The natural homogeneous coordinate system is the analog of the Cartesian coordinate system for projective geometry. Roughly speaking a projective geometry adds an axiom that parallel lines meet at a point at infinity. This removes the impediment to line-point duality that is found in traditional Euclidean geometry. The natural homogeneous coordinate system is surprisingly useful in a number of applications including computer graphics and statistical data visualization. In this article, we describe the axioms of projective geometry, introduce the formalism of natural homogeneous coordinates, and illustrate their use with four applications. WIREs Comp Stat 2010 2 678–685 DOI: 10.1002/wics.122
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