Question

The usual transformations on homogeneous coordinates for

2D computer graphics involve \({\bf{3}} \times {\bf{3}}\) matrices of the form \(\left[ {\begin{array}{*{20}{c}}A&p\\{{0^T}}&1\end{array}} \right]\) where Ais a \(2 \times {\bf{2}}\) matrix and p is in \({\mathbb{R}^{\bf{2}}}\). Show that such a transformation amounts to a linear transformation on \({\mathbb{R}^{\bf{2}}}\) followed by a translation. [Hint:Find an appropriate matrix factorization involving partitioned matrices.]

Short answer

A linear transformation on\({\mathbb{R}^{\bf{2}}}\)is followed by a translation by p in the provided matrix.

Step-by-step solution

Write the translation matrix

The transformation onhomogeneous coordinates for graphics has the matrix of the form\(\left[ {\begin{array}{*{20}{c}}A&{\bf{p}}\\{{0^T}}&1\end{array}} \right]\).

In thehomogeneous coordinates, the matrix representation of the linear transformation is\(\left[ {\begin{array}{*{20}{c}}A&{\bf{p}}\\{{0^T}}&1\end{array}} \right]\).

As p is in\({\mathbb{R}^{\bf{2}}}\), let\({\bf{p}} = \left( {h,k} \right)\). Also, as A is a\(2 \times 2\)matrix, let\(A = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]\).

In thehomogeneous coordinates, the matrix representation of the transformation is\(\left[ {\begin{array}{*{20}{c}}I&{\bf{p}}\\{{0^T}}&1\end{array}} \right]\).

Thetranslation matrix is represented as

\(\left[ {\begin{array}{*{20}{c}}1&0&h\\0&1&k\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&{\bf{p}}\\{{0^T}}&1\end{array}} \right]\).

Compute the product

Obtain the product of\(\left[ {\begin{array}{*{20}{c}}I&{\bf{p}}\\{{0^T}}&1\end{array}} \right]\)and \(\left[ {\begin{array}{*{20}{c}}A&{\bf{p}}\\{{0^T}}&1\end{array}} \right]\) .

\(\left[ {\begin{array}{*{20}{c}}I&{\bf{p}}\\{{0^T}}&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&0\\{{0^T}}&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}A&{\bf{p}}\\{{0^T}}&1\end{array}} \right]\)

Alinear transformation on\({\mathbb{R}^{\bf{2}}}\)is followed by a translation by p in the provided matrix.