Scaling

From our discussion about the transformation of points, it follows that the magnitude scale is determined by the elements of the original diagonal matrix. If the matrix

(1.38)

is used as the operator of the impact on the vertices of a triangle, then the "double" expansion of scale with respect to the origin (uniform scaling) is used. If the element values ​​are not equal, then the triangle is distorted (differential scaling) as illustrated in Fig.1.5. Triangle ABC, transformed by the matrix 1.38, changes in proportion to the increased triangle A*B*C*. The same triangle, but transformed by the matrix

(1.39)

changes in the triangle DEF, having distortion caused by different scaling factors:

Fig.1.5. Uniform and differential scaling

In general, for the matrix

(1.40)

where a = d, b = c = 0 is the uniform scaling, and if a <> d, b = c = 0, the scaling is differential. In the first case a = d> 1 there is an expansion, i.e. zooming. If a = d <1, then it is a uniform reduction. Disproportionate expansion and reduction can arise depending on the values ​​of a and d, which may be less than or greater than 1, independently of each other.

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