = {}
for (coef, base) in zip(coefs, bases):
cdict[str(base)] = coef
comp = []
for base in basis:
if base in cdict:
comp.append(cdict[base])
else:
comp.append(0)
return comp
def __init__(self, x, coords, debug=False, I=None):
"""
coords: list of coordinate variables
x: vector fuction of coordinate variables (parametric surface)
"""
self.I = I
self.x = x
self.coords = coords
self.basis = []
self.basis_str = []
self.embedded_basis = []
for u in coords:
tv = x.diff(u)
self.basis.append(tv)
(coefs, bases) = linear_expand(tv.obj)
tc = {}
for (coef, base) in zip(coefs, bases):
str_base = str(base)
tc[str_base] = coef
if str_base not in self.embedded_basis:
self.embedded_basis.append(str_base)
self.basis_str.append(tc)
self.gij = []
for base1 in self.basis:
tmp = []
for base2 in self.basis:
tmp.append(simplify(trigsimp((base1 | base2).scalar())))
self.gij.append(tmp)
for tv in self.basis_str:
for base in self.embedded_basis:
if base not in tv:
tv[base] = 0
self.dim = len(self.basis)
indexes = tuple(range(self.dim))
self.index = [()]
for i in indexes:
self.index.append(tuple(combinations(indexes, i + 1)))
self.index = tuple(self.index)
self.MFbasis = [[MV.ONE], self.basis]
for igrade in self.index[2:]:
grade = []
for iblade in igrade:
blade = MV(1, 'scalar')
for ibasis in iblade:
blade ^= self.basis[ibasis]
blade = blade.trigsimp(deep=True, recursive=True)
grade.append(blade)
self.MFbasis.append(grade)
self.E = self.MFbasis[-1][0]
self.E_sq = trigsimp((self.E * self.E).scalar(),
deep=True,
recursive=True)
duals = copy.copy(self.MFbasis[-2])
duals.reverse()
sgn = 1
self.rbasis = []
for dual in duals:
recpv = (sgn * dual * self.E).trigsimp(deep=True, recursive=True)
self.rbasis.append(recpv)
sgn = -sgn
self.dbasis = []
for base in self.basis:
dbase = []
for coord in self.coords:
d = base.diff(coord).trigsimp(deep=True, recursive=True)
dbase.append(d)
self.dbasis.append(dbase)
self.surface = {}
(coefs, bases) = linear_expand(self.x.obj)
for (coef, base) in zip(coefs, bases):
self.surface[str(base)] = coef
self.grad = MV()
self.grad.is_grad = True
self.grad.blade_rep = True
self.grad.igrade = 1
self.grad.rcpr_bases_MV = []
for rbase in self.rbasis:
self.grad.rcpr_bases_MV.append(rbase / self.E_sq)
self.grad.rcpr_bases_MV = tuple(self.grad.rcpr_bases_MV)
self.grad.coords = self.coords
self.grad.norm = self.E_sq
self.grad.connection = {}
if debug:
oprint('x', self.x, 'coords', self.coords, 'basis vectors',
self.basis, 'index', self.index, 'basis blades',
self.MFbasis, 'E', self.E, 'E**2', self.E_sq, '*basis',
duals, 'rbasis', self.rbasis, 'basis derivatives',
self.dbasis, 'surface', self.surface, 'basis strings',
self.basis_str, 'embedding basis', self.embedded_basis,
'metric tensor', self.gij)
def __init__(self, x, coords, debug=False, I=None):
"""
coords: list of coordinate variables
x: vector fuction of coordinate variables (parametric surface)
"""
self.I = I
self.x = x
self.coords = coords
self.basis = []
self.basis_str = []
self.embedded_basis = []
for u in coords:
tv = x.diff(u)
self.basis.append(tv)
(coefs, bases) = linear_expand(tv.obj)
tc = {}
for (coef, base) in zip(coefs, bases):
str_base = str(base)
tc[str_base] = coef
if str_base not in self.embedded_basis:
self.embedded_basis.append(str_base)
self.basis_str.append(tc)
self.gij = []
for base1 in self.basis:
tmp = []
for base2 in self.basis:
tmp.append(simplify(trigsimp((base1 | base2).scalar())))
self.gij.append(tmp)
for tv in self.basis_str:
for base in self.embedded_basis:
if base not in tv:
tv[base] = 0
self.dim = len(self.basis)
indexes = tuple(range(self.dim))
self.index = [()]
for i in indexes:
self.index.append(tuple(combinations(indexes, i + 1)))
self.index = tuple(self.index)
self.MFbasis = [[MV.ONE], self.basis]
for igrade in self.index[2:]:
grade = []
for iblade in igrade:
blade = MV(1, 'scalar')
for ibasis in iblade:
blade ^= self.basis[ibasis]
blade = blade.trigsimp(deep=True, recursive=True)
grade.append(blade)
self.MFbasis.append(grade)
self.E = self.MFbasis[-1][0]
self.E_sq = trigsimp((self.E * self.E).scalar(), deep=True, recursive=True)
duals = copy.copy(self.MFbasis[-2])
duals.reverse()
sgn = 1
self.rbasis = []
for dual in duals:
recpv = (sgn * dual * self.E).trigsimp(deep=True, recursive=True)
self.rbasis.append(recpv)
sgn = -sgn
self.dbasis = []
for base in self.basis:
dbase = []
for coord in self.coords:
d = base.diff(coord).trigsimp(deep=True, recursive=True)
dbase.append(d)
self.dbasis.append(dbase)
self.surface = {}
(coefs, bases) = linear_expand(self.x.obj)
for (coef, base) in zip(coefs, bases):
self.surface[str(base)] = coef
self.grad = MV()
self.grad.is_grad = True
self.grad.blade_rep = True
self.grad.igrade = 1
self.grad.rcpr_bases_MV = []
for rbase in self.rbasis:
self.grad.rcpr_bases_MV.append(rbase / self.E_sq)
self.grad.rcpr_bases_MV = tuple(self.grad.rcpr_bases_MV)
self.grad.coords = self.coords
self.grad.norm = self.E_sq
self.grad.connection = {}
if debug:
oprint('x', self.x,
'coords', self.coords,
'basis vectors', self.basis,
'index', self.index,
'basis blades', self.MFbasis,
'E', self.E,
'E**2', self.E_sq,
'*basis', duals,
'rbasis', self.rbasis,
'basis derivatives', self.dbasis,
'surface', self.surface,
'basis strings', self.basis_str,
'embedding basis', self.embedded_basis,
'metric tensor', self.gij)
