Source code for

Geometric Algebra (inherits Metric)
import sys
import warnings
import operator
import copy
from collections import OrderedDict
from itertools import combinations
import functools
from functools import reduce

from sympy import (
    diff, Rational, Symbol, S, Mul, Add,
    expand, simplify, eye, trigsimp,
    symbols, sqrt, Function

from . import printer
from . import metric
from . import mv
from . import lt
from . import utils

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 lazy_dict(dict): """ A dictionary that creates missing entries on the fly. When the dictionary is indexed and the key used is not one of the existing keys, ``self.f_value(key)`` is called to evaluate the key. The result is then added to the dictionary so that ``self.f_value`` is not used to evaluate the same key again. Parameters ---------- d : Arguments to pass on to the :class:`dict` constructor, typically a regular dictionary f_value : function The function to call to generate a value for a given key """ def __init__(self, d, f_value): dict.__init__(self, d) self.f_value = f_value def __missing__(self, key): value = self.f_value(key) self[key] = value return value def __repr__(self): return '{}({}, f_value={!r})'.format( type(self).__qualname__, dict.__repr__(self), self.f_value) def _repr_pretty_(self, p, cycle): # ipython support p_open, p_close = type(self).__qualname__ + '(', ')' with, p_open, p_close): p.type_pprinters[dict](self, p, cycle) p.text(',') p.breakable() p.text('f_value={}'.format(self.f_value))
[docs]def update_and_substitute(expr1, expr2, 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] * mul_dict[bases1[i], bases2[j]] where ``mul_dict[bases1[i], bases2[j]]`` contains the appropriate product of ``bases1[i]*bases2[j]`` as a linear combination of scalars and bases of the geometric algebra. """ (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): expr += coef1 * coef2 * mul_dict[base1, base2] 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 :class:`~galgebra.metric.Metric`. The instanciating the class :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 :class:`Ga` class via the :class:`` class or any use of enhanced printing (text or latex) will cause the bases and multiplication table entries to be incorrectly labeled . .. rubric:: Inherited from Metric class .. autosummary:: ~galgebra.metric.Metric.g ~galgebra.metric.Metric.g_inv ~galgebra.metric.Metric.norm ~galgebra.metric.Metric.coords ~galgebra.metric.Metric.is_ortho ~galgebra.metric.Metric.connect_flg ~galgebra.metric.Metric.basis ~galgebra.metric.Metric.r_symbols ~galgebra.metric.Metric.n ~galgebra.metric.Metric.n_range .. rubric:: Basis, basis bases, and basis blades data structures .. attribute:: indexes Index list for multivector bases and blades by grade (tuple of tuples). Tuple so that indexes can be used to index dictionaries. .. attribute:: bases List of bases (non-commutative sympy symbols) by grade. Only created for non-orthogonal basis vectors. .. attribute:: blades List of basis blades (non-commutative sympy symbols) by grade. For orthogonal basis vectors the same as bases. .. attribute:: coord_vec Linear combination of coordinates and basis vectors. For example in orthogonal 3D :math:`x*e_x+y*e_y+z*e_z`. .. attribute:: blades_to_indexes_dict Map basis blades to index tuples (dictionary). .. attribute:: indexes_to_blades_dict Map index tuples to basis blades (dictionary). .. attribute:: bases_to_indexes_dict Map basis bases to index tuples (dictionary). .. attribute:: indexes_to_bases_dict Map index tuples to basis bases (dictionary). .. rubric:: 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). These dictionaries are lazy, meaning they may be empty until an attempt is made to index them. .. attribute:: mul_table_dict Geometric products of basis blades as a :class:`lazy_dict`, ``{base1*base2: Expansion of base1*base2,...}`` .. attribute:: wedge_table_dict Outer products of basis blades as a :class:`lazy_dict`, ``{base1*base2: Expansion of base1^base2,...}` .. attribute:: dot_table_dict Hestenes inner products of basis blades as a :class:`lazy_dict`, ``{base1*base2: Expansion of base1|base2,...}`` .. attribute:: left_contract_table_dict Left contraction of basis blades as a :class:`lazy_dict`, ``{base1*base2: Expansion of base1<base2,...}`` .. attribute:: right_contract_table_dict Right contraction of basis blades as a :class:`lazy_dict`, ``{base1*base2: Expansion of base1>base2,...}`` .. rubric:: Reciprocal basis data structures .. attribute:: r_symbols Reciprocal basis vector symbols (list of non-commutative sympy variables) .. attribute:: r_basis List of reciprocal basis vectors expanded as linear combination of basis vector symbols. .. attribute:: 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, ...}`` .. attribute:: r_basis_mv List of reciprocal basis vectors in terms of basis multivectors (elements of list can be used in multivector expressions.) .. rubric:: Derivative data structures .. attribute:: 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. .. attribute:: grad Geometric derivative operator from left. ``grad*F`` returns multivector derivative, ``F*grad`` returns differential operator. .. attribute:: rgrad Geometric derivative operator from right. ``rgrad*F`` returns differential operator, ``F*rgrad`` returns multivector derivative. .. Sphinx adds all the other members below this docstring .. rubric:: Other members .. attribute:: dot_mode Controls the behavior of :meth:`dot` ======= ====================== value ``dot`` aliases ======= ====================== ``'|'`` :meth:`hestenes_dot` ``'<'`` :meth:`left_contract` ``'>'`` :meth:`right_contract` ======= ====================== """ 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(*args, **kwargs): """ Static method to instantiate geometric algebra and return geometric algebra, basis vectors, and grad operator as a tuple. """ GA = Ga(*args, **kwargs) basis = list( 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]) args = [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(*args, **kwargs) def __hash__(self): return hash( def __eq__(self, ga): return == if sys.version_info.major < 3: def __ne__(self, other): return not (self == other) 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._build_basis_product_tables() if self.coords is not None: self.coords = list(self.coords) self.e = mv.Mv(self._all_blades_lst[-1], 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._build_grads() else: self.r_basis_mv = None if self.connect_flg: self._build_connection() self._lt_flg = False # cache for `` # 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.debug: print('Exit Ga.__init__()') self.a = [] # List of dummy vectors for Mlt calculations self._agrads = {} # cache of gradient operator with respect to vector a self.dslot = -1 # args slot for dervative, -1 for coordinates self._XOX ='XOX','vector') # cached vector for use in is_versor 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 def E(self): # Unnoromalized pseudo-scalar return self.e def I(self): # Noromalized pseudo-scalar return self.i @property def mv_I(self): # This exists for backwards compatibility. Note this is not `I()`! # galgebra 0.4.5 warnings.warn( "`ga.mv_I` is deprecated, use `ga.E()` instead, or perhaps `ga.I()`", DeprecationWarning, stacklevel=2) # default pseudoscalar return self.E() @property def mv_x(self): # This exists for backwards compatibility. # galgebra 0.4.5 warnings.warn( "`ga.mv_x` is deprecated, use `, 'vector')` instead", DeprecationWarning, stacklevel=2) # testing vectors return mv.Mv('XxXx', 'vector', ga=self) def X(self): return[coord*base for (coord, base) in zip(self.coords, self.basis)])) @property def Pdiffs(self): # galgebra 0.4.5 warnings.warn( "ga.Pdiffs[x] is deprecated, use `ga.pdop(x)` instead", DeprecationWarning, stacklevel=2) return {x: self.pdop(x) for x in self.coords} @property def sPds(self): # galgebra 0.4.5 warnings.warn( "ga.sPds[x] is deprecated, use `ga.sdop(x)` instead", DeprecationWarning, stacklevel=2) return {x: self.sdop(x) for x in self.coords} @property def Pdop_identity(self): # galgebra 0.4.5 warnings.warn( "ga.Pdop_identity is deprecated, use `ga.pdop({})` instead", DeprecationWarning, stacklevel=2) return self.pdop({})
[docs] def mv(self, root=None, *args, **kwargs): """ Instanciate and return a multivector for this, 'self', geometric algebra. """ if root is None: # Return ga basis and compute grad and rgrad return self.mv_basis # ensure that ga is not already in kwargs kwargs = dict(ga=self, **kwargs) if not utils.isstr(root): return mv.Mv(root, *args, **kwargs) if ' ' in root and ' ' not in args[0]: root_lst = root.split(' ') mv_lst = [] for root in root_lst: mv_lst.append(mv.Mv(root, *args, **kwargs)) return tuple(mv_lst) if ' ' in root and ' ' in args[0]: root_lst = root.split(' ') mvtype_lst = args[0].split(' ') if len(root_lst) != len(mvtype_lst): raise ValueError('In 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): args = list(args) args[0] = mv_type args = tuple(args) mv_lst.append(mv.Mv(root, *args, **kwargs)) return tuple(mv_lst) return mv.Mv(root, *args, **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 .. math:: {i}\cdot e^{j} = \delta_{i}^{j}. If the basis is not orthogonal and norm=False then .. math:: 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._all_mv_blades_lst[1:] # do not include the scalar, which is not named 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 _build_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.pdop(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) def grads(self): if self.coords is None: raise ValueError("Ga must have been initialized with coords to compute grads") return self.grad, self.rgrad
[docs] def pdop(self, *args, **kwargs): """ Shorthand to construct a :class:`` for this algebra """ return mv.Pdop(*args, ga=self, **kwargs)
[docs] def dop(self, *args, **kwargs): """ Shorthand to construct a :class:`` for this algebra """ return mv.Dop(*args, ga=self, **kwargs)
[docs] def sdop(self, *args, **kwargs): """ Shorthand to construct a :class:`` for this algebra """ return mv.Sdop(*args, ga=self, **kwargs)
[docs] def lt(self, *args, **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) return lt.Lt(*args, ga=self, **kwargs)
[docs] def sm(self, *args, **kwargs): """ Instanciate and return a submanifold for this geometric algebra. See :class:`Sm` for instantiation inputs. """ return Sm(*args, ga=self, **kwargs)
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) def _build_basis_base_symbol(self, base_index): """ Build a symbol used for the `base_rep` from the given tuple """ if not base_index: return S(1) symbol_str = '*'.join([str(self.basis[i]) for i in base_index]) return Symbol(symbol_str, commutative=False) def _build_basis_blade_symbol(self, base_index): """ Build a symbol used for the `blade_rep` from the given tuple """ if not base_index: return S(1) if self.wedge_print: symbol_str = '^'.join([str(self.basis[i]) for i in base_index]) 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!!!!') return Symbol(symbol_str, commutative=False) def _build_bases(self): r""" 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 .. math:: e_{i_{1}}e_{i_{2}}...e_{i_{r}} where :math:`0 < i_{1} < i_{2} < ... < i_{r}` and :math:`0 < r \le n` the dimension of the vector space and :math:`0 < i_{j} \le n`. The total number of all symbols of this form plus the scalars is :math:`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 - .. math:: e_{i_{1}}\wedge e_{i_{2}}\wedge ...\wedge 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._all_indexes_lst = [] for i in range(len(basis_indexes) + 1): base_tuple = tuple(combinations(basis_indexes, i)) self.indexes.append(base_tuple) self._all_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._all_blades_lst = [] for grade_index in self.indexes: blades = [] super_scripts = [] for base_index in grade_index: blade_symbol = self._build_basis_blade_symbol(base_index) blades.append(blade_symbol) self._all_blades_lst.append(blade_symbol) self.blades.append(blades) self.blades_to_indexes = [] self.indexes_to_blades = [] for (index, blade) in zip(self._all_indexes_lst, self._all_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 = {} for igrade, grade in enumerate(self.blades): for blade in grade: self.blades_to_grades_dict[blade] = igrade if not self.is_ortho: self.bases = [] self._all_bases_lst = [] for grade_index in self.indexes: bases = [] for base_index in grade_index: base_symbol = self._build_basis_base_symbol(base_index) bases.append(base_symbol) self._all_bases_lst.append(base_symbol) self.bases.append(bases) self.bases_to_indexes = [] self.indexes_to_bases = [] for (index, base) in zip(self._all_indexes_lst, self._all_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 = {} for igrade, grade in enumerate(self.bases): for base in grade: self.bases_to_grades_dict[base] = igrade 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._all_indexes_lst, 'blades', self.blades, 'list(blades)', self._all_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._all_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) # create the Mv wrappers self._all_mv_blades_lst = [ mv.Mv(obj, ga=self) for obj in self._all_blades_lst ] self.mv_basis = [ mv.Mv(obj, ga=self) for obj in self.basis ] # TODO[gh-64]: For compatibility with old behavior, the public # properties do not include the scalar. We should consider making the # breaking change such that they do. self.indexes_lst = self._all_indexes_lst[1:] self.blades_lst = self._all_blades_lst[1:] self.mv_blades_lst = self._all_mv_blades_lst[1:] if not self.is_ortho: self.bases_lst = self._all_bases_lst[1:] def _build_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_dict = lazy_dict({}, f_value=self.geometric_product_basis_blades) # Geometric product (*) of blades if not self.is_ortho: self._build_non_orthogonal_mul_table() # Fully populated geometric product (*) multiplication table self._build_base_blade_conversions() # Generates conversion dictionaries between bases and blades self.wedge_table_dict = lazy_dict({}, f_value=self.wedge_product_basis_blades) # Outer product (^) # 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 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 self.dot_table_dict = lazy_dict({}, f_value=functools.partial(dot_product_basis_blades, mode='|')) self.left_contract_table_dict = lazy_dict({}, f_value=functools.partial(dot_product_basis_blades, mode='<')) self.right_contract_table_dict = lazy_dict({}, f_value=functools.partial(dot_product_basis_blades, mode='>')) if self.debug: print('Exit _build_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): r""" blst is a list of integers :math:`[i_{1},...,i_{r}]` representing the geometric product of r basis vectors :math:`a_{{i_1}}*...*a_{{i_r}}`. :meth:`reduce_basis_loop` searches along the list :math:`[i_{1},...,i_{r}]` untill it finds :math:`i_{j} = i_{j+1}` and in this case contracts the list, or if :math:`i_{j} > i_{j+1}` it revises the list (:math:`\sim i_{j}` means remove :math:`i_{j}` from the list) * Case 1: If :math:`i_{j} = i_{j+1}`, return :math:`a_{i_{j}}^2` and :math:`[i_{1},..,\sim i_{j},\sim i_{j+1},...,i_{r}]` * Case 2: If :math:`i_{j} > i_{j+1}`, return :math:`a_{i_{j}}.a_{i_{j+1}}`, :math:`[i_{1},..,\sim i_{j},\sim i_{j+1},...,i_{r}]`, and :math:`[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 for jstep in range(1, 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 return True # revision complete, blst in normal order
#******************* Outer/wedge (^) product **********************# @staticmethod def blade_reduce(lst): sgn = S(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 S(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 S(0) (sgn, wedge12) = Ga.blade_reduce(index12) if sgn != 0: return(sgn * self.indexes_to_blades_dict[tuple(wedge12)]) else: return S(0) #****** Dot (|) product, reft (<) and right (>) contractions ******# def _dot_product_grade(self, grade1, grade2, mode): """ Get the grade to select from the geometric product, for a given dot product """ if mode == '|': if grade1 == 0 or grade2 == 0: return None return abs(grade1 - grade2) elif mode == '<': grade = grade2 - grade1 if grade < 0: return None return grade elif mode == '>': grade = grade1 - grade2 if grade < 0: return None return grade else: raise ValueError('mode={!r} not allowed'.format(mode)) def dot_product_basis_blades(self, blade12, mode): # 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) grade = self._dot_product_grade(len(index1), len(index2), mode=mode) if grade is None: return zero n = len(index) sgn = S(1) result = S(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 zero 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 zero 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, mode): # 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] grade = self._dot_product_grade(grade1, grade2, mode=mode) if grade is None: return zero return grade_dict.get(grade, zero) ############# Non-Orthogonal Tables and Dictionaries ############### def _build_non_orthogonal_mul_table(self): mul_table = [] self.basic_mul_keys = [] self.basic_mul_values = [] for base1 in self._all_bases_lst: for base2 in self._all_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 _build_base_blade_conversions(self): blade_expansion = [] blade_index = [] # expand blade basis in terms of base basis for blade in self._all_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._all_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._all_bases_lst, self._all_blades_lst, self._all_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._all_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) elif mode == '|': return self.hestenes_dot(A, B) elif mode == '<': return self.left_contract(A, B) elif mode == '>': return self.right_contract(A, B) else: raise ValueError('Unknown multiplication operator {!r}', mode) def mul(self, A, B): # geometric (*) product of blade representations return update_and_substitute(A, B, 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 return update_and_substitute(A, B, self.wedge_table_dict)
[docs] def hestenes_dot(self, A, B): r""" compute the hestenes dot product, :math:`A \bullet B` """ return update_and_substitute(A, B, self.dot_table_dict)
[docs] def left_contract(self, A, B): r""" compute the left contraction, :math:`A \rfloor B` """ return update_and_substitute(A, B, self.left_contract_table_dict)
[docs] def right_contract(self, A, B): r""" compute the right contraction, :math:`A \lfloor B` """ return update_and_substitute(A, B, self.right_contract_table_dict)
[docs] def dot(self, A, B): r""" Inner product ``|``, ``<``, or ``>``. The :attr:`dot_mode` attribute determines which of these is used. """ # forbid something silly like setting dot_mode to the wedge or geometric # product if self.dot_mode in '^*': raise ValueError('"' + str(self.dot_mode) + '" not a legal mode in dot') return self.Mul(A, B, mode=self.dot_mode)
######################## 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] =[grade]) return grade_dict
[docs] def split_multivector(self, A): """ Split multivector :math:`A` into commutative part :math:`a` and non-commutative part :math:`A'` so that :math:`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 ###################### def _build_reciprocal_basis(self,gsym): r""" Calculate reciprocal basis vectors :math:`e^{j}` where .. math:: e^{j}\cdot e_{k} = \delta_{k}^{j} and :math:`\delta_{k}^{j}` is the kronecker delta. We use the formula from Doran and Lasenby 4.94: .. math:: e^{j} = (-1)^{j-1}e_{1} \wedge ...e_{j-1} \wedge e_{j+1} \wedge ... \wedge e_{n}*E_{n}^{-1} where :math:`E_{n} = e_{1}\wedge ...\wedge e_{n}`. For non-orthogonal basis :math:`e^{j}` is not normalized and must be divided by :math:`E_{n}^2` (``self.e_sq``) in any relevant calculations. If ``gsym = True`` then :math:`E_{n}^2` is not evaluated, but is represented as :math:`E_{n}^2 = (-1)^{n*(n-1)/2}\operatorname{det}(g)` where :math:`\operatorname{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: 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) duals = list(self.blades[self.n - 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} """ print('blades list =',self._all_blades_lst) print('debug =',expand(self.base_to_blade_rep(self.mul(sgn * dual_base_rep, self.e.obj)))) print('collect arg =',expand(self.base_to_blade_rep(self.mul(sgn * dual_base_rep, self.e.obj)))) """ r_basis_j = metric.collect(expand(self.base_to_blade_rep(self.mul(sgn * dual_base_rep, self.e.obj))), self._all_blades_lst) self.r_basis.append(r_basis_j) # sgn = (-1)**{j-1} sgn = -sgn if self.debug: printer.oprint('E', self.e, 'E**2', self.e_sq, 'unnormalized reciprocal basis =\n', self.r_basis) print('reciprocal basis test =') for ei in self.basis: for ej in self.r_basis: ei_dot_ej = self.hestenes_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))) # 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:[x_i][jb] = metric.Simp.apply([x_i][jb].subs(self.r_basis_dict) / self.e_sq) else:[x_i][jb] = metric.Simp.apply([x_i][jb].subs(self.r_basis_dict)) g_inv = eye(self.n) # 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.hestenes_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): r""" 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: .. math:: e^{j}*(e_{i_{1}}\wedge ...\wedge 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: if left: return self.Mul(er, blade, mode=mode) else: return self.Mul(blade, er, mode=mode)
[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' caches the results in a dictionary, ``self._dbases``, so that the derivation for a given blade and coordinate is never calculated more that once. Note that the return value is not a multivector, but linear combination of basis blade symbols. """ 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 =[ib][index[0]] elif grade == 2: db = self.wedge([ib][index[0]], self.basis[index[1]]) + \ self.wedge(self.basis[index[0]],[ib][index[1]]) else: db = self.wedge([ib][index[0]], self.indexes_to_blades[index[1:]]) + \ self.wedge(self.indexes_to_blades[index[:-1]],[ib][index[-1]]) for i in range(1, grade - 1): db += self.wedge(self.wedge(self.indexes_to_blades[index[:i]],[ib][index[i]]), self.indexes_to_blades[index[i + 1:]]) self._dbases[key] = db return db
[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 =, 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: dA += reduce(operator.mul, c, x) return dA
[docs] def grad_sqr(self, A, grad_sqr_mode, mode, left): r""" Calculate :math:`(grad *_{1} grad) *_{2} A` or :math:`A *_{2} (grad *_{1} grad)` where ``grad_sqr_mode`` = :math:`*_{1}` = ``*``, ``^``, or ``|`` and ``mode`` = :math:`*_{2}` = ``*``, ``^``, or ``|``. """ (Sop, Bop) = Ga.DopFop[(grad_sqr_mode, mode)] print('(Sop, Bop) =', Sop, Bop) print('grad_sqr:A =', A) 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 +=[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 +=[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) :math:`e^{j}*(D_{j}e_{i_{1}...i_{r}})` and :math:`(D_{j}e_{i_{1}...i_{r}})*e^{j}` where :math:`*` could be ``*``, ``^``, ``|``, ``<``, or ``>`` depending upon the mode, and :math:`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 ='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,*args,**kwargs): return lt.Mlt(args[0], self, *args[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: Parameters ---------- u : (``args[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 (``args[0]``) is a parametric vector function of the basis vectors of the base manifold. The coefficients of the bases are functions of the coordinates (``args[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 : (``args[1]``) The coordinate list for the submanifold, for example ``[u, v]``. Notes ----- 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 =[sin(u)*cos(v),sin(u)*sin(v),cos(u)],coords) (eu,ev) = 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, *args, **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 = args[0] # Coordinate map or vector embedding to define submanifold coords = args[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 = 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