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Natural homogeneous coordinates

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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

Figure 1.

Representation of the projective plane by a hemisphere which can be deformed into a crosscap.

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Figure 2.

Partially deformed hemisphere so that antipodal points along the equator are approaching each other.

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Figure 3.

The completely deformed hemisphere with antipodal points identified. In this rendition, 2D view of a 3D structure, the surfaces penetrate each other. However, embedded in a higher dimensional space these surfaces do not intersect.

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Figure 4.

Crosscap rendered as a color shaded figure. (Reprinted with permission from Professor Paul Bourke, University of Western Australia. http://local.wasp.uwa.edu.au/∼pbourke/geometry/).

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Figure 5.

Rotations in polar coordinates.

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Figure 6.

Perspective representation using natural homogeneous coordinates.

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Figure 7.

Mapping Cartesian points into lines in parallel coordinate space and Cartesian lines into points in parallel coordinate space.

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Figure 8.

A model for two‐dimensional projective plane for special relativity using natural homogeneous coordinates.

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