How do I get the inverse of a homogeneous transformation matrix?

Question

I want to get the inverse of this homogeneous transformation matrix:

iab = {
 {1, 0, 0, 0},
 {0, 0, -1, 0},
 {0, 1, 0, 3},
 {0, 0, 0, 1}
      }

using the inverse function, but the problem is that I cannot directly use this function. Somehow I have to convert first the matrix to a transformation function.

iba = InverseFunction[iab] // MatrixForm

Using that I don't get any result. I can use TranslationTransform and RotationTransform to compute the inverse, but the problem with that approach is that I can only create a rotation matrix or a translation matrix, but not both in one transformation like in the above matrix I posted.

Answer 1

If you have a homogenous transformation matrix of the form $$\begin{bmatrix} \mathrm{R_{3 \times 3}} & \mathrm{d}_{3 \times 1} \\ 0_{1\times 3} & 1_{1\times 1} \end{bmatrix}$$ Then the inverse is given by $$\begin{bmatrix} \mathrm{R}^{-1} & -\mathrm{R}^{-1}\mathrm{d} \\ 0 & 1 \end{bmatrix}$$

Therefore, if your homogeneous matrix is (I have added the 1 in the lower corner that I think should be there)

iab = {{1, 0, 0, 0},
       {0, 0, -1, 0},
       {0, 1, 0, 3},
       {0, 0, 0, 1}
      };

then the inverse can be written as (note that for rotation matrices, the inverse is the transpose)

homogeneousTransformationInverse[mat_] /; Dimensions[mat] == {4, 4} :=
  Module[{
     rot = Transpose[mat[[1 ;; 3, 1 ;; 3]]],
     vec = mat[[1 ;; 3, -1]],
     inv = mat
    },
   inv[[1 ;; 3, 1 ;; 3]] = rot;
   inv[[1 ;; 3, -1]] = -rot.vec;
   inv
  ]

---

I think the above is cleaner than the original version:

homogeneousTransformationInverse[mat_] /; Dimensions[mat] == {4, 4} :=
  Module[
   {rot = Transpose[mat[[1 ;; 3, 1 ;; 3]]], vec = mat[[1 ;; 3, -1]]},
   ArrayFlatten[{{rot, Map[List, -rot.vec]}, {{{0, 0, 0}}, {{mat[[-1, -1]]}}}}]
  ]
homogeneousTransformationInverse[iab] // MatrixForm

Answer 2

Here's a nice one-liner:

TransformationFunction[{{1, 0, 0, 0}, {0, 0, -1, 0}, {0, 1, 0, 3}, {0, 0, 0, 1}}]
// InverseFunction
   TransformationFunction[{{1, 0, 0, 0}, {0, 0, 1, -3}, {0, -1, 0, 0}, {0, 0, 0, 1}}]

Note that TransformationFunction[] is the head of the results returned by geometric *Transform functions, which take a homogeneous transformation matrix as an argument. Since you have the matrix already, you merely need to add the wrapper and then use InverseFunction[] to invert the transformation.

---

For future reference: Composition[] is a handy way to chain together more than one TransformationFunction:

Composition[TranslationTransform[{0, 0, 3}], RotationTransform[π/2, {1, 0, 0}]]

Alternatively, AffineTransform[] allows a direct construction:

AffineTransform[{RotationMatrix[π/2, {1, 0, 0}], {0, 0, 3}}]

Answer 3

This one's not orthogonal but it is invertible with det = 1, am I missing something?

Dan Bach - www.dansmath.com