# homogeneous coordinates

**homogeneous coordinates** A coordinate system that algebraically treats all points in the projective plane (both Euclidean and ideal) equally. For example, the standard homogeneous coordinates [*p*_{1},*p*_{2},*p*_{3}] of a point P in the projective plane are of the form [*x*,*y*,1] if P is a point in the Euclidean plane *z*=1 whose Cartesian coordinates are (*x*,*y*,1), or are of the form [*a*,*b*,0] if P is the ideal point – the point at infinity – associated to all lines in the Euclidean plane *z*=1 with direction numbers *a*,*b*,0. Homogeneous coordinates are so called because they treat Euclidean and ideal points in the same way.

Homogeneous coordinates are widely used in computer graphics because they enable affine and projective transformations to be described as matrix manipulations in a coherent way.

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**homogeneous coordinates**