homogeneous_transform.h File Reference

Inverse of 4×4 homogeneous transformation matrices. More...

#include "config.h"
#include "fused/fused_matrix.h"
#include "fused/fused_vector.h"

Include dependency graph for homogeneous_transform.h:

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Namespaces
namespace	matrix_algorithms

Functions
template<typename T >
FusedMatrix< T, 4, 4 >	matrix_algorithms::homogeneous_inverse (const FusedMatrix< T, 4, 4 > &H)
	Compute the inverse of a 4×4 homogeneous transformation matrix.

Detailed Description

Inverse of 4×4 homogeneous transformation matrices.

A homogeneous transform encodes a rigid-body transformation in SE(3):

  T = [ R | t ]
      [ 0 | 1 ]

where R ∈ SO(3) is a 3×3 rotation and t ∈ ℝ³ is a translation.

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ALGORITHM

The inverse exploits the orthogonality of R (R⁻¹ = Rᵀ):

  T⁻¹ = [ Rᵀ  | -Rᵀ·t ]
        [ 0   |   1    ]

This avoids the general 4×4 inverse (which requires cofactor expansion or LU decomposition) and uses only a 3×3 transpose + 3×3·3×1 matmul.

Complexity: O(1) — 9 copies (transpose) + 9 multiplies + 6 adds (matmul).

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PRECONDITIONS

The caller is responsible for ensuring T has valid homogeneous form:

• R must be orthogonal (Rᵀ·R = I, det(R) = +1)
• Bottom row must be [0 0 0 1]

No runtime validation is performed. Passing a non-homogeneous matrix produces a mathematically meaningless result.

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