Homogeneous Coordinates

Any two-dimensional linear transformation can be expressed as a three-by-three matrix, and any three-dimensional linear transformation can be expressed as a four-by-four matrix. In two dimensions, any coordinate (x, y) can be assigned to the vector x y , i and in three dimensions, any point (x, y, z) can be assigned to the vector x y z i

In each case, the extra final element can be viewed as a pragmatic placeholder that makes the mathematics work properly, though in fact, it turns out to have a meaningful geometric interpretation that is discussed later in the section on perspective transformations. In the event that a matrix computation generates a vector ending with a nonzero value other than unity, the entire vector can be rescaled by a scalar value and then interpreted as a homogeneous coordinate. For example, the three-dimensional vector

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can be rescaled to

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