3.3 Transformations

Transformation of coordinates from one two-dimensional coordinate system to another can be specified by means of a 3x3 transformation matrix. The matrix specifies how a point in one coordinate system is transformed into the corresponding point in another coordinate system.

The subsequent definitions of the geometric transformations are illustrated with the example polygon in the margin.

A transformation matrix(TM) specifies a transformation of point (x,y) to point (x', y') in the following way:

The common transformations: scaling, moving (translation), and rotation can easily be described by transformation matrices.

Scaling by factor s_x in the x dimension and s_y in the y dimension is accomplished by:

Moving (translation) by a specified displacement (t_x,t_y) is obtained by

Rotating counterclockwise, around the origin of the coordinate system, by an angle is described by the following matrix:

The most powerful feature of the matrix application, is that composition of geometric transformations can be expressed as multiplications of the corresponding matrices. That is, a combination of a rotate, move, and scale transformation can be combined into one matrix, and thus reduce the time of calculation of a complex transformation:

The user must be aware of the order of transformations, since matrix multiplication is not commutative. For a more thorough explanation of matrix operations and transformations, see e.g. [Newman 81].