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3D Programming for Windows® by Charles Petzold

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Affine Transforms and Beyond

Transforms that incorporate translation as well as scaling, rotation, and shear are known as affine transforms. The word affine derives from the same root as affinity, but in mathematics it usually refers to something that preserves finiteness. An affine transform never transforms a point in finite 3D space to a point with one or more infinite coordinates. That's a very broad definition, but where transforms are concerned, affine transforms are basically linear transforms with an additional translation factor.

Figuring out how to represent affine transforms with matrices is easier if we drop back to the relative comfort of two dimensions.

The Two-Dimensional Analogy

In two dimensions, a linear transform can be represented ...

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