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brainhops.datamodel.kinds

This module defines a transformation

The class A is a subclass of B if A can be converted to B without loss. For example, Linear is a subclass of Affine. This is akin to set theory in mathematics.

This differs from the usual "type hierarchy", where inheriting classes can be more specialized then their parents.

We implement both types of hierarchy, so that users can easily check whether a transformation can be conceptually thought as a given class of transformation (e.g. "is this transformation a linear transform?"), without it being conflated with type inheritance (e.g. "is this transformation an instance of the Linear type?").

In our hierarchy, transformation kinds correspond to sets, and inheritance describes set inclusion: issubclass(A, B) implies A ⊆ B. Transformations map from ℝⁿ to ℝᵐ and can be composed when their domains and codomains match. Homeomorphisms require n = m.

Some of these transformation sets (but not all!) are groups under the composition operator. This is indicated by the group decorator. The group structure refers to transformations of a fixed space ℝⁿ under composition; these groups act on ℝⁿ. If a set A is a subset of set B, and both A and B are groups, then A is a subgroup of B.

Many linear transformations, when constrained to be invertible, can be thought of as members of a Lie group, i.e. of a smooth manifold,c and their group operations (composition and inversion) are smooth maps. This is indicated by the liegroup decorator. Note that the set A may be a subgroup of a Lie group B without being a Lie group itself!

Lie groups 
G | > 0 | = 1 | ⋉ T
======================
GL ------------- Aff             General Linear     | Affine
 | \             |
 |   GL+ ------- Aff+            Positive Linear    | Positive Affine
 |    | \        |
 |    |  SL ---- SAff            Special Linear     | Special Affine
 |    |   |      |
CO ------------- Sim             Conformal          | Affine Conformal
 | \  |   |      |
 |   CO+ ------- Sim+            Special Conformal  | Special Affine Conformal
 |    |   |      |
 O ------------- E               Orthogonal         | Euclidean
 | \  | /        |
 S   SO -------- SE              Special Orthogonal | Special Euclidean
 | /             |
 I ------------- T               Identity           | Translation

General Lie groups (GL/CO/O) are in general not "connected", and instead composed of two disconnected components. One that contains the identity, and one that does not (the "flipped" version).

Positive Lie groups (SO/CO+/GL+) are restricted to transformations with positive determinant, and therefore exclude flips. They can be defined as the component from the corresponding general group that contains the identity transform.

All classical linear group can be extended with the translation group (⋉ T) to define affine groups.

Info

  • The Special Euclidean group (SE) contains transformations that are classicaly referred to as "rigid-body" transformations. They preserve angles and volumes.

  • The Special Conformal Euclidean group (Sim+) contains transformations that are classicaly referred to as "similitude" (or conformal) transformations. They only preserve (oriented) angles.

Warning

Python preserves the declared order of bases in the method resolution order (MRO), whereas set inclusion imposes no order between unrelated supersets. These ordering constraints can conflict, even when the inclusion hierarchy is acyclic. Resolving a conflict may require reordering bases throughout the hierarchy or explicitly listing ancestors that are already inherited indirectly. Neither change alters the intended set inclusions.

Classes

TransformationFamily magic 

TransformationFamily(
    kind: Kind, ndim: Dim = MISSING, odim: Dim = MISSING
)

Bases: Sequence, Magic

A parametric family of transformations.

A family is parameterized by a base type and, optionally, input and/or output dimensions.

Attributes

kind instance-attribute 
kind: Kind

All transformations in this family are instances of this type.

ndim class-attribute instance-attribute 
ndim: Dim = MISSING

The dimensionality of the transformations in this family, if known.

odim class-attribute instance-attribute 
odim: Dim = MISSING

The output dimensionality of the transformations in this family, if known.

name property 
name: str

The preferred name of the transformation family.

names property 
names: tuple[str, ...]

The names of the transformation family.

symbol property 
symbol: str | None

The symbol of the transformation family.

fsymbol property 
fsymbol: str | None

The symbol of the transformation family, with its dimension.

If the family dimensionality is unknown, the dimension is the placeholder {n}.

Methods:

to_tuple 
to_tuple() -> tuple[Kind, Dim, Dim]

Convert to a (kind, ndim, odim) tuple.

parse classmethod 
parse(
    repr: FamilyLike,
    ndim: Dim = MISSING,
    odim: Dim = MISSING,
) -> Self

Return a transformation family from its name, symbol or type.

An explicit ndim wins over one carried by repr; ndim=None means "whatever repr says", so parsing a family is idempotent.

The cases are ordered so that no branch is reached with a value it cannot type-check: a bare issubclass would raise on an int or a str.

from_tuple classmethod 
from_tuple(
    values: FamilyTupleLike,
    ndim: Dim = MISSING,
    odim: Dim = MISSING,
) -> Self

Return a transformation family from its (kind, ndim) pair.

from_name classmethod 
from_name(
    name: str, ndim: Dim = MISSING, odim: Dim = MISSING
) -> Self

Return a transformation family from its name.

from_symbol classmethod 
from_symbol(
    symbol: str, ndim: Dim = MISSING, odim: Dim = MISSING
) -> Self

Return a transformation family from its symbol.

TransformationKind

Bases: ABC

The root of the transformation

A subclass declares its SYMBOL (a short mathematical symbol, such as "SO") and its FSYMBOL (the same symbol with a dimension placeholder, such as "SO({n})"), and its ALIAS (one or more plain-text names). These are recorded automatically on subclassing, and are what TransformationFamily.parse resolves a string against.

The class name is automatically registered as a name, and is the preferred name of the transformation kind.

Aliases can also be registered with the alias decorator.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Transformation

Bases: TransformationKind

Any coordinate transformation.

alias: Morphism, Map

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

VolumePreserving

Bases: Transformation

A transformation that preserves volumes locally .

If the map is smooth, this means |det Df| = 1.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

OrientationPreserving

Bases: Transformation

A transformation that preserves orientation.

If the map is smooth, this means det Df > 0.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

AnglePreserving

Bases: Transformation

A transformation that preserves angles.

If the map is smooth, this means (Df).T @ (Df) = λ(x) I.

alias: Conformal

wiki: https://en.wikipedia.org/wiki/Conformal_map

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Injection

Bases: Transformation

A one-to-one transformation (distinct inputs map to distinct outputs).

alias: Injective

wiki: https://en.wikipedia.org/wiki/Injective_function

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Surjection

Bases: Transformation

An onto transformation (every output is reached).

alias: Surjective

wiki: https://en.wikipedia.org/wiki/Surjective_function

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Bijection

Bases: Injection, Surjection

An invertible transformation.

A bijection is both injective and surjective.

alias: Bijective, Invertible

wiki: https://en.wikipedia.org/wiki/Bijection

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Homeomorphism

Bases: Bijection

A continuous bijection with a continuous inverse.

wiki: https://en.wikipedia.org/wiki/Homeomorphism

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Diffeomorphism

Bases: Homeomorphism

A smooth homeomorphism, with a smooth inverse.

wiki: https://en.wikipedia.org/wiki/Diffeomorphism

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

VolumePreservingDiffeomorphism

Bases: VolumePreserving, Diffeomorphism

A diffeomorphism that preserves volumes.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

OrientationPreservingDiffeomorphism

Bases: OrientationPreserving, Diffeomorphism

A diffeomorphism that preserves orientation.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

ConformalDiffeomorphism

Bases: AnglePreserving, Diffeomorphism

A diffeomorphism that preserves angles.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialDiffeomorphism

Bases: VolumePreservingDiffeomorphism, OrientationPreservingDiffeomorphism

A diffeomorphism that preserves both volume and orientation.

Preserves the standard volume form: det(Df(x)) = 1 everywhere. Includes all connected components satisfying this condition.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Matrix

Bases: Transformation

Transformation that can be represented as a matrix.

This includes linear operators, as well as more general affine operators that can be represented as linear operators acting on homogeneous coordinates.

wiki: https://en.wikipedia.org/wiki/Transformation_matrix

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

InvertibleMatrix

Bases: Matrix, Diffeomorphism

A matrix transformation that is invertible.

A matrix map is smooth, and an invertible one has a smooth (matrix) inverse, so this is where the smooth end of the lattice attaches: every invertible affine transformation is a diffeomorphism, and a volume preserving one (SL, SAff) is one of those.

wiki: https://en.wikipedia.org/wiki/Invertible_matrix

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

OrientationPreservingMatrix

Bases: OrientationPreservingDiffeomorphism, InvertibleMatrix

An invertible matrix transformation with positive determinant.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

VolumePreservingMatrix

Bases: VolumePreservingDiffeomorphism, InvertibleMatrix

An invertible matrix transformation with determinant ± 1.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Affine

Bases: Matrix

An affine transformation. May not be invertible.

wiki: https://en.wikipedia.org/wiki/Affine_transformation

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

InvertibleAffine

Bases: Affine, InvertibleMatrix

An invertible affine transformation

Invertible affine transformations form a Lie group, named Aff.

This group includes reflections, and has therefore two connected components.

symbol: Aff = GL ⋉ T

wiki: https://en.wikipedia.org/wiki/Affine_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

PositiveAffine

Bases: InvertibleAffine, OrientationPreservingMatrix

Affine transformation with a positive determinant.

This group does not include reflections, and has therefore a single connected component.

symbol: Aff+ = GL+ ⋉ T

wiki: https://en.wikipedia.org/wiki/Affine_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

VolumePreservingAffine

Bases: InvertibleAffine, VolumePreservingMatrix

An affine transformation with determinant ±1 -- preserves volumes.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialAffine

Bases: PositiveAffine, VolumePreservingAffine, SpecialDiffeomorphism

An affine transformation with determinant +1 -- preserves volumes

This group does not include reflections, and has therefore a single connected component.

symbol: SAff = SL ⋉ T

wiki: https://en.wikipedia.org/wiki/Affine_group#Special_affine_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

ConformalEuclidean

Bases: InvertibleAffine, ConformalDiffeomorphism

An affine transformation that preserves angles.

This group includes reflections, and has therefore two connected components.

symbol: Sim = CO ⋉ T

alias: Similarity, Similitude

wiki: https://en.wikipedia.org/wiki/Similarity_(geometry)#In_Euclidean_space

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialConformalEuclidean

Bases: ConformalEuclidean, PositiveAffine

An affine transformation that preserves angles and orientation.

This group does not include reflections, and has therefore a single connected component.

symbol: Sim+ = CO+ ⋉ T

alias: DirectSimilarity, DirectSimilitude

wiki: https://en.wikipedia.org/wiki/Similarity_(geometry)#In_Euclidean_space

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Euclidean

Bases: ConformalEuclidean, VolumePreservingAffine

A euclidean transformation with determinant ± 1

symbol: E = O ⋉ T

wiki: https://en.wikipedia.org/wiki/Euclidean_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialEuclidean

Bases: SpecialConformalEuclidean, Euclidean, SpecialAffine

A euclidean transformation with determinant +1

symbol: SE = SO ⋉ T

alias: RigidTransformation

wiki: https://en.wikipedia.org/wiki/Euclidean_group#Direct_and_indirect_isometries

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Dilation

Bases: ConformalEuclidean

A dilation is a scaling and a translation.

symbol: ℝ* ⋉ T

alias: HomothetyTranslation

wiki: https://en.wikipedia.org/wiki/Homothety wiki: https://en.wikipedia.org/wiki/Dilation_(metric_space)

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

PositiveDilation

Bases: Dilation, SpecialConformalEuclidean

A dilation with a positive scaling factor.

symbol: ℝ+ ⋉ T

wiki: https://en.wikipedia.org/wiki/Dilation_(metric_space)

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Translation

Bases: PositiveDilation, SpecialEuclidean

A translation.

symbol: T

wiki: https://en.wikipedia.org/wiki/Translation_(geometry)#As_a_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Linear

Bases: Affine

A linear transformation. May not be invertible.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

InvertibleLinear

Bases: Linear, InvertibleAffine

An invertible linear transformation.

Invertible linear transformation form a Lie group, named the General Linear group (GL).

This group includes reflections, and has therefore two connected components.

symbol: GL

wiki: https://en.wikipedia.org/wiki/General_linear_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

PositiveLinear

Bases: InvertibleLinear, PositiveAffine

Linear transformation with a positive determinant.

This group does not include reflections, and has therefore a single connected component.

symbol: GL+

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialLinear

Bases: PositiveLinear, SpecialAffine

A linear transformation with determinant 1 -- preserves volumes .

A linear map is an affine map whose translation vanishes, so SL sits inside SAff = SL ⋉ T -- the horizontal link of the diagram above -- and reaches VolumePreservingDiffeomorphism through it.

symbol: SL

wiki: https://en.wikipedia.org/wiki/Special_linear_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

ConformalOrthogonal

Bases: InvertibleLinear, ConformalEuclidean

A linear transformation that preserves angles (CO)

symbol: CO = O x ℝ+

wiki: https://en.wikipedia.org/wiki/Orthogonal_group#Conformal_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialConformalOrthogonal

Bases: ConformalOrthogonal, PositiveLinear, SpecialConformalEuclidean

A linear transformation that preserves angles and orientation (CO+)

symbol: CO+ = SO x ℝ+

wiki: https://en.wikipedia.org/wiki/Orthogonal_group#Conformal_group

Note

A scaled rotation c R has determinant c**n, which is positive but only equal to one when c is. CO+ therefore sits under GL+ (PositiveLinear), in the positive-determinant column of the diagram above, and not under SL: it does not preserve volumes. The rotations do -- see SpecialOrthogonal, which declares SL for itself.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Orthogonal

Bases: ConformalOrthogonal, Euclidean

A orthogonal matrix (AA' = I) with determinant ±1

symbol: O

wiki: https://en.wikipedia.org/wiki/Orthogonal_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialOrthogonal

Bases: Orthogonal, SpecialConformalOrthogonal, SpecialEuclidean, SpecialLinear

A orthogonal matrix (AA' = I) with determinant +1

symbol: SO

alias: Rotation

wiki: https://en.wikipedia.org/wiki/Orthogonal_group#Special_orthogonal_group

Note

SL is declared here rather than inherited: a rotation preserves volumes, but the conformal group it also belongs to does not (see SpecialConformalOrthogonal), so no ancestor can carry the fact.

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

GeneralizedPermutation

Bases: InvertibleLinear

A generalized permutation.

Permutations are invertible by definition.

The generalized permutation group is the semidirect product of the symmetric group (permutations) and the group of invertible diagonal matrices.

symbol: Δ ⋊ S

wiki: https://en.wikipedia.org/wiki/Generalized_permutation_matrix

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SignedPermutation

Bases: GeneralizedPermutation, Orthogonal

A signed permutation.

A generalized permutation with non-zero entries ±1.

symbol: B = C₂ⁿ ⋊ S

alias: HyperoctahedralGroup

wiki: https://en.wikipedia.org/wiki/Generalized_permutation_matrix#Signed_permutation_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Permutation

Bases: SignedPermutation, Orthogonal

A permutation.

A generalized permutation with non-zero entries +1.

Permutations form the symmetric group, named S.

symbol: S

wiki: https://en.wikipedia.org/wiki/Permutation_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

EvenPermutation

Bases: Permutation, SpecialOrthogonal

An even permutation.

A permutation with determinant +1.

A permutation matrix is orthogonal, and an even one has determinant +1, so the even permutations are exactly the permutations that lie in SO(n). That edge is what makes SO(n) closed under composition with an even permutation -- and hence what lets a subspace transform that reindexes its axes by an even permutation stay a rotation.

Even permutations form the (finite discrete) Alternating Group A.

symbol: A

wiki: https://en.wikipedia.org/wiki/Alternating_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

OddPermutation

Bases: Permutation

An odd permutation.

A permutation with determinant -1.

Note

This is the coset S \ A, not a group: composing two odd permutations gives an even one, so the set is not closed under composition and does not contain the identity.

symbol: S \ A

wiki: https://en.wikipedia.org/wiki/Permutation_group wiki: https://en.wikipedia.org/wiki/Alternating_group

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Diagonal

Bases: Linear

A diagonal matrix, may not be invertible.

wiki: https://en.wikipedia.org/wiki/Diagonal_matrix wiki: https://en.wikipedia.org/wiki/Scaling_(geometry)

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

InvertibleDiagonal

Bases: Diagonal, GeneralizedPermutation

An invertible diagonal matrix.

Invertible diagonal matrices form a Lie group.

This group includes reflections, and has therefore 2**n connected components.

symbol: Δ

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

PositiveDiagonal

Bases: InvertibleDiagonal, PositiveLinear

A diagonal matrix with positive entries.

This group does not include reflections, and has therefore a single connected component.

symbol: Δ+

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialDiagonal

Bases: InvertibleDiagonal, SpecialLinear

A diagonal matrix with determinant +1.

symbol: Δ_S

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

OrthogonalDiagonal

Bases: InvertibleDiagonal, SignedPermutation

A diagonal matrix with entries ±1.

symbol: Δ_O

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

SpecialOrthogonalDiagonal

Bases: OrthogonalDiagonal, SpecialDiagonal, SpecialOrthogonal

An orthogonal diagonal matrix with determinant +1.

symbol: Δ_SO

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Multiplicative

Bases: Diagonal

A scaling with the same factor in all dimensions.

symbol: ℝ = {c I : c ∈ ℝ}

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

InvertibleMultiplicative

Bases: Multiplicative, InvertibleDiagonal, ConformalOrthogonal

A scaling with the same nonzero factor in all dimensions.

An isotropic scaling preserves angles: c I is |c| times the orthogonal sign(c) I, so ℝ* lies in CO.

symbol: ℝ = {c I : c ∈ ℝ}

alias: Homothety, HomogeneousDilation

wiki: https://en.wikipedia.org/wiki/Multiplicative_group wiki: https://en.wikipedia.org/wiki/Homothety

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

PositiveMultiplicative

Bases: InvertibleMultiplicative, PositiveDiagonal, SpecialConformalOrthogonal

A scaling with the same positive factor in all dimensions.

A positive factor is a non-zero one, so this is a subgroup of ℝ* (InvertibleMultiplicative) and not merely a subset of ℝ (Multiplicative), which it reaches through it.

symbol: ℝ+ = {c I : c ∈ ℝ+*}

alias: PositiveHomothety

wiki: https://en.wikipedia.org/wiki/Scaling_(geometry)

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Identity

Bases: Translation, EvenPermutation, SpecialOrthogonalDiagonal, PositiveMultiplicative, SpecialOrthogonal

The identity transformation.

symbol: I

wiki: https://en.wikipedia.org/wiki/Identity_function

Methods:

__new__ 
__new__(subcls: type) -> type

Register a concrete transformation type to this set.

parse classmethod 
parse(repr: KindLike) -> Type[Self]

Return a transformation kind from its name, symbol or type.

Functions:

is_group 

is_group(cls: _Type) -> bool

Return whether a set of transformations forms a group.

A class is recognized as a group when it was marked with the group or liegroup decorator.

is_lie_group 

is_lie_group(cls: _Type) -> bool

Return whether a set of transformations forms a Lie group.

A class is recognized as a Lie group when it was marked with the liegroup decorator.

is_connected 

is_connected(cls: _Type) -> bool

Return whether a set of transformations is connected.

A class is recognized as connected when it was marked with the connected or simplyconnected decorator.

is_simplyconnected 

is_simplyconnected(cls: _Type) -> bool

Return whether a set of transformations is simply connected.

A class is recognized as simply connected when it was marked with the simplyconnected decorator.

is_invertible 

is_invertible(cls: _Type) -> bool

Return whether a set of transformations is known to be invertible.

is_closedunder cached 

is_closedunder(cls: _Type, subcls: _Type) -> bool

Return whether a set of transformations is closed under composition with another set.

This is always the case when subcls is a subgroup of cls, or when subcls is a subset of cls and cls is closed.

There can be more special cases, which are registered with the closedunder decorator.

is_closed 

is_closed(cls: _Type) -> bool

Return whether a set of transformations is closed under composition.

is_transformation_set 

is_transformation_set(cls: Any) -> bool

Return whether an object is a node of this hierarchy, i.e. whether it denotes a set of maps rather than a representation of one.

True for a class defined in this module (a real subclass of TransformationKind), False for a concrete transformation, a field, a wrapper or a container. A concrete transformation is only ever a virtual subclass of the node it registered to, and virtual registration does not touch the MRO, so the test separates the two cleanly.

is_embeddable 

is_embeddable(cls: _Type) -> bool

Return whether a set of transformations is known to embed into a higher-dimensional space by padding with the identity.

The embedding is T -> T ⊕ I, block-diagonal: T acts on the axes it already acted on, and the new ones pass through. The set is embeddable when the result stays in the same set one dimension up, G(n) ⊕ I ⊆ G(n+k). For example:

  • Every element of SO(3) is an element of SO(4) that way;
  • Most elements of ℝ(3) (a scaling by the same factor along every axis) are not elements of ℝ(4), since e.g. diag(2, 2, 2, 1) is not isotropic in ℝ⁴.

Note

This is an embedding: an injective homomorphism placing one group inside a bigger one.

all_sets cached 

all_sets() -> FrozenSet[type]

Return all known transformation sets.

all_groups cached 

all_groups() -> FrozenSet[type]

Return all known transformation groups.

all_lie_groups cached 

all_lie_groups() -> FrozenSet[type]

Return all known Lie groups.

as_invertible 

as_invertible(cls: _Type) -> _Type

Return the invertible subset of a set of transformations, if any.

Raises:

Type Description
TypeError

If the set is not invertible and has no known invertible subset.

as_unrestricted 

as_unrestricted(cls: _Type) -> _Type

Return the superset of a set of transformations that does not eclude non-invertible transformations (if any).

group 

group(cls: _Type) -> _Type

Mark the set of transformations as forming a group under composition.

wiki: https://en.wikipedia.org/wiki/Group_(mathematics)

liegroup 

liegroup(cls: _Type) -> _Type

Mark the set of transformations as forming a Lie group under composition.

wiki: https://en.wikipedia.org/wiki/Lie_group

connected 

connected(cls: _Type) -> _Type

Mark the set of transformations as being connected.

wiki: https://en.wikipedia.org/wiki/Connected_space

simplyconnected 

simplyconnected(cls: _Type) -> _Type

Mark the set of transformations as being simply connected.

wiki: https://en.wikipedia.org/wiki/Simply_connected_space

invertible_subset_of 

invertible_subset_of(cls: _Type) -> _Decorator

Mark the set of transformations as the invertible subset of another set.

wiki: https://en.wikipedia.org/wiki/Inverse_element

closedunder 

closedunder(cls: _Type) -> _Decorator

Mark the set of transformations as being closed under compatible composition with another set.

The argument is a set of transformations that is a superset of cls.

wiki: https://en.wikipedia.org/wiki/Closure_(mathematics)

closed 

closed(cls: _Type) -> _Type

Mark the set of transformations as being closed under compatible composition.

wiki: https://en.wikipedia.org/wiki/Closure_(mathematics)

nonembeddable 

nonembeddable(cls: _Type) -> _Type

Mark the set of transformations as not embedding into a higher-dimensional space by padding with the identity.

See is_embeddable for what the embedding is.

alias 

alias(cls: _Type, *names: str) -> _Type

Register a transformation set under an additional name.

This is useful for sets that have multiple common names.

is_family_tuple 

is_family_tuple(value: tuple) -> bool

Whether a tuple is a well-formed (kind, ndim) pair.

Exactly two elements, whose second one is a dimension: an int or None. A bool is an int in Python but not a dimension, so it is excluded.