W-curve

In geometry, a W-curve is a curve in projective n-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the real projective plane can be constructed with straightedge alone. Many well-known curves are W-curves, among them conics, logarithmic spirals, powers (like y = x3), logarithms and the helix, but not e.g. the sine. W-curves occur widely in the realm of plants.

caption
A typical plane W-curve with source O and sink Y

Name

The 'W' stands for the German 'Wurf' a throw which in this context refers to a series of four points on a line. A 1-dimensional W-curve (read: the motion of a point on a projective line) is determined by such a series.

The German "W-Kurve" sounds almost exactly like "Weg-Kurve" and the last can be translated by "path curve". That is why in the English literature one often finds "path curve" or "pathcurve".

See also

Further Reading

This article is issued from Wikipedia - version of the Thursday, May 22, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.