Scaling Rotating and Transforming

A single Matrix object can store a single transformation or a sequence of transformations. The latter is called a composite  transformation. The matrix of a composite transformation is obtained by multiplying the matrices of the individual transformations.

In a composite transformation, the order of the individual transformations is important. For example, if you first rotate, then scale, then translate, you get a different result than if you first translate, then rotate, then scale. In Windows GDI+, composite transformations are built from left to right. If S, R, and T are scale, rotation, and translation matrices respectively, then the product SRT (in that order) is the matrix of the composite transformation that first scales, then rotates, then translates. The matrix produced by the product SRT is different from the matrix produced by the product TRS.

One reason order is significant is that transformations like rotation and scaling are done with respect to the origin of the coordinate system. Scaling an object that is centered at the origin produces a different result than scaling an object that has been moved away from the origin. Similarly, rotating an object that is centered at the origin produces a different result than rotating an object that has been moved away from the origin.