Matrix Representation of Transformations on Two-Dimensional Space

Abstract

In chapter 2 we saw the need to translate pictures of objects about the screen. Rather than perpetually to change the screen coordinate system, it is conceptually much easier to define an object in the simplest terms possible (as vertices in the form of pixel or coordinate values, together with line and area information that is related to the vertices), and then transform the object to various parts of the screen but keeping the screen coordinate system fixed. We shall restrict ourselves to linear transformations (see below). It will often be necessary to transform a large number of vertices, and to do this efficiently we use matrices. Before looking at such matrix representations we should explain exactly what we mean by a matrix, and also by a column vector. In fact we restrict ourselves to square matrices: to 3 × 3 (said 3 by 3) for the study of two-dimensional space, and later we use 4 × 4 matrices when considering three-dimensional space. Such a 3 × 3 matrix (A say) is simply a group of real numbers placed in a block of 3 rows by 3 columns: a column vector (D say) is a group of numbers placed in a column of 3 rows:

$$ \left({\begin{array}{*{20}{c}} {{{\text{A}}_{11}}}&{{{\text{A}}_{12}}}&{{{\text{A}}_{13}}}\\ {{{\text{A}}_{21}}}&{{{\text{A}}_{22}}}&{{{\text{A}}_{23}}}\\ {{{\text{A}}_{31}}}&{{{\text{A}}_{32}}}&{{{\text{A}}_{33}}} \end{array}}\right)\;\;\;\;\;\;\;\;{\text{and}}\;\;\;\;\left({\begin{array}{*{20}{c}} {{{\text{D}}_1}}\\ {{{\text{D}}_2}}\\ {{{\text{D}}_3}} \end{array}}\right) $$

A general entry in the matrix is usually written A _ij ; the first subscript denotes the i^th row, and the second subscript the j^th column (for example, A₂₃ represents the value in the second row of the third column). The entry in the column vector, D _i , denotes the value in the i^th row. All these named entries will be explicitly replaced by numerical values and it is important to realise that the information stored in a matrix or column vector is not just the individual values but it is also the position of these values within the matrix or vector. Naturally