Matrix Concatenation

One advantage of using matrices is that you can combine the effects of two or more matrices by multiplying them. This means that, to rotate a model and then translate it to some location, you don't need to apply two matrices. Instead, you multiply the rotation and translation matrices to produce a composite matrix that contains all their effects. This process, called matrix concatenation, can be written with the following formula.

In this formula, C is the composite matrix being created, and M₁ through M_n are the individual transformations that matrix C contains. In most cases, only two or three matrices are concatenated, but there is no limit.

The order in which the matrix multiplication is performed is crucial. The preceding formula reflects the left-to-right rule of matrix concatenation. That is, the visible effects of the matrices that you use to create a composite matrix occur in left-to-right order. A typical world transformation matrix is shown the following example. Imagine that you are creating the world transformation matrix for a stereotypical flying saucer. You would probably want to spin the flying saucer around its center—the y-axis of model space—and translate it to some other location in your scene. To accomplish this effect, you first create a rotation matrix, and then multiply it by a translation matrix, as shown in the following formula.

In this formula, R_y is a matrix for rotation about the y-axis, and T_w is a translation to some position in world coordinates.

The order in which you multiply the matrices is important because, unlike multiplying two scalar values, matrix multiplication is not commutative. Multiplying the matrices in the opposite order has the visual effect of translating the flying saucer to its world space position, and then rotating it around the world origin.

No matter what type of matrix you are creating, remember the left-to-right rule to ensure that you achieve the expected effects.