Matrices and Linear Transformations
Fletcher Dunn, Ian Parberry · 2011
This chapter discusses the implementation of linear transformations via matrices and shows how a sequence of primitive transformations may be combined by using matrix multiplication to form a more complicated transformation. It shows how to take a sequence of transformation matrices and combine them into one single transformation matrix. This new matrix represents the cumulative result of applying all of the original transformations in order. The chapter describes various interesting categories of transformations, including linear, affine, invertible, angle-preserving, orthogonal, and rigid-body transforms. The simplest scale operation applies a separate scale factor along each cardinal axis. The scale along an axis is applied about the perpendicular axis or plane. Translation, rotation, and reflection are the only orthogonal transformations. All orthogonal transformations are affine and invertible. Rigid body transformations are also known as proper transformations. All rigid body transformations are orthogonal, angle-preserving, invertible, and affine.