Crate apex_manifolds 

Manifold representations for optimization on non-Euclidean spaces.

This module provides manifold representations commonly used in computer vision and robotics:

• SE(3): Special Euclidean group (rigid body transformations)
• SO(3): Special Orthogonal group (rotations)
• Sim(3): Similarity transformations (rotation + translation + scale)
• SGal(3): Special Galilean group (rotation + translation + velocity + time)
• SE_2(3): Extended Special Euclidean group (rotation + translation + velocity)
• SE(2): Rigid transformations in 2D
• SO(2): Rotations in 2D

Lie group M,°	size	dim	X ∈ M	Constraint	T_E M	T_X M	Exp(T)	Comp.	Action
n-D vector	Rⁿ,+	n	n	v ∈ Rⁿ		v-v	=0	v ∈ Rⁿ	v ∈ Rⁿ
Circle	S¹,.	2	1	z ∈ C	z*z = 1	iθ ∈ iR	θ ∈ R	z = exp(iθ)	z₁z₂
Rotation	SO(2),.	4	1	R	RᵀR = I	[θ]x ∈ so(2)	[θ] ∈ R²	R = exp([θ]x)	R₁R₂
Rigid motion	SE(2),.	9	3	M = [R t; 0 1]	RᵀR = I	[v̂] ∈ se(2)	[v̂] ∈ R³	Exp([v̂])	M₁M₂
3-sphere	S³,.	4	3	q ∈ H	q*q = 1	θ/2 ∈ Hp	θ ∈ R³	q = exp(uθ/2)	q₁q₂
Rotation	SO(3),.	9	3	R	RᵀR = I	[θ]x ∈ so(3)	[θ] ∈ R³	R = exp([θ]x)	R₁R₂
Rigid motion	SE(3),.	16	6	M = [R t; 0 1]	RᵀR = I	[v̂] ∈ se(3)	[v̂] ∈ R⁶	Exp([v̂])	M₁M₂
Similarity	Sim(3),.	16	7	M = [sR t; 0 1]	RᵀR=I, s>0	[v̂] ∈ sim(3)	[ρ,θ,σ] ∈ R⁷	Exp([v̂])	M₁M₂
Galilean	SGal(3),.	25	10	(R,t,v,s)	RᵀR = I	[v̂] ∈ sgal(3)	[ρ,ν,θ,s] ∈ R¹⁰	Exp([v̂])	M₁M₂
Extended pose	SE_2(3),.	25	9	(R,t,v)	RᵀR = I	[v̂] ∈ se_2_3	[ρ,θ,ν] ∈ R⁹	Exp([v̂])	M₁M₂

The design is inspired by the manif C++ library and provides:

• Analytic Jacobian computations for all operations
• Right and left perturbation models
• Composition and inverse operations
• Exponential and logarithmic maps
• Tangent space operations

Mathematical Background

This module implements Lie group theory for robotics applications. Each manifold represents a Lie group with its associated tangent space (Lie algebra). Operations are differentiated with respect to perturbations on the local tangent space.

Modules

rn

Rn - n-dimensional Euclidean Space

se2

SE(2) - Special Euclidean Group in 2D

se3

SE(3) - Special Euclidean Group in 3D

se23

SE_2(3) - Extended Special Euclidean Group (Rotation + Translation + Velocity)

sgal3

SGal(3) - Special Galilean Group

sim3

Sim(3) - Similarity Transformations in 3D

so2

SO(2) - Special Orthogonal Group in 2D

so3

SO3 - Special Orthogonal Group in 3D

Enums

ManifoldError

Errors that can occur during manifold operations.

ManifoldType

Constants

SMALL_ANGLE_THRESHOLD

Threshold for switching between exact formulas and Taylor approximations in small-angle computations.

Traits

Interpolatable

Trait for Lie groups that support interpolation.

LieGroup

Core trait for Lie group operations.

Tangent

Trait for Lie algebra operations.

Type Aliases

ManifoldResult

Result type for manifold operations.