galgebra.ga¶
Geometric Algebra (inherits Metric)
Members¶

class
galgebra.ga.
Ga
(bases, **kwargs)[source]¶ 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 nonorthogonal 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 theMv
class or any use of enhanced printing (text or latex) will cause the bases and multiplication table entries to be incorrectly labeled .Inherited from Metric class
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 (noncommutative sympy symbols) by grade. Only created for nonorthogonal basis vectors.

blades
¶ List of basis blades (noncommutative sympy symbols) by grade. 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).
Multiplication tables data structures
Keys in all multiplication tables (
*
,^
,
,<
,>
) are alwayssymbol1*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.
mul_table_dict
¶ Geometric products of basis blades as a
lazy_dict
,{base1*base2: Expansion of base1*base2,...}

wedge_table_dict
¶ Outer products of basis blades as a
lazy_dict
, ``{base1*base2: Expansion of base1^base2,…}`

dot_table_dict
¶ Hestenes inner products of basis blades as a
lazy_dict
,{base1*base2: Expansion of base1base2,...}

left_contract_table_dict
¶ Left contraction of basis blades as a
lazy_dict
,{base1*base2: Expansion of base1<base2,...}

right_contract_table_dict
¶ Right contraction of basis blades as a
lazy_dict
,{base1*base2: Expansion of base1>base2,...}
Reciprocal basis data structures

r_symbols
¶ Reciprocal basis vector symbols (list of noncommutative 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.

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.
rgrad*F
returns differential operator,F*rgrad
returns multivector derivative.
Other members

dot_mode
¶ Controls the behavior of
dot()
value dot
aliases''
hestenes_dot()
'<'
left_contract()
'>'
right_contract()

Christoffel_symbols
(mode=1)¶ mode = 1 Christoffel symbols of the first kind mode = 2 Christoffel symbols of the second kind

bases_dict
(prefix=None)[source]¶ 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())

blade_derivation
(blade, ib)[source]¶ 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.

static
build
(*args, **kwargs)[source]¶ Static method to instantiate geometric algebra and return geometric algebra, basis vectors, and grad operator as a tuple.

connection
(rbase, key_base, mode, left)[source]¶ 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.

dop
(*args, **kwargs)[source]¶ Instanciate and return a multivector differential operator for this, ‘self’, geometric algebra.

dot
(A, B)[source]¶ Inner product

,<
, or>
.The
dot_mode
attribute determines which of these is used.

static
dot_orthogonal
(V1, V2, g=None)¶ Returns the dot product of two vectors in an orthogonal coordinate system. V1 and V2 are lists of sympy expressions. g is a list of constants that gives the signature of the vector space to allow for noneuclidian vector spaces.
This function is only used to form the dot product of vectors in the embedding space of a vector manifold or in the case where the basis vectors are explicitly defined by vector fields in the embedding space.
A g of None is for a Euclidian embedding space.

static
dual_mode
(mode='I+')[source]¶ 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

er_blade
(er, blade, mode='*', left=True)[source]¶ 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:\[e^{j}*(e_{i_{1}}\wedge ...\wedge e_{i_{r}})\]

grad_sqr
(A, grad_sqr_mode, mode, left)[source]¶ Calculate \((grad *_{1} grad) *_{2} A\) or \(A *_{2} (grad *_{1} grad)\) where
grad_sqr_mode
= \(*_{1}\) =*
,^
, or
andmode
= \(*_{2}\) =*
,^
, or
.

grade_decomposition
(A)[source]¶ 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 noncommutative symbols) the dictionary entries are sympy expressions.

lt
(*args, **kwargs)[source]¶ Instanciate and return a linear transformation for this, ‘self’, geometric algebra.

metric_symbols_list
(s=None)¶ rows of metric tensor are separated by “,” and elements of each row separated by ” “. If the input is a single row it is assummed that the metric tensor is diagonal.
Output is a square matrix.

mv
(root=None, *args, **kwargs)[source]¶ Instanciate and return a multivector for this, ‘self’, geometric algebra.

mvr
(norm=True)[source]¶ Returns tumple of reciprocal basis vectors. If norm=True or basis vectors are orthogonal the reciprocal basis is normalized in the sense that
\[{i}\cdot e^{j} = \delta_{i}^{j}.\]If the basis is not orthogonal and norm=False then
\[e_{i}\cdot e^{j} = I^{2}\delta_{i}^{j}.\]

pDiff
(A, coord)[source]¶ Compute partial derivative of multivector function ‘A’ with respect to coordinate ‘coord’.

reduce_basis
(blst)[source]¶ Repetitively applies reduce_basis_loop to blst product representation until normal form is realized for nonorthogonal basis

static
reduce_basis_loop
(g, blst)[source]¶ 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 (\(\sim i_{j}\) means remove \(i_{j}\) from the list) Case 1: If \(i_{j} = i_{j+1}\), return \(a_{i_{j}}^2\) and \([i_{1},..,\sim i_{j},\sim i_{j+1},...,i_{r}]\)
 Case 2: If \(i_{j} > i_{j+1}\), return \(a_{i_{j}}.a_{i_{j+1}}\), \([i_{1},..,\sim i_{j},\sim i_{j+1},...,i_{r}]\), and \([i_{1},..,i_{j+1},i_{j},...,i_{r}]\)


class
galgebra.ga.
Sm
(*args, **kwargs)[source]¶ 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 = o3d.sm([sin(u)*cos(v),sin(u)*sin(v),cos(u)],coords) (eu,ev) = sm_example.mv() sm_grad = sm_example.grad

Christoffel_symbols
(mode=1)¶ mode = 1 Christoffel symbols of the first kind mode = 2 Christoffel symbols of the second kind

bases_dict
(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())

blade_derivation
(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.

static
build
(*args, **kwargs)¶ Static method to instantiate geometric algebra and return geometric algebra, basis vectors, and grad operator as a tuple.

connection
(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.

dop
(*args, **kwargs)¶ Instanciate and return a multivector differential operator for this, ‘self’, geometric algebra.

dot
(A, B)¶ Inner product

,<
, or>
.The
dot_mode
attribute determines which of these is used.

static
dot_orthogonal
(V1, V2, g=None)¶ Returns the dot product of two vectors in an orthogonal coordinate system. V1 and V2 are lists of sympy expressions. g is a list of constants that gives the signature of the vector space to allow for noneuclidian vector spaces.
This function is only used to form the dot product of vectors in the embedding space of a vector manifold or in the case where the basis vectors are explicitly defined by vector fields in the embedding space.
A g of None is for a Euclidian embedding space.

static
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

er_blade
(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:\[e^{j}*(e_{i_{1}}\wedge ...\wedge e_{i_{r}})\]

grad_sqr
(A, grad_sqr_mode, mode, left)¶ Calculate \((grad *_{1} grad) *_{2} A\) or \(A *_{2} (grad *_{1} grad)\) where
grad_sqr_mode
= \(*_{1}\) =*
,^
, or
andmode
= \(*_{2}\) =*
,^
, or
.

grade_decomposition
(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 noncommutative symbols) the dictionary entries are sympy expressions.

hestenes_dot
(A, B)¶ compute the hestenes dot product, \(A \bullet B\)

left_contract
(A, B)¶ compute the left contraction, \(A \rfloor B\)

lt
(*args, **kwargs)¶ Instanciate and return a linear transformation for this, ‘self’, geometric algebra.

metric_symbols_list
(s=None)¶ rows of metric tensor are separated by “,” and elements of each row separated by ” “. If the input is a single row it is assummed that the metric tensor is diagonal.
Output is a square matrix.

mv
(root=None, *args, **kwargs)¶ Instanciate and return a multivector for this, ‘self’, geometric algebra.

mvr
(norm=True)¶ Returns tumple of reciprocal basis vectors. If norm=True or basis vectors are orthogonal the reciprocal basis is normalized in the sense that
\[{i}\cdot e^{j} = \delta_{i}^{j}.\]If the basis is not orthogonal and norm=False then
\[e_{i}\cdot e^{j} = I^{2}\delta_{i}^{j}.\]

pDiff
(A, coord)¶ Compute partial derivative of multivector function ‘A’ with respect to coordinate ‘coord’.

reduce_basis
(blst)¶ Repetitively applies reduce_basis_loop to blst product representation until normal form is realized for nonorthogonal basis

static
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 (\(\sim i_{j}\) means remove \(i_{j}\) from the list) Case 1: If \(i_{j} = i_{j+1}\), return \(a_{i_{j}}^2\) and \([i_{1},..,\sim i_{j},\sim i_{j+1},...,i_{r}]\)
 Case 2: If \(i_{j} > i_{j+1}\), return \(a_{i_{j}}.a_{i_{j+1}}\), \([i_{1},..,\sim i_{j},\sim i_{j+1},...,i_{r}]\), and \([i_{1},..,i_{j+1},i_{j},...,i_{r}]\)

remove_scalar_part
(A)¶ Return noncommutative part (sympy object) of
A.obj
.

right_contract
(A, B)¶ compute the right contraction, \(A \lfloor B\)

sm
(*args, **kwargs)¶ Instanciate and return a submanifold for this geometric algebra. See
Sm
for instantiation inputs.

split_multivector
(A)¶ Split multivector \(A\) into commutative part \(a\) and noncommutative part \(A'\) so that \(A = a+A'\)
 u –

class
galgebra.ga.
lazy_dict
(d, f_value)[source]¶ 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 thatself.f_value
is not used to evaluate the same key again.Parameters:  d – Arguments to pass on to the
dict
constructor, typically a regular dictionary  f_value (function) – The function to call to generate a value for a given key

clear
() → None. Remove all items from D.¶

copy
() → a shallow copy of D¶

fromkeys
()¶ Create a new dictionary with keys from iterable and values set to value.

get
()¶ Return the value for key if key is in the dictionary, else default.

items
() → a setlike object providing a view on D's items¶

keys
() → a setlike object providing a view on D's keys¶

pop
(k[, d]) → v, remove specified key and return the corresponding value.¶ If key is not found, d is returned if given, otherwise KeyError is raised

popitem
() → (k, v), remove and return some (key, value) pair as a¶ 2tuple; but raise KeyError if D is empty.

setdefault
()¶ Insert key with a value of default if key is not in the dictionary.
Return the value for key if key is in the dictionary, else default.

update
([E, ]**F) → None. Update D from dict/iterable E and F.¶ If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k]

values
() → an object providing a view on D's values¶
 d – Arguments to pass on to the

galgebra.ga.
nc_subs
(expr, base_keys, base_values=None)[source]¶ 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 noncommutative keys for long lists of keys.

galgebra.ga.
update_and_substitute
(expr1, expr2, mul_dict)[source]¶ Linear expand expr1 and expr2 to get (summation convention):
expr1 = coefs1[i] * bases1[i] expr2 = coefs2[j] * bases2[j]
where
coefs1
andcoefs2
are lists of are commutative expressions andbases1
andbases2
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 ofbases1[i]*bases2[j]
as a linear combination of scalars and bases of the geometric algebra.