Matrix Applications in Computer Graphics

Abstract

Column matrices can be used to represent points in$$2D$$ or$$3D$$, while matrices of dimension$$2\times n$$ and$$3\times n$$ can be used to represent sets of points in$$2D$$ or$$3D$$. Matrices allow arbitrary linear transformations to be represented in a consistent format ($$T(\mathbf x ) = \mathbf{A }\mathbf x $$ for some$$2\times n$$ (or$$3\times n$$ ) matrix$$\mathbf{A }$$, called the transformation matrix of T), suitable for computation. This format allows transformations to be conveniently combined with each other by multiplying their matrices. In this chapter we first use matrices to represent points, lines and polygons. We then discuss in detail some linear transformations such as translation, scaling, rotation, reflections and shearing in 2D, and examine how transformations can be concatenated using matrix multiplication.