Trait Tangent 

pub trait Tangent<Group>: Clone + PartialEq
where
    Group: LieGroup,
{
    const DIM: usize;

    // Required methods
    fn exp(
        &self,
        jacobian: Option<&mut <Group as LieGroup>::JacobianMatrix>,
    ) -> Group;
    fn right_jacobian(&self) -> <Group as LieGroup>::JacobianMatrix;
    fn left_jacobian(&self) -> <Group as LieGroup>::JacobianMatrix;
    fn right_jacobian_inv(&self) -> <Group as LieGroup>::JacobianMatrix;
    fn left_jacobian_inv(&self) -> <Group as LieGroup>::JacobianMatrix;
    fn hat(&self) -> <Group as LieGroup>::LieAlgebra;
    fn small_adj(&self) -> <Group as LieGroup>::JacobianMatrix;
    fn lie_bracket(&self, other: &Self) -> <Group as LieGroup>::TangentVector;
    fn is_approx(&self, other: &Self, tolerance: f64) -> bool;
    fn generator(&self, i: usize) -> <Group as LieGroup>::LieAlgebra;
    fn zero() -> <Group as LieGroup>::TangentVector;
    fn random() -> <Group as LieGroup>::TangentVector;
    fn is_zero(&self, tolerance: f64) -> bool;
    fn normalize(&mut self);
    fn normalized(&self) -> <Group as LieGroup>::TangentVector;

    // Provided method
    fn is_dynamic() -> bool { ... }
}

Trait for Lie algebra operations.

This trait provides operations for vectors in the Lie algebra of a Lie group, including vector space operations, adjoint actions, and conversions to matrix form.

Type Parameters

• G: The associated Lie group type

Required Associated Constants

const DIM: usize

Dimension of the tangent space.

For fixed-size manifolds (SE2, SE3, SO2, SO3), this is the compile-time constant. For dynamic-size manifolds (Rn), this is 0 as a sentinel value — use the is_dynamic() method to check, and the LieGroup::tangent_dim() instance method to get the actual runtime dimension.

Required Methods

fn exp( &self, jacobian: Option<&mut <Group as LieGroup>::JacobianMatrix>, ) -> Group

Exponential map to Lie group: exp(φ^∧).

Arguments

• jacobian - Optional Jacobian ∂exp(φ^∧)/∂φ

fn right_jacobian(&self) -> <Group as LieGroup>::JacobianMatrix

Right Jacobian Jr.

Matrix Jr such that for small δφ: exp((φ + δφ)^∧) ≈ exp(φ^∧) ∘ exp((Jr δφ)^∧)

fn left_jacobian(&self) -> <Group as LieGroup>::JacobianMatrix

Left Jacobian Jl.

Matrix Jl such that for small δφ: exp((φ + δφ)^∧) ≈ exp((Jl δφ)^∧) ∘ exp(φ^∧)

fn right_jacobian_inv(&self) -> <Group as LieGroup>::JacobianMatrix

Inverse of right Jacobian Jr⁻¹.

fn left_jacobian_inv(&self) -> <Group as LieGroup>::JacobianMatrix

Inverse of left Jacobian Jl⁻¹.

fn hat(&self) -> <Group as LieGroup>::LieAlgebra

Hat operator: φ^∧ (vector to matrix).

Maps the tangent vector to its matrix representation in the Lie algebra. For SO(3): 3×1 vector → 3×3 skew-symmetric matrix For SE(3): 6×1 vector → 4×4 transformation matrix

fn small_adj(&self) -> <Group as LieGroup>::JacobianMatrix

Small adjugate operator: adj(φ) = φ^∧.

Maps the tangent vector to its matrix representation in the Lie algebra. For SO(3): 3×1 vector → 3×3 skew-symmetric matrix For SE(3): 6×1 vector → 4×4 transformation matrix

fn lie_bracket(&self, other: &Self) -> <Group as LieGroup>::TangentVector

Lie bracket: [φ, ψ] = φ ∘ ψ - ψ ∘ φ.

Computes the Lie bracket of two tangent vectors in the Lie algebra. For SO(3): 3×1 vector → 3×1 vector For SE(3): 6×1 vector → 6×1 vector

fn is_approx(&self, other: &Self, tolerance: f64) -> bool

Check if the tangent vector is approximately equal to another tangent vector.

Arguments

• other - The other tangent vector to compare with
• tolerance - The tolerance for the comparison

fn generator(&self, i: usize) -> <Group as LieGroup>::LieAlgebra

Get the i-th generator of the Lie algebra.

fn zero() -> <Group as LieGroup>::TangentVector

Zero tangent vector.

fn random() -> <Group as LieGroup>::TangentVector

Random tangent vector (useful for testing).

fn is_zero(&self, tolerance: f64) -> bool

Check if the tangent vector is approximately zero.

fn normalize(&mut self)

Normalize the tangent vector to unit norm.

fn normalized(&self) -> <Group as LieGroup>::TangentVector

Return a unit tangent vector in the same direction.

Provided Methods

fn is_dynamic() -> bool

Whether this tangent type has dynamic (runtime-determined) dimension.

Returns true for RnTangent where DIM == 0 is used as a sentinel. Returns false for all fixed-size tangent types (SE2, SE3, SO2, SO3).

Dyn Compatibility

This trait is not dyn compatible.

In older versions of Rust, dyn compatibility was called "object safety", so this trait is not object safe.

Implementors

impl Tangent<Rn> for RnTangent

const DIM: usize = 0

impl Tangent<SE2> for SE2Tangent

const DIM: usize = 3

impl Tangent<SE3> for SE3Tangent

const DIM: usize = 6

impl Tangent<SO2> for SO2Tangent

const DIM: usize = 1

impl Tangent<SO3> for SO3Tangent

const DIM: usize = 3