"""
Geometric Algebra (inherits Metric)
"""
import operator
import copy
from sympy import diff, Rational, Symbol, S, Mul, Pow, Add, \
collect, expand, simplify, eye, trigsimp, sin, cos, sinh, cosh, \
symbols, sqrt, Abs, numbers, Integer, Function
import sympy
from collections import OrderedDict
#from sympy.core.compatibility import combinations
from itertools import combinations
from . import printer
from . import metric
from . import mv
from . import lt
from . import utils
from functools import reduce
half = Rational(1, 2)
one = S(1)
zero = S(0)
def all_same(items):
return all(x == items[0] for x in items)
def is_bases_product(w):
nc_w = w.args_cnc()
nc = nc_w[1]
return len(nc) == 2 or len(nc) == 1 and nc[0].is_Pow and nc[0].exp == 2
[docs]class auto_update_dict(dict):
"""
auto_update_dict creats entries to a dictionary on the fly. When the
dictionary is called and the key used is not one of the existing keys
The function self.f_update(key) is called to evaluate the key. The
result is then added to the dictionary so that self.f_update is not
used to evaluate the same key again.
The __init__ is used to input self.f_update for a given dictionary.
"""
def __init__(self, f_update, instance=None):
self.f_update = f_update
self.instance = instance
self._dict = {}
def __getitem__(self, key):
try:
return dict.__getitem__(self, key)
except KeyError:
try:
if self.instance is None:
f_key = self.f_update(key)
else:
f_key = self.f_update(self.instance, key)
self[key] = f_key
dict.__setitem__(self, key, f_key)
return dict.__getitem__(self, key)
except ValueError:
raise ValueError('"f_update(' + str(key) + ')" not defined '
+ 'for key')
[docs]def update_and_substitute(expr1, expr2, func, mul_dict):
"""
Linear expand expr1 and expr2 to get (summation convention)
expr1 = coefs1[i]*bases1[i]
expr2 = coefs2[j]*bases2[j]
where coefs1 and coefs2 are lists of are commutative expressions and
bases1 and bases2 are lists of bases for the geometric algebra.
Then evaluate
expr = coefs1[i]*coefs2[j]*F(bases1[i],bases2[j])
where F(bases1[i],bases2[j]) is a function that returns the appropriate
product of bases1[i]*bases2[j] as a linear combination of scalars and
bases of the geometric algebra.
"""
if (isinstance(expr1, numbers.Number) or expr1.is_commutative) \
or (isinstance(expr2, numbers.Number) or expr2.is_commutative):
return expr1 * expr2
(coefs1, bases1) = metric.linear_expand(expr1)
(coefs2, bases2) = metric.linear_expand(expr2)
expr = S(0)
for (coef1, base1) in zip(coefs1, bases1):
for (coef2, base2) in zip(coefs2, bases2):
#Special cases where base1 and/or base2 is scalar
if base1 == 1 and base2 == 1:
expr += coef1 * coef2
elif base1 == 1:
expr += coef1 * coef2 * base2
elif base2 == 1:
expr += coef1 * coef2 * base1
else:
key = (base1, base2)
#Update mul dictionary for future
if key not in mul_dict:
mul_dict[key] = func(key)
expr += coef1 * coef2 * mul_dict[key]
return expr
[docs]def nc_subs(expr, base_keys, base_values=None):
"""
See if expr contains nc keys in base_keys and substitute corresponding
value in base_values for nc key. This was written since standard
sympy subs was very slow in performing this operation for non-commutative
keys for long lists of keys.
"""
if base_values is None:
[base_keys, base_values] = list(zip(*base_keys))
if expr.is_commutative:
return expr
if isinstance(expr, Add):
args = expr.args
else:
args = [expr]
s = zero
for term in args:
if term.is_commutative:
s += term
else:
c, nc = term.args_cnc(split_1=False)
key = Mul._from_args(nc)
coef = Mul._from_args(c)
if key in base_keys:
base = base_values[base_keys.index(key)]
s += coef * base
else:
s += term
return s
[docs]class Ga(metric.Metric):
r"""
The vector space (basis, metric, derivatives of basis vectors) is
defined by the base class 'Metric'.
The instanciating the class 'Ga' constructs the geometric algebra of
the vector space defined by the metric.
The construction includes the multivector bases, multiplication
tables or functions for the geometric (*), inner (|), outer (^)
products, plus the left (<) and right (>) contractions. The
geometric derivative operator and any required connections for the
derivative are also calculated.
Except for the geometric product in the case of a non-orthogonal
set of basis vectors all products and connections (if needed) are
calculated when needed and place in dictionaries (lists of tuples)
to be used when needed. This greatly speeds up evaluations of
multivector expressions over previous versions of this code since
the products of multivector bases and connection are not calculated
unless they are actually needed in the current calculation.
Only instantiate the Ga class via the Mv class or any use of enhanced
printing (text or latex) will cause the bases and multiplication
table entries to be incorrectly labeled .
Data Variables -
Inherited from Metric class -
g[,]: Metric tensor (sympy matrix)
g_inv[,]: Inverse of metric tensor (sympy matirx)
norm: Normalized diagonal metric tensor (list of sympy numbers)
coords[]: Coordinate variables (list of sympy symbols)
is_ortho: True if basis is orthogonal (bool)
connect_flg: True if connection is non-zero (bool)
basis[]: Basis vector symbols (list of non-commutative sympy variables)
r_symbols[]: Reciprocal basis vector symbols (list of non-commutative sympy variables)
n: Dimension of vector space/manifold (integer)
n_range: List of basis indices
de[][]: Derivatives of basis functions. Two dimensional list. First
entry is differentiating coordiate. Second entry is basis
vector. Quantities are linear combinations of basis vector
symbols.
Basis, basis bases, and basis blades data structures -
indexes[]: Index list for multivector bases and blades by grade (tuple of tuples). Tuple
so that indexes can be used to index dictionaries.
bases[]: List of bases (non-commutative sympy symbols). Only created for
non-orthogonal basis vectors.
blades[]: List of basis blades (non-commutative sympy symbols). For
orthogonal basis vectors the same as bases.
coord_vec: Linear combination of coordinates and basis vectors. For
example in orthogonal 3D x*e_x+y*e_y+z*e_z.
blades_to_indexes_dict{}: Map basis blades to index tuples (dictionary).
indexes_to_blades_dict{}: Map index tuples to basis blades (dictionary).
bases_to_indexes_dict{}: Map basis bases to index tuples (dictionary).
indexes_to_bases_dict{}: Map index tuples to basis bases (dictionary).
pseudoI: Symbol for pseudo scalar (non-commutative sympy symbol).
Multiplication tables data structures -
Keys in all multiplication tables (*,^,|,<,>) are always symbol1*symbol2. The correct operation is known
by the context (name) of the relevant list or dictionary)
mul_table[]: Geometric products of basis blades as list of [(base1*base2, Expansion of base1*base2),...]
mul_table_dict{}: Geometric products of basis blades as dicitionary {base1*base2: Expansion of base1*base2,...}
wedge_table[]: Outer products of basis blades as list of [(base1*base2, Expansion of base1^base2),...]
wedge_table_dict{}: Outer products of basis blades as dicitionary {base1*base2: Expansion of base1^base2,...}
Reciprocal basis data structures -
r_symbols[]: Reciprocal basis vector symbols (list of non-commutative sympy variables)
r_basis[]: List of reciprocal basis vectors expanded as linear combination of basis vector symbols.
r_basis_dict{}: Dictionary to map reciprocal basis symbols to reciprocal basis expanded in terms of basis symbols
{reciprocal basis symbol: linear combination of basis symbols,...}
r_basis_mv[]: List of reciprocal basis vectors in terms of basis multivectors (elements of list can be used in
multivector expressions.)
Derivative data structures -
de[][]: Derivatives of basis functions. Two dimensional list. First entry is differentiating coordinate index.
Second entry is basis vector index. Quantities are linear combinations of basis vector symbols.
dbases{}: Dictionary of derivatives of basis blades with respect to coordinate {(coordinate index, basis blade):
derivative of basis blade with respect to coordinate,...} (Note that values in dictionary are not
multivectors, but linear combinations of basis blade symbols).
Pdop_identity: Partial differential operator identity (operates on multivector function to return function).
Pdiffs{}: Dictionary of partial differential operators (operates on multivector functions) for each coordinate
{x: \partial_{x}, ...}
sPds{}: Dictionary of scalar partial differential operators (operates on scalar functions) for each coordinate
{x: \partial_{x}, ...}
grad: Geometric derivative operator from left. grad*F returns multivector derivative, F*grad returns differential
operator.
rgrad: Geometric derivative operator from right. grad*F returns differential operator, F*grad returns multivector
derivative.
"""
dual_mode_value = 'I+'
dual_mode_lst = ['+I', 'I+', '-I', 'I-', '+Iinv', 'Iinv+', '-Iinv', 'Iinv-']
restore = False
a = []
presets = {'o3d': 'x,y,z:[1,1,1]:[1,1,0]',
'cyl3d': 'r,theta,z:[1,r**2,1]:[1,1,0]:norm=True',
'sph3d': 'r,theta,phi:[1,X[0]**2,X[0]**2*cos(X[1])**2]:[1,1,0]:norm=True',
'para3d': 'u,v,z:[u**2+v**2,u**2+v**2,1]:[1,1,0]:norm=True'}
[docs] @staticmethod
def dual_mode(mode='I+'):
"""
Sets mode of multivector dual function for all geometric algebras
in users program.
If Ga.dual_mode(mode) not called the default mode is 'I+'.
mode return value
+I I*self
-I -I*self
I+ self*I
I- -self*I
+Iinv Iinv*self
-Iinv -Iinv*self
Iinv+ self*Iinv
Iinv- -self*Iinv
"""
if mode not in Ga.dual_mode_lst:
raise ValueError('mode = ' + mode + ' not allowed for Ga.dual_mode.')
Ga.dual_mode_value = mode
return
@staticmethod
def com(A, B):
return half * (A * B - B * A)
[docs] @staticmethod
def build(*kargs, **kwargs):
"""
Static method to instantiate geometric algebra and return geometric
algebra, basis vectors, and grad operator as a tuple.
"""
GA = Ga(*kargs, **kwargs)
basis = list(GA.mv())
return tuple([GA] + basis)
@staticmethod
def preset(setting, root='e', debug=False):
if setting not in Ga.presets:
raise ValueError(str(setting) + 'not in Ga.presets.')
set_lst = Ga.presets[setting].split(':')
X = symbols(set_lst[0], real=True)
g = eval(set_lst[1])
simps = eval(set_lst[2])
kargs = [root]
kwargs = {'g': g, 'coords': X, 'debug': debug, 'I': True, 'gsym': False}
if len(set_lst) > 3:
args_lst = set_lst[-1].split(';')
for arg in args_lst:
[name, value] = arg.split('=')
kwargs[name] = eval(value)
Ga.set_simp(*simps)
return Ga(*kargs, **kwargs)
def __eq__(self, ga):
if self.name == ga.name:
return True
return False
def __init__(self, bases, **kwargs):
# Each time a geometric algebra is intialized in setup of append
# the printer must be restored to the simple text mode (not
# enhanced text of latex printing) so that when 'str' is used to
# create symbol names the names are not mangled.
kwargs = metric.test_init_slots(metric.Metric.init_slots, **kwargs)
self.wedge_print = kwargs['wedge']
if printer.GaLatexPrinter.latex_flg:
printer.GaLatexPrinter.restore()
Ga.restore = True
metric.Metric.__init__(self, bases, **kwargs)
self.par_coords = None
self.build_bases()
self.dot_mode = '|'
self.basis_product_tables()
if self.coords is not None:
self.coords = list(self.coords)
self.e = mv.Mv(self.iobj, ga=self) # Pseudo-scalar for geometric algebra
self.e_sq = simplify(expand((self.e*self.e).scalar()))
if self.coords is not None:
self.coord_vec = sum([coord * base for (coord, base) in zip(self.coords, self.basis)])
self.build_reciprocal_basis(self.gsym)
self.Pdop_identity = mv.Pdop({},ga=self) # Identity Pdop = 1
self.Pdiffs = {}
self.sPds = {}
for x in self.coords: # Partial derivative operator for each coordinate
self.Pdiffs[x] = mv.Pdop({x:1}, ga=self)
self.sPds[x] = mv.Sdop([(S(1), self.Pdiffs[x])], ga=self)
self.grad, self.rgrad = self.grads()
else:
self.r_basis_mv = None
if self.connect_flg:
self.build_connection()
self.lt_flg = False
# Calculate normalized pseudo scalar (I**2 = +/-1)
self.sing_flg = False
if self.e_sq.is_number:
if self.e_sq == S(0):
self.sing_flg = True
print('!!!!If I**2 = 0, I cannot be normalized!!!!')
#raise ValueError('!!!!If I**2 = 0, I cannot be normalized!!!!')
if self.e_sq > S(0):
self.i = self.e/sqrt(self.e_sq)
self.i_inv = self.i
else: # I**2 = -1
self.i = self.e/sqrt(-self.e_sq)
self.i_inv = -self.i
else:
if self.Isq == '+': # I**2 = 1
self.i = self.e/sqrt(self.e_sq)
self.i_inv = self.i
else: # I**2 = -1
self.i = self.e/sqrt(-self.e_sq)
self.i_inv = -self.i
if Ga.restore: # restore printer to appropriate enhanced mode after ga is instantiated
printer.GaLatexPrinter.redirect()
if self.coords is not None:
self.grads()
if self.debug:
print('Exit Ga.__init__()')
self.a = [] # List of dummy vectors for Mlt calculations
self.agrads = {} # Gradient operator with respect to vector a
self.dslot = -1 # kargs slot for dervative, -1 for coordinates
self.XOX = self.mv('XOX','vector') # Versor test vector
def make_grad(self, a, cmpflg=False): # make gradient operator with respect to vector a
if isinstance(a,(list,tuple)):
for ai in a:
self.make_grad(ai)
return
if a in list(self.agrads.keys()):
return self.agrads[a]
if isinstance(a, mv.Mv):
ai = a.get_coefs(1)
else:
ai = a
coefs = []
pdiffs = []
for (base, coord) in zip(self.r_basis_mv, ai):
coefs.append(base)
pdiffs.append(mv.Pdop({coord: 1}, ga=self))
self.agrads[a] = mv.Dop(coefs, pdiffs, ga=self, cmpflg=cmpflg)
self.a.append(a)
return self.agrads[a]
def __str__(self):
return self.name
def E(self): # Unnoromalized pseudo-scalar
return self.e
def I(self): # Noromalized pseudo-scalar
return self.i
def X(self):
return self.mv(sum([coord*base for (coord, base) in zip(self.coords, self.basis)]))
def sdop(self, coefs, pdiffs=None):
return mv.Sdop(coefs, pdiffs, ga=self)
[docs] def mv(self, root=None, *kargs, **kwargs):
"""
Instanciate and return a multivector for this, 'self',
geometric algebra.
"""
(self.mv_I, self.mv_basis, self.mv_x) = mv.Mv.setup(ga=self)
if root is None: # Return ga basis and compute grad and rgrad
if self.coords is not None:
self.grads()
return self.mv_basis
kwargs['ga'] = self
if not utils.isstr(root):
return mv.Mv(root, *kargs, **kwargs)
if ' ' in root and ' ' not in kargs[0]:
root_lst = root.split(' ')
mv_lst = []
for root in root_lst:
mv_lst.append(mv.Mv(root, *kargs, **kwargs))
return tuple(mv_lst)
if ' ' in root and ' ' in kargs[0]:
root_lst = root.split(' ')
mvtype_lst = kargs[0].split(' ')
if len(root_lst) != len(mvtype_lst):
raise ValueError('In Ga.mv() for multiple multivectors and ' +
'multivector types incompatible args ' +
str(root_lst) + ' and ' + str(mvtype_lst))
mv_lst = []
for (root, mv_type) in zip(root_lst, mvtype_lst):
kargs = list(kargs)
kargs[0] = mv_type
kargs = tuple(kargs)
mv_lst.append(mv.Mv(root, *kargs, **kwargs))
return tuple(mv_lst)
return mv.Mv(root, *kargs, **kwargs)
[docs] def mvr(self,norm=True):
r"""
Returns tumple of reciprocal basis vectors. If norm=True or
basis vectors are orthogonal the reciprocal basis is normalized
in the sense that
e_{i}\cdot e^{j} = \delta_{i}^{j}.
If the basis is not orthogonal and norm=False then
e_{i}\cdot e^{j} = I^{2}\delta_{i}^{j}.
"""
if self.r_basis_mv is None:
self.build_reciprocal_basis(self.gsym)
if norm and not self.is_ortho:
return tuple([self.r_basis_mv[i] / self.e_sq for i in self.n_range])
else:
return tuple(self.r_basis_mv)
[docs] def bases_dict(self, prefix=None):
'''
returns a dictionary mapping basis element names to their MultiVector
instances, optionally for specific grades
if you are lazy, you might do this to populate your namespace
with the variables of a given layout.
>>> locals().update(ga.bases())
'''
if prefix is None:
prefix='e'
bl = self.mv_blades_lst
var_names = [prefix+''.join([k for k in str(b) if k.isdigit()]) for b in bl]
return {key:val for key,val in zip(var_names, bl)}
def grads(self):
if not self.is_ortho:
r_basis = [x / self.e_sq for x in self.r_basis_mv]
else:
r_basis = self.r_basis_mv
if self.norm:
r_basis = [x / e_norm for (x, e_norm) in zip(self.r_basis_mv, self.e_norm)]
pdx = [self.Pdiffs[x] for x in self.coords]
self.grad = mv.Dop(r_basis, pdx, ga=self)
self.rgrad = mv.Dop(r_basis, pdx, ga=self, cmpflg=True)
return self.grad, self.rgrad
[docs] def dop(self, *kargs, **kwargs):
"""
Instanciate and return a multivector differential operator for
this, 'self', geometric algebra.
"""
kwargs['ga'] = self
return mv.Dop(*kargs, **kwargs)
[docs] def lt(self, *kargs, **kwargs):
"""
Instanciate and return a linear transformation for this, 'self',
geometric algebra.
"""
if not self.lt_flg:
self.lt_flg = True
(self.lt_coords, self.lt_x) = lt.Lt.setup(ga=self)
kwargs['ga'] = self
return lt.Lt(*kargs, **kwargs)
[docs] def sm(self, *kargs, **kwargs):
"""
Instanciate and return a submanifold for this, 'self',
geometric algebra. See 'Sm' class for instantiation inputs.
"""
kwargs['ga'] = self
SM = Sm(*kargs, **kwargs)
return SM
def parametric(self, coords):
if not isinstance(coords, list):
raise TypeError('In Ga.parametric coords = ' + str(coords) +
' is not a list.')
if len(coords) != self.n:
raise ValueError('In Ga.parametric number of parametric functions' +
' not equal to number of coordinates.')
self.par_coords = {}
for (coord, par_coord) in zip(self.coords, coords):
self.par_coords[coord] = par_coord
return
def basis_vectors(self):
return tuple(self.basis)
[docs] def build_bases(self):
"""
The bases for the multivector (geometric) algebra are formed from
all combinations of the bases of the vector space and the scalars.
Each base is represented as a non-commutative symbol of the form -
e_{i_{1}}e_{i_{2}}...e_{i_{r}}
where 0 < i_{1} < i_{2} < ... < i_{r} and 0 < r <= n the
dimension of the vector space and 0 < i_{j} <= n. The total
number of all symbols of this form plus the scalars is 2^{n}.
Any multivector can be represented as a linear combination
of these bases and the scalars.
If the basis vectors are not orthogonal a second set of symbols
is required given by -
e_{i_{1}}^e_{i_{2}}^...^e_{i_{r}}.
These are called the blade basis for the geometric algebra and
and multivector can also be represented by a linears combination
of these blades and the scalars. The number of basis vectors
that are in the symbol for the blade is call the grade of the
blade.
Representing the multivector as a linear combination of blades
gives a blade decomposition of the multivector.
There is a linear mapping from bases to blades and blades to
bases so that one can easily convert from one representation to
another. For the case of an orthogonal set of basis vectors the
bases and blades are identical.
"""
# index list for multivector bases and blades by grade
basis_indexes = tuple(self.n_range)
self.indexes = [()]
self.indexes_lst = []
for i in basis_indexes:
base_tuple = tuple(combinations(basis_indexes, i + 1))
self.indexes.append(base_tuple)
self.indexes_lst += list(base_tuple)
self.indexes = tuple(self.indexes)
# list of non-commutative symbols for multivector bases and blades
# by grade and as a flattened list
self.blades = []
self.blades_lst = []
for grade_index in self.indexes:
blades = []
super_scripts = []
for base_index in grade_index:
if self.wedge_print:
symbol_str = (''.join([str(self.basis[i]) + '^' for i in base_index]))[:-1]
else:
sub_str = []
root_str = []
for i in base_index:
basis_vec_str = str(self.basis[i])
split_lst = basis_vec_str.split('_')
if len(split_lst) != 2:
raise ValueError('!!!!Incompatible basis vector '+basis_vec_str+' for wedge_print = False!!!!')
else:
sub_str.append(split_lst[1])
root_str.append(split_lst[0])
if all_same(root_str):
symbol_str = root_str[0] + '_' + ''.join(sub_str)
else:
raise ValueError('!!!!No unique root symbol for wedge_print = False!!!!')
blade_symbol = Symbol(symbol_str, commutative=False)
blades.append(blade_symbol)
self.blades_lst.append(blade_symbol)
self.blades.append(blades)
self.blades_lst0 = [S(1)] + self.blades_lst
self.iobj = self.blades_lst[-1]
self.blades_to_indexes = []
self.indexes_to_blades = []
for (index, blade) in zip(self.indexes_lst, self.blades_lst):
self.blades_to_indexes.append((blade, index))
self.indexes_to_blades.append((index, blade))
self.blades_to_indexes_dict = OrderedDict(self.blades_to_indexes)
self.indexes_to_blades_dict = OrderedDict(self.indexes_to_blades)
self.blades_to_grades_dict = {}
igrade = 0
for grade in self.blades:
for blade in grade:
self.blades_to_grades_dict[blade] = igrade
igrade += 1
if not self.is_ortho:
self.bases = []
self.bases_lst = []
for grade_index in self.indexes:
bases = []
for base_index in grade_index:
symbol_str = (''.join([str(self.basis[i]) + '*' for i in base_index]))[:-1]
base_symbol = Symbol(symbol_str, commutative=False)
bases.append(base_symbol)
self.bases_lst.append(base_symbol)
self.bases.append(bases)
self.pseudoI = self.bases_lst[-1]
self.bases_to_indexes = []
self.indexes_to_bases = []
for (index, base) in zip(self.indexes_lst, self.bases_lst):
self.bases_to_indexes.append((base, index))
self.indexes_to_bases.append((index, base))
self.bases_to_indexes_dict = OrderedDict(self.bases_to_indexes)
self.indexes_to_bases_dict = OrderedDict(self.indexes_to_bases)
self.bases_to_grades_dict = {}
igrade = 0
for grade in self.bases:
for base in grade:
self.bases_to_grades_dict[base] = igrade
igrade += 1
if self.coords is None:
base0 = str(self.basis[0])
if '_' in base0:
sub_index = base0.index('_')
self.basis_super_scripts = [str(base)[sub_index + 1:] for base in self.basis]
else:
self.basis_super_scripts = [str(i + 1) for i in self.n_range]
else:
self.basis_super_scripts = [str(coord) for coord in self.coords]
self.blade_super_scripts = []
for grade_index in self.indexes:
super_scripts = []
for base_index in grade_index:
super_scripts.append(''.join([self.basis_super_scripts[i]
for i in base_index]))
self.blade_super_scripts.append(super_scripts)
if self.debug:
printer.oprint('indexes', self.indexes, 'list(indexes)', self.indexes_lst,
'blades', self.blades, 'list(blades)', self.blades_lst,
'blades_to_indexes_dict', self.blades_to_indexes_dict,
'indexes_to_blades_dict', self.indexes_to_blades_dict,
'blades_to_grades_dict', self.blades_to_grades_dict,
'blade_super_scripts', self.blade_super_scripts)
if not self.is_ortho:
printer.oprint('bases', self.bases, 'list(bases)', self.bases_lst,
'bases_to_indexes_dict', self.bases_to_indexes_dict,
'indexes_to_bases_dict', self.indexes_to_bases_dict,
'bases_to_grades_dict', self.bases_to_grades_dict)
self.mv_blades_lst = []
for obj in self.blades_lst:
self.mv_blades_lst.append(self.mv(obj))
return
[docs] def basis_product_tables(self):
"""
For the different products of geometric algebra bases/blade
initialize auto-updating of bases/blades product lists. For
orthogonal bases all basis product lists are generated on the
fly using functions and the base and blade representations
are identical. For a non-orthogonal basis the multiplication
table for the geometric product is pre-calcuated for base pairs.
The tables for all other products (including the geometric
product) are calulated on the fly and updated and are for blade
pairs.
All tables are of the form
[ (blade1*blade2,f(blade1,blade1)),... ]
"""
self.mul_table = [] # Geometric product (*) of blades
self.mul_table_dict = {}
if not self.is_ortho:
self.non_orthogonal_mul_table() # Fully populated geometric product (*) multiplication table
self.base_blade_conversions() # Generates conversion dictionaries between bases and blades
self.wedge_table = [] # Outer product (^)
self.wedge_table_dict = {}
# All three (|,<,>) types of contractions use the same generation function
# self.dot_product_basis_blades. The type of dictionary entry generated depend
# on self.dot_mode = '|', '<', or '>' as set in self.dot.
self.dot_table = [] # Inner product (|)
self.dot_table_dict = {}
self.left_contract_table = [] # Left contraction (<)
self.left_contract_table_dict = {}
self.right_contract_table = [] # Right contraction (>)
self.right_contract_table_dict = {}
self.dot_mode = '|'
if self.debug:
print('Exit basis_product_tables.\n')
return
def build_connection(self):
# Partial derivatives of multivector bases multiplied (*,^,|,<,>)
# on left and right (True and False) by reciprocal basis vectors.
self.connect = {('*', True): [], ('^', True): [], ('|', True): [],
('<', True): [], ('>', True): [], ('*', False): [],
('^', False): [], ('|', False): [], ('<', False): [],
('>', False): []}
# Partial derivatives of multivector bases
self.dbases = {}
return
######## Functions for Calculation products of blades/bases ########
#******************** Geometric Product (*) ***********************#
def geometric_product_basis_blades(self, blade12):
# geometric (*) product for orthogonal basis
if self.is_ortho:
(blade1, blade2) = blade12
index1 = self.blades_to_indexes_dict[blade1]
index2 = self.blades_to_indexes_dict[blade2]
blade_index = list(index1 + index2)
repeats = []
sgn = 1
for i in range(1, len(blade_index)):
save = blade_index[i]
j = i
while j > 0 and blade_index[j - 1] > save:
sgn = -sgn
blade_index[j] = blade_index[j - 1]
j -= 1
blade_index[j] = save
if blade_index[j] == blade_index[j - 1]:
repeats.append(save)
result = S(sgn)
for i in repeats:
blade_index.remove(i)
blade_index.remove(i)
result *= self.g[i, i]
if len(blade_index) > 0:
result *= self.indexes_to_blades_dict[tuple(blade_index)]
return result
else:
(blade1, blade2) = blade12
base1 = self.blade_to_base_rep(blade1)
base2 = self.blade_to_base_rep(blade2)
base12 = expand(base1 * base2)
base12 = nc_subs(base12, self.basic_mul_keys, self.basic_mul_values)
return self.base_to_blade_rep(base12)
[docs] def reduce_basis(self, blst):
"""
Repetitively applies reduce_basis_loop to blst
product representation until normal form is
realized for non-orthogonal basis
"""
blst = list(blst)
if blst == []: # blst represents scalar
blst_coef = [1]
blst_expand = [[]]
return blst_coef, blst_expand
blst_expand = [blst]
blst_coef = [1]
blst_flg = [False]
# reduce untill all blst revise flgs are True
while not reduce(operator.and_, blst_flg):
for i in range(len(blst_flg)):
if not blst_flg[i]: # keep revising if revise flg is False
tmp = Ga.reduce_basis_loop(self.g, blst_expand[i])
if isinstance(tmp, bool):
blst_flg[i] = tmp # revision of blst_expand[i] complete
elif len(tmp) == 3: # blst_expand[i] contracted
blst_coef[i] = tmp[0] * blst_coef[i]
blst_expand[i] = tmp[1]
blst_flg[i] = tmp[2]
else: # blst_expand[i] revised
blst_coef[i] = -blst_coef[i]
#if revision force one more pass in case revision
#causes repeated index previous to revised pair of
#indexes
blst_flg[i] = False
blst_expand[i] = tmp[3]
blst_coef.append(-blst_coef[i] * tmp[0])
blst_expand.append(tmp[1])
blst_flg.append(tmp[2])
new_blst_coef = []
new_blst_expand = []
for (coef, xpand) in zip(blst_coef, blst_expand):
if xpand in new_blst_expand:
i = new_blst_expand.index(xpand)
new_blst_coef[i] += coef
else:
new_blst_expand.append(xpand)
new_blst_coef.append(coef)
return new_blst_coef, new_blst_expand
[docs] @staticmethod
def reduce_basis_loop(g, blst):
"""
blst is a list of integers [i_{1},...,i_{r}] representing the geometric
product of r basis vectors a_{{i_1}}*...*a_{{i_r}}. reduce_basis_loop
searches along the list [i_{1},...,i_{r}] untill it finds i_{j} == i_{j+1}
and in this case contracts the list, or if i_{j} > i_{j+1} it revises
the list (~i_{j} means remove i_{j} from the list)
Case 1: If i_{j} == i_{j+1}, return a_{i_{j}}**2 and
[i_{1},..,~i_{j},~i_{j+1},...,i_{r}]
Case 2: If i_{j} > i_{j+1}, return a_{i_{j}}.a_{i_{j+1}},
[i_{1},..,~i_{j},~i_{j+1},...,i_{r}], and
[i_{1},..,i_{j+1},i_{j},...,i_{r}]
"""
nblst = len(blst) # number of basis vectors
if nblst <= 1:
return True # a scalar or vector is already reduced
jstep = 1
while jstep < nblst:
istep = jstep - 1
if blst[istep] == blst[jstep]: # basis vectorindex is repeated
i = blst[istep] # save basis vector index
if len(blst) > 2:
blst = blst[:istep] + blst[jstep + 1:] # contract blst
else:
blst = []
if len(blst) <= 1 or jstep == nblst - 1:
blst_flg = True # revision of blst is complete
else:
blst_flg = False # more revision needed
return g[i, i], blst, blst_flg
if blst[istep] > blst[jstep]: # blst not in normal order
blst1 = blst[:istep] + blst[jstep + 1:] # contract blst
a1 = 2 * g[blst[jstep], blst[istep]] # coef of contraction
blst = blst[:istep] + [blst[jstep]] + [blst[istep]] + blst[jstep + 1:] # revise blst
if len(blst1) <= 1:
blst1_flg = True # revision of blst is complete
else:
blst1_flg = False # more revision needed
return a1, blst1, blst1_flg, blst
jstep += 1
return True # revision complete, blst in normal order
#******************* Outer/wedge (^) product **********************#
@staticmethod
def blade_reduce(lst):
sgn = 1
for i in range(1, len(lst)):
save = lst[i]
j = i
while j > 0 and lst[j - 1] > save:
sgn = -sgn
lst[j] = lst[j - 1]
j -= 1
lst[j] = save
if lst[j] == lst[j - 1]:
return 0, None
return sgn, lst
def wedge_product_basis_blades(self, blade12): # blade12 = blade1*blade2
# outer (^) product of basis blades
# this method works for both orthogonal and non-orthogonal basis
(blade1, blade2) = blade12
index1 = self.blades_to_indexes_dict[blade1]
index2 = self.blades_to_indexes_dict[blade2]
index12 = list(index1 + index2)
if len(index12) > self.n:
return 0
(sgn, wedge12) = Ga.blade_reduce(index12)
if sgn != 0:
return(sgn * self.indexes_to_blades_dict[tuple(wedge12)])
else:
return 0
#****** Dot (|) product, reft (<) and right (>) contractions ******#
def dot_product_basis_blades(self, blade12):
# dot (|), left (<), and right (>) products
# dot product for orthogonal basis
(blade1, blade2) = blade12
index1 = self.blades_to_indexes_dict[blade1]
index2 = self.blades_to_indexes_dict[blade2]
index = list(index1 + index2)
grade1 = len(index1)
grade2 = len(index2)
if self.dot_mode == '|':
grade = abs(grade1 - grade2)
elif self.dot_mode == '<':
grade = grade2 - grade1
if grade < 0:
return 0
elif self.dot_mode == '>':
grade = grade1 - grade2
if grade < 0:
return 0
n = len(index)
sgn = 1
result = 1
ordered = False
while n > grade:
ordered = True
i2 = 1
while i2 < n:
i1 = i2 - 1
index1 = index[i1]
index2 = index[i2]
if index1 == index2:
n -= 2
if n < grade:
return 0
result *= self.g[index1, index1]
index = index[:i1] + index[i2 + 1:]
elif index1 > index2:
ordered = False
index[i1] = index2
index[i2] = index1
sgn = -sgn
i2 += 1
else:
i2 += 1
if ordered:
break
if n > grade:
return 0
else:
if index == []:
return sgn * result
else:
return sgn * result * self.indexes_to_blades_dict[tuple(index)]
def non_orthogonal_dot_product_basis_blades(self, blade12): # blade12 = (blade1,blade2)
# dot product of basis blades if basis vectors are non-orthogonal
# inner (|), left (<), and right (>) products of basis blades
# blade12 is the sympy product of two basis blades
(blade1, blade2) = blade12
# Need base rep for blades since that is all we can multiply
base1 = self.blade_expansion_dict[blade1]
base2 = self.blade_expansion_dict[blade2]
# geometric product of basis blades
base12 = self.mul(base1, base2)
# blade rep of geometric product
blade12 = self.base_to_blade_rep(base12)
# decompose geometric product by grades
grade_dict = self.grade_decomposition(blade12)
# grades of input blades
grade1 = self.blades_to_grades_dict[blade1]
grade2 = self.blades_to_grades_dict[blade2]
if self.dot_mode == '|':
grade_dot = abs(grade2 - grade1)
if grade_dot in grade_dict:
return grade_dict[grade_dot]
else:
return zero
elif self.dot_mode == '<':
grade_contract = grade2 - grade1
if grade_contract in grade_dict:
return grade_dict[grade_contract]
else:
return zero
elif self.dot_mode == '>':
grade_contract = grade1 - grade2
if grade_contract in grade_dict:
return grade_dict[grade_contract]
else:
return zero
else:
raise ValueError('"' + str(self.dot_mode) + '" not allowed '
'dot mode in non_orthogonal_dot_basis')
############# Non-Orthogonal Tables and Dictionaries ###############
def non_orthogonal_mul_table(self):
mul_table = []
self.basic_mul_keys = []
self.basic_mul_values = []
for base1 in self.bases_lst:
for base2 in self.bases_lst:
key = base1 * base2
value = self.non_orthogonal_bases_products((base1, base2))
mul_table.append((key, value))
self.basic_mul_keys.append(key)
self.basic_mul_values.append(value)
self.basic_mul_table = mul_table
self.basic_mul_table_dict = OrderedDict(mul_table)
if self.debug:
print('basic_mul_table =\n', self.basic_mul_table)
return
def non_orthogonal_bases_products(self, base12): # base12 = (base1,base2)
# geometric product of bases for non-orthogonal basis vectors
(base1, base2) = base12
index = self.bases_to_indexes_dict[base1] + self.bases_to_indexes_dict[base2]
(coefs, indexes) = self.reduce_basis(index)
s = 0
if [] in indexes: # extract scalar part from multivector expansion
iscalar = indexes.index([])
s += coefs[iscalar]
del indexes[iscalar]
del coefs[iscalar]
for (coef, index) in zip(coefs, indexes):
s += coef * self.indexes_to_bases_dict[tuple(index)]
return s
def base_blade_conversions(self):
blade_expansion = []
blade_index = []
# expand blade basis in terms of base basis
for blade in self.blades_lst:
index = self.blades_to_indexes_dict[blade]
grade = len(index)
if grade == 1:
blade_expansion.append(blade)
blade_index.append(index)
else:
a = self.indexes_to_blades_dict[(index[0],)]
Aexpand = blade_expansion[blade_index.index(index[1:])]
# Formula for outer (^) product of a vector and grade-r multivector
# a^A_{r} = (a*A + (-1)^{r}*A*a)/2
# The folowing evaluation takes the most time for setup it is the due to
# the substitution required for the multiplications
a_W_A = half * (self.basic_mul(a, Aexpand) - ((-1) ** grade) * self.basic_mul(Aexpand, a))
blade_index.append(index)
blade_expansion.append(expand(a_W_A))
self.blade_expansion = blade_expansion
self.blade_expansion_dict = OrderedDict(list(zip(self.blades_lst, blade_expansion)))
if self.debug:
print('blade_expansion_dict =', self.blade_expansion_dict)
# expand base basis in terms of blade basis
base_expand = []
for (base, blade, index) in zip(self.bases_lst, self.blades_lst, self.indexes_lst):
grade = len(index)
if grade == 1:
base_expand.append((base, base))
else: # back substitution of tridiagonal system
tmp = self.blade_expansion_dict[blade]
tmp = tmp.subs(base, -blade)
tmp = -tmp.subs(base_expand)
base_expand.append((base, expand(tmp)))
self.base_expand = base_expand
self.base_expansion_dict = OrderedDict(base_expand)
if self.debug:
print('base_expansion_dict =', self.base_expansion_dict)
return
def base_to_blade_rep(self, A):
if self.is_ortho:
return A
else:
#return(expand(A).subs(self.base_expansion_dict))
return nc_subs(expand(A), self.base_expand)
def blade_to_base_rep(self, A):
if self.is_ortho:
return A
else:
#return(expand(A).subs(self.blade_expansion_dict))
return nc_subs(expand(A), self.blades_lst, self.blade_expansion)
###### Products (*,^,|,<,>) for multivector representations ########
def basic_mul(self, A, B): # geometric product (*) of base representations
# only multiplicative operation to assume A and B are in base representation
AxB = expand(A * B)
AxB = nc_subs(AxB, self.basic_mul_keys, self.basic_mul_values)
return expand(AxB)
def Mul(self, A, B, mode='*'): # Unifies all products into one function
if mode == '*':
return self.mul(A, B)
elif mode == '^':
return self.wedge(A, B)
else:
self.dot_mode = mode
return self.dot(A, B)
def mul(self, A, B): # geometric (*) product of blade representations
if A == 0 or B == 0:
return 0
return update_and_substitute(A, B, self.geometric_product_basis_blades, self.mul_table_dict)
def wedge(self, A, B):
# wedge assumes A and B are in blade rep
# wedge product is same for both orthogonal and non-orthogonal for A and B in blade rep
if A == 0 or B == 0:
return 0
return update_and_substitute(A, B, self.wedge_product_basis_blades, self.wedge_table_dict)
[docs] def dot(self, A, B): # inner products |, <, and >
"""
Let A = a + A' and B = b + B' where a and b are the scalar parts of
A and B and A' and B' are the remaining parts of A and B. Then
we have:
(a+A')<(b+B') = a(b+B') + A'<B'
(a+A')>(b+B') = b(a+A') + A'>B'
We use these relations to reduce A<B and A>B.
"""
if A == 0 or B == 0:
return 0
if self.is_ortho:
dot_product_basis_blades = self.dot_product_basis_blades
else:
dot_product_basis_blades = self.non_orthogonal_dot_product_basis_blades
if self.dot_mode == '|': # Hestenes dot product
A = self.remove_scalar_part(A)
B = self.remove_scalar_part(B)
return update_and_substitute(A, B, dot_product_basis_blades, self.dot_table_dict)
elif self.dot_mode == '<' or self.dot_mode == '>':
(a, Ap) = self.split_multivector(A) # Ap = A'
(b, Bp) = self.split_multivector(B) # Bp = B'
if self.dot_mode == '<': # Left contraction
if Ap != 0 and Bp != 0: # Neither nc part of A or B is zero
prod = update_and_substitute(Ap, Bp, dot_product_basis_blades, self.left_contract_table_dict)
return prod + a * B
else: # Ap or Bp is zero
return a * B
elif self.dot_mode == '>': # Right contraction
if Ap != 0 and Bp != 0: # Neither nc part of A or B is zero
prod = update_and_substitute(Ap, Bp, dot_product_basis_blades, self.right_contract_table_dict)
return prod + b * A
else: # Ap or Bp is zero
return b * A
else:
raise ValueError('"' + str(self.dot_mode) + '" not a legal mode in dot')
######################## Helper Functions ##########################
[docs] def grade_decomposition(self, A):
"""
Returns dictionary with grades as keys of grades of A. For example
if A is a rotor the dictionary keys would be 0 and 2. For a vector
the single key would be 1. Note A can be input as a multivector or
an multivector object (sympy expression). If A is a multivector the
dictionary entries are multivectors. If A is a sympy expression
(in this case a linear combination of non-commutative symbols) the
dictionary entries are sympy expressions.
"""
if isinstance(A,mv.Mv):
A.blade_rep()
A.characterise_Mv()
Aobj = expand(A.obj)
else:
Aobj = A
coefs,blades = metric.linear_expand(Aobj)
grade_dict = {}
for (coef,blade) in zip(coefs,blades):
if blade == one:
if 0 in list(grade_dict.keys()):
grade_dict[0] += coef
else:
grade_dict[0] = coef
else:
grade = self.blades_to_grades_dict[blade]
if grade in grade_dict:
grade_dict[grade] += coef * blade
else:
grade_dict[grade] = coef * blade
if isinstance(A, mv.Mv):
for grade in list(grade_dict.keys()):
grade_dict[grade] = self.mv(grade_dict[grade])
return grade_dict
[docs] def split_multivector(self, A):
"""
Split multivector A into commutative part a and non-commutative
part A' so that A = a+A'
"""
if isinstance(A, mv.Mv):
return self.split_multivector(A.obj)
else:
A = expand(A)
if isinstance(A, Add):
a = sum([x for x in A.args if x.is_commutative])
Ap = sum([x for x in A.args if not x.is_commutative])
return (a, Ap)
elif isinstance(A, Symbol):
if A.is_commutative:
return (A, 0)
else:
return (0, A)
else:
if A.is_commutative:
return (A, 0)
else:
return (0, A)
[docs] def remove_scalar_part(self, A):
"""
Return non-commutative part (sympy object) of A.obj.
"""
if isinstance(A, mv.Mv):
return self.remove_scalar_part(A.obj)
else:
if isinstance(A, Add):
A = expand(A)
return(sum([x for x in A.args if not x.is_commutative]))
elif isinstance(A, Symbol):
if A.is_commutative:
return 0
else:
return A
else:
if A.is_commutative:
return 0
else:
return A
def scalar_part(self, A):
if isinstance(A, mv.Mv):
return self.scalar_part(A.obj)
else:
A = expand(A)
if isinstance(A, Add):
return(sum([x for x in A.args if x.is_commutative]))
elif isinstance(A, Symbol):
if A.is_commutative:
return A
else:
return 0
else:
if A.is_commutative:
return A
else:
return 0
"""
else:
if A.is_commutative:
return A
else:
return zero
"""
def grades(self, A): # Return list of grades present in A
A = self.base_to_blade_rep(A)
A = expand(A)
blades = []
if isinstance(A, Add):
args = A.args
else:
args = [A]
for term in args:
blade = term.args_cnc()[1]
l_blade = len(blade)
if l_blade > 0:
if blade[0] not in blades:
blades.append(blade[0])
else:
if one not in blades:
blades.append(one)
grade_lst = []
if one in blades:
grade_lst.append(0)
for blade in blades:
if blade != one:
grade = self.blades_to_grades_dict[blade]
if grade not in grade_lst:
grade_lst.append(grade)
grade_lst.sort()
return(grade_lst)
def reverse(self, A): # Calculates reverse of A (see documentation)
A = expand(A)
blades = {}
if isinstance(A, Add):
args = A.args
else:
if A.is_commutative:
return A
else:
args = [A]
for term in args:
if term.is_commutative:
if 0 in blades:
blades[0] += term
else:
blades[0] = term
else:
_c, nc = term.args_cnc()
blade = nc[0]
grade = self.blades_to_grades_dict[blade]
if grade in blades:
blades[grade] += term
else:
blades[grade] = term
s = zero
for grade in blades:
if (grade * (grade - 1)) / 2 % 2 == 0:
s += blades[grade]
else:
s -= blades[grade]
return s
def get_grade(self, A, r): # Return grade r of A, <A>_{r}
if r == 0:
return self.scalar_part(A)
coefs, bases = metric.linear_expand(A)
s = zero
for (coef, base) in zip(coefs, bases):
if base != one and self.blades_to_grades_dict[base] == r:
s += coef * base
return s
def even_odd(self, A, even=True): # Return even or odd part of A
A = expand(A)
if A.is_commutative and even:
return A
if isinstance(A, Add):
args = A.args
else:
args = [A]
s = zero
for term in args:
if term.is_commutative:
if even:
s += term
else:
c, nc = term.args_cnc(split_1=False)
blade = nc[0]
grade = self.blades_to_grades_dict[blade]
if even and grade % 2 == 0:
s += Mul._from_args(c) * blade
elif not even and grade % 2 == 1:
s += Mul._from_args(c) * blade
return s
##################### Multivector derivatives ######################
[docs] def build_reciprocal_basis(self,gsym):
r"""
Calculate reciprocal basis vectors e^{j} where
e^{j}\cdot e_{k} = \delta_{k}^{j}
and \delta_{k}^{j} is the kronecker delta. We use the formula
from Doran and Lasenby 4.94 -
e^{j} = (-1)**{j-1}e_{1}^...e_{j-1}^e_{j+1}^...^e_{n}*E_{n}**{-1}
where E_{n} = e_{1}^...^e_{n}.
For non-orthogonal basis e^{j} is not normalized and must be
divided by E_{n}**2 (self.e_sq) in any relevant calculations.
If gsym = True then (E_{n})**2 is not evaluated, but is represented
as (E_{n})**2 = (-1)**(n*(n-1)/2)*det(g) where det(g) the determinant
of the metric tensor can be general scalar function of the coordinates.
"""
if self.debug:
print('Enter build_reciprocal_basis.\n')
if self.is_ortho:
self.r_basis = [self.basis[i] / self.g[i, i] for i in self.n_range]
else:
self.e_obj = self.e.obj
if gsym is not None:
# Define name of metric tensor determinant as sympy symbol
if printer.GaLatexPrinter.latex_flg:
det_str = r'\det\left ( ' + gsym + r'\right ) '
else:
det_str = 'det(' + gsym + ')'
# Define square of pseudo-scalar in terms of metric tensor
# determinant
n = self.n
if self.coords is None: # Metric tensor is constant
self.e_sq = (-1) ** (n*(n - 1)/2) * Symbol(det_str,real=True)
else: # Metric tensor is function of coordinates
n = len(self.coords)
self.e_sq = (-1) ** (n*(n - 1)/2) * Function(det_str,real=True)(*self.coords)
else:
self.e_sq = simplify((self.e * self.e).obj)
if self.debug:
print('E**2 =', self.e_sq)
# Take all (n-1)-blades
duals = list(self.blades_lst[-(self.n + 1):-1])
# After reverse, the j-th of them is exactly e_{1}^...e_{j-1}^e_{j+1}^...^e_{n}
duals.reverse()
sgn = 1
self.r_basis = []
for dual in duals:
dual_base_rep = self.blade_to_base_rep(dual)
# {E_n}^{-1} = \frac{E_n}{{E_n}^{2}}
# r_basis_j = sgn * duals[j] * E_n so it's not normalized, missing a factor of {E_n}^{-2}
r_basis_j = collect(expand(self.base_to_blade_rep(self.mul(sgn * dual_base_rep, self.e_obj))), self.blades_lst)
self.r_basis.append(r_basis_j)
# sgn = (-1)**{j-1}
sgn = -sgn
if self.debug:
printer.oprint('E', self.iobj, 'E**2', self.e_sq, 'unnormalized reciprocal basis =\n', self.r_basis)
self.dot_mode = '|'
print('reciprocal basis test =')
for ei in self.basis:
for ej in self.r_basis:
ei_dot_ej = self.dot(ei, ej)
if ei_dot_ej == zero:
print('e_{i}|e_{j} = ' + str(ei_dot_ej))
else:
print('e_{i}|e_{j} = ' + str(expand(ei_dot_ej / self.e_sq)))
self.e_obj = self.blades_lst[-1]
# Dictionary to represent reciprocal basis vectors as expansions
# in terms of basis vectors.
self.r_basis_dict = {}
self.r_basis_mv = []
for (r_symbol, r_base) in zip(self.r_symbols, self.r_basis):
self.r_basis_dict[r_symbol] = r_base
self.r_basis_mv.append(mv.Mv(r_base, ga=self))
# Replace reciprocal basis vectors with expansion in terms of
# basis vectors in derivatives of basis vectors
if self.connect_flg:
for x_i in self.n_range:
for jb in self.n_range:
if not self.is_ortho:
self.de[x_i][jb] = metric.Simp.apply(self.de[x_i][jb].subs(self.r_basis_dict) / self.e_sq)
else:
self.de[x_i][jb] = metric.Simp.apply(self.de[x_i][jb].subs(self.r_basis_dict))
g_inv = eye(self.n)
self.dot_mode = '|'
# Calculate inverse of metric tensor, g^{ij}
for i in self.n_range:
rx_i = self.r_symbols[i]
for j in self.n_range:
rx_j = self.r_symbols[j]
if j >= i:
g_inv[i, j] = self.dot(self.r_basis_dict[rx_i], self.r_basis_dict[rx_j])
if not self.is_ortho:
g_inv[i, j] /= self.e_sq**2
else:
g_inv[i, j] = g_inv[j, i]
self.g_inv = simplify(g_inv)
if self.debug:
print('reciprocal basis dictionary =\n', self.r_basis_dict)
# True is for left derivative and False is for right derivative
self.deriv = {('*', True): [], ('^', True): [], ('|', True): [],
('<', True): [], ('>', True): [], ('*', False): [],
('^', False): [], ('|', False): [], ('<', False): [],
('>', False): []}
return
[docs] def er_blade(self, er, blade, mode='*', left=True):
"""
Product (*,^,|,<,>) of reciprocal basis vector 'er' and basis
blade 'blade' needed for application of derivatives to
multivectors. left is 'True' means 'er' is multiplying 'blade'
on the left, 'False' is for 'er' multiplying 'blade' on the
right. Symbolically for left geometric product -
e^{j}*(e_{i_{1}}^...^e_{i_{r}})
"""
if mode == '*':
base = self.blade_to_base_rep(blade)
if left:
return self.base_to_blade_rep(self.mul(er, base))
else:
return self.base_to_blade_rep(self.mul(base, er))
elif mode == '^':
if left:
return self.wedge(er, blade)
else:
return self.wedge(blade, er)
else:
self.dot_mode = mode
if left:
return self.dot(er, blade)
else:
return self.dot(blade, er)
[docs] def blade_derivation(self, blade, ib):
"""
Calculate derivatives of basis blade 'blade' using derivative of
basis vectors calculated by metric. 'ib' is the index of the
coordinate the derivation is with respect to or the coordinate
symbol. These are requried for the calculation of the geometric
derivatives in curvilinear coordinates or for more general
manifolds.
'blade_derivation' saves the results in a dictionary, 'self.dbases',
so that the derivation for a given blade and coordinate is never
calculated more that once.
"""
if isinstance(ib, int):
coord = self.coords[ib]
else:
coord = ib
ib = self.coords.index(coord)
key = (coord, blade)
if key in self.dbases:
return self.dbases[key]
index = self.blades_to_indexes_dict[blade]
grade = len(index)
if grade == 1:
db = self.de[ib][index[0]]
elif grade == 2:
db = self.wedge(self.de[ib][index[0]], self.basis[index[1]]) + \
self.wedge(self.basis[index[0]], self.de[ib][index[1]])
else:
db = self.wedge(self.de[ib][index[0]], self.indexes_to_blades[index[1:]]) + \
self.wedge(self.indexes_to_blades[index[:-1]], self.de[ib][index[-1]])
for i in range(1, grade - 1):
db += self.wedge(self.wedge(self.indexes_to_blades[index[:i]], self.de[ib][index[i]]),
self.indexes_to_blades[index[i + 1:]])
self.dbases[key] = db
return db
def pdop(self,*kargs):
return mv.Pdop(kargs,ga=self)
[docs] def pDiff(self, A, coord):
"""
Compute partial derivative of multivector function 'A' with
respect to coordinate 'coord'.
"""
if isinstance(coord, list):
# Perform multiple partial differentiation where coord is
# a list of differentiation orders for each coordinate and
# the coordinate is determinded by the list index. If the
# element in the list is zero no differentiation is to be
# performed for that coordinate index.
dA = copy.copy(A) # Make copy of A
for i in self.n_range:
x = self.coords[i]
xn = coord[i]
if xn > 0: # Differentiate with respect to coordinate x
for _j in range(xn): # xn > 1 multiple differentiation
dA = self.pDiff(dA, x)
return dA
# Simple partial differentiation, once with respect to a single
# variable, but including case of non-constant basis vectors
dA = self.mv(expand(diff(A.obj, coord)))
if self.connect_flg and self.dslot == -1 and not A.is_scalar(): # Basis blades are function of coordinates
B = self.remove_scalar_part(A)
if B != zero:
if isinstance(B, Add):
args = B.args
else:
args = [B]
for term in args:
if not term.is_commutative:
c, nc = term.args_cnc(split_1=False)
x = self.blade_derivation(nc[0], coord)
if x != zero:
if len(c) == 1:
dA += c[0] * x
elif len(c) == 0:
dA += x
else:
dA += reduce(operator.mul, c, one) * x
return dA
[docs] def grad_sqr(self, A, grad_sqr_mode, mode, left):
"""
Caclulate '(grad *_{1} grad) *_{2} A' or 'A *_{2} (grad *_{1} grad)'
where grad_sqr_mode = *_{1} = *, ^, or | and
mode = *_{2} = *, ^, or |.
"""
(Sop, Bop) = Ga.DopFop[(grad_sqr_mode, mode)]
print('(Sop, Bop) =', Sop, Bop)
print('grad_sqr:A =', A)
self.dot_mode == '|'
s = zero
if Sop is False and Bop is False:
return s
dA_i = []
for coord_i in self.coords:
dA_i.append(self.pDiff(A, coord_i))
print('dA_i =', dA_i)
if Sop:
for i in self.n_range:
coord_i = self.coords[i]
if self.connect_flg:
s += self.grad_sq_scl_connect[coord_i] * dA_i[i]
for j in self.n_range:
d2A_j = self.pDiff(dA_i[i], self.coords[j])
s += self.g_inv[i, j] * d2A_j
if Bop and self.connect_flg:
for i in self.n_range:
coord_i = self.coords[i]
print('mode =', mode)
print('dA_i[i] =', dA_i[i])
if left:
if mode == '|':
s += self.dot(self.grad_sq_mv_connect[coord_i], dA_i[i])
if mode == '^':
s += self.wedge(self.grad_sq_mv_connect[coord_i], dA_i[i])
if mode == '*':
s += self.mul(self.grad_sq_mv_connect[coord_i], dA_i[i])
else:
if mode == '|':
s += self.dot(dA_i[i], self.grad_sq_mv_connect[coord_i])
if mode == '^':
s += self.wedge(dA_i[i], self.grad_sq_mv_connect[coord_i])
if mode == '*':
s += self.mul(dA_i[i], self.grad_sq_mv_connect[coord_i])
return s
[docs] def connection(self, rbase, key_base, mode, left):
"""
Compute required multivector connections of the form
(Einstein summation convention) e^{j}*(D_{j}e_{i_{1}...i_{r}})
and (D_{j}e_{i_{1}...i_{r}})*e^{j} where * could be *, ^, |,
<, or > depending upon the mode and e^{j} are reciprocal
basis vectors
"""
mode_key = (mode, left)
keys = [i for i, j in self.connect[mode_key]]
if left:
key = rbase * key_base
else:
key = key_base * rbase
if key not in keys:
keys.append(key)
C = zero
for ib in self.n_range:
x = self.blade_derivation(key_base, ib)
if self.norm:
x /= self.e_norm[ib]
C += self.er_blade(self.r_basis[ib], x, mode, left)
# Update connection dictionaries
self.connect[mode_key].append((key, C))
return C
def ReciprocalFrame(self, basis, mode='norm'):
dim = len(basis)
indexes = tuple(range(dim))
index = [()]
for i in indexes[-2:]:
index.append(tuple(combinations(indexes, i + 1)))
MFbasis = []
for igrade in index[-2:]:
grade = []
for iblade in igrade:
blade = self.mv('1', 'scalar')
for ibasis in iblade:
blade ^= basis[ibasis]
blade = blade.trigsimp()
grade.append(blade)
MFbasis.append(grade)
E = MFbasis[-1][0]
E_sq = trigsimp((E * E).scalar())
duals = copy.copy(MFbasis[-2])
duals.reverse()
sgn = 1
rbasis = []
for dual in duals:
recpv = (sgn * dual * E).trigsimp()
rbasis.append(recpv)
sgn = -sgn
if mode != 'norm':
rbasis.append(E_sq)
else:
for i in range(dim):
rbasis[i] = rbasis[i] / E_sq
return tuple(rbasis)
def Mlt(self,*kargs,**kwargs):
return lt.Mlt(kargs[0], self, *kargs[1:], **kwargs)
[docs]class Sm(Ga):
"""
Submanifold is a geometric algebra defined on a submanifold of a
base geometric algebra defined on a manifold. The submanifold is
defined by a mapping from the coordinates of the base manifold to
the coordinates of the submanifold. The inputs required to define
the submanifold are:
u {kargs[0]} The coordinate map defining the submanifold
which is a list of functions of coordinates of the base
manifold in terms of the coordinates of the submanifold.
for example if the manifold is a unit sphere then -
'u = [sin(u)*cos(v),sin(u)*sin(v),cos(u)]'.
Alternatively {kargs[0]} is a parametric vector function
of the basis vectors of the base manifold. The
coefficients of the bases are functions of the coordinates
{kargs[1]}. In this case we would call the submanifold
a "vector" manifold and additional characteristics of the
manifold can be calculated since we have given an explicit
embedding of the manifold in the base manifold.
coords {kargs[1]} The coordinate list for the submanifold, for
example '[u,v]'.
See 'init_slots' for possible other inputs. The 'Ga' member function
'sm' can be used to instantiate the submanifold via (o3d is the base
manifold)
coords = (u,v) = symbols(',v',real=True)
sm_example = o3d.sm([sin(u)*cos(v),sin(u)*sin(v),cos(u)],coords)
(eu,ev) = sm_example.mv()
sm_grad = sm_example.grad
"""
init_slots = {'debug': (False, 'True for debug output'),
'root': ('e', 'Root symbol for basis vectors'),
'name': (None, 'Name of submanifold'),
'norm': (False, 'Normalize basis if True'),
'ga': (None, 'Base Geometric Algebra')}
def __init__(self, *kargs, **kwargs):
#print '!!!Enter Sm!!!'
if printer.GaLatexPrinter.latex_flg:
printer.GaLatexPrinter.restore()
Ga.restore = True
kwargs = metric.test_init_slots(Sm.init_slots, **kwargs)
u = kargs[0] # Coordinate map or vector embedding to define submanifold
coords = kargs[1] # List of cordinates
ga = kwargs['ga'] # base geometric algebra
if ga is None:
raise ValueError('Base geometric algebra must be specified for submanifold.')
g_base = ga.g_raw
n_base = ga.n
n_sub = len(coords)
# Construct names of basis vectors
root = kwargs['root']
"""
basis_str = ''
for x in coords:
basis_str += root + '_' + str(x) + ' '
basis_str = basis_str[:-1]
"""
#print 'u =', u
if isinstance(u,mv.Mv): #Define vector manifold
self.ebasis = []
for coord in coords:
#Partial derivation of vector function to get basis vectors
self.ebasis.append(u.diff(coord))
#print 'sm ebasis =', self.ebasis
self.g = []
for b1 in self.ebasis:
#Metric tensor from dot products of basis vectors
tmp = []
for b2 in self.ebasis:
tmp.append(b1 | b2)
self.g.append(tmp)
else:
if len(u) != n_base:
raise ValueError('In submanifold dimension of base manifold' +
' not equal to dimension of mapping.')
dxdu = []
for x_i in u:
tmp = []
for u_j in coords:
tmp.append(diff(x_i, u_j))
dxdu.append(tmp)
#print 'dxdu =', dxdu
sub_pairs = list(zip(ga.coords, u))
#Construct metric tensor form coordinate maps
g = eye(n_sub) #Zero n_sub x n_sub sympy matrix
n_range = list(range(n_sub))
for i in n_range:
for j in n_range:
s = zero
for k in ga.n_range:
for l in ga.n_range:
s += dxdu[k][i] * dxdu[l][j] * g_base[k, l].subs(sub_pairs)
g[i, j] = trigsimp(s)
norm = kwargs['norm']
debug = kwargs['debug']
if Ga.restore: # restore printer to appropriate enhanced mode after sm is instantiated
printer.GaLatexPrinter.redirect()
Ga.__init__(self, root, g=g, coords=coords, norm=norm, debug=debug)
if isinstance(u,mv.Mv): #Construct additional functions for vector manifold
#self.r_basis_mv under construction
pass
self.ga = ga
self.u = u
if debug:
print('Exit Sm.__init__()')
def vpds(self):
if not self.is_ortho:
r_basis = [x / self.e_sq for x in self.r_basis_mv]
else:
r_basis = self.r_basis_mv
if self.norm:
r_basis = [x / e_norm for (x, e_norm) in zip(self.r_basis_mv, self.e_norm)]
pdx = [self.Pdiffs[x] for x in self.coords]
self.vpd = mv.Dop(r_basis, pdx, ga=self)
self.rvpd = mv.Dop(r_basis, pdx, ga=self, cmpflg=True)
return self.vpd, self.rvpd
if __name__ == "__main__":
pass