Page 117 - Computer Graphics Handout

Rotation and translation are known as rigid-body transformations. No combination of rotations and translations can alter the shape
or volume of an object; they can alter only the object’s location and orientation. Consequently, rotation and translation alone cannot
give us all possible affine transformations. The transformations shown in Figure 3.37 are affine, but they are not rigid-body
transformations.

3.8.3 Scaling
Scaling is an affine non–rigid-body transformation by which we can make an object bigger or smaller. Figure 3.38 illustrates both
uniform scaling in all directions and scaling in a single direction. We need nonuniform scaling to build up the full set of affine
transformations that we use in modeling and viewing by combining a properly
chosen sequence of scalings, translations, and rotations.

Scaling transformations have a fixed point, as we can see from Figure 3.39. Hence, to specify a scaling, we can specify the fixed point,
a direction in which we wish to scale, and a scale factor (α). For α >1, the object gets longer in the specified direction; for 0 ≤α <1,
the object gets smaller in that direction. Negative values of α give us reflection (Figure 3.40) about the fixed point, in the scaling

direction. Scaling has six degrees of freedom because we can specify an arbitrary fixed point and three independent scaling factors.

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