This Title All WIREs
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.

Abstract 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 This article is categorized under: Applications of Computational Statistics > Computational Mathematics

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

[ Normal View | Magnified View ]

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

[ Normal View | Magnified View ]

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

[ Normal View | Magnified View ]

Perspective representation using natural homogeneous coordinates.

[ Normal View | Magnified View ]

Rotations in polar coordinates.

[ Normal View | Magnified View ]

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 ]

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 ]

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

[ Normal View | Magnified View ]

Related Articles

Scientific Visualization

Browse by Topic

Applications of Computational Statistics > Computational Mathematics

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