Home
This Title All WIREs
WIREs RSS Feed
How to cite this WIREs title:
WIREs Comp Stat

Natural homogeneous coordinates

Full article on Wiley Online Library:   HTML PDF

Can't access this content? Tell your librarian.

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.

[ Normal View | Magnified View ]
Figure 2.

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

[ Normal View | Magnified View ]
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.

[ Normal View | Magnified View ]
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/).

[ Normal View | Magnified View ]
Figure 5.

Rotations in polar coordinates.

[ Normal View | Magnified View ]
Figure 6.

Perspective representation using natural homogeneous coordinates.

[ Normal View | Magnified View ]
Figure 7.

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

[ Normal View | Magnified View ]
Figure 8.

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

[ Normal View | Magnified View ]

Related Articles

Scientific Visualization

Browse by Topic

Statistical Methods > Mathematical Methods
Applications of Computational Statistics > Computational Mathematics
blog comments powered by Disqus

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

The latest WIREs articles in your inbox

Sign Up for Article Alerts

Twitter: WileyCompSci Follow us on Twitter

    Journal of the Assn for Information Science and Technology's impact factor ranks in top 10 for Information Science http://t.co/IZ4fLZVqlA
    Prosanta Chakrabarty of LSU explains the dos & don'ts for your 1st month in a grad school lab. http://t.co/dH8Vq0VUMI via @WileyExchanges
    Computer-Aided Civil & Infrastructure Engineering retains no.1 ranking in 3 categories with an Impact Factor of 5.625 http://t.co/EN5ivfMPNk