VII.1 - DECOMPOSING A MATRIX INTO SIMPLE TRANSFORMATIONS

Abstract

This chapter discusses decomposition of a matrix into simple transformations. Given any 4 × 4 transformation matrix M, it will compute the arguments to the sequence of transformations, such that concatenating the transformations will reproduce the original matrix. This routine has been used for tasks such as removing the shears from a rotation matrix, for feeding an arbitrary transformation to a graphics system that only understands a particular sequence of transformations, or for any other application in which one wants just part of the transformation sequence. The algorithm works by undoing the transformation sequence in reverse order. It first determines perspective elements that, when removed from the matrix, will leave the last column as (0, 0, 0, 1)T. Then it extracts the translations. This leaves a 3 × 3 matrix comprising the scales, shears, and rotations. It is decomposed from the left, extracting first the scaling factors and then the shearing components, leaving a pure rotation matrix. This is broken down into three consecutive rotations.