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'' = ''x''<sup>3</sup>), logarithms and the helix, but not e.g. the sine. W-curves occur widely in the realm of plants.
thumb|right|alt=caption|300px|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== * Homography
==Further reading== * Felix Klein and Sophus Lie: ''Ueber diejenigen ebenen Curven...'' in Mathematische Annalen, Band 4, 1871; online available at the [https://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0004 University of Goettingen] * For an introduction on W-curves and how to draw them, see Lawrence Edwards ''Projective Geometry'', Floris Books 2003, {{ISBN|0-86315-393-3}} * On the occurrence of W-curves in nature see Lawrence Edwards ''The vortex of life'', Floris Books 1993, {{ISBN|0-86315-148-5}} * For an algebraic classification of 2- and 3-dimensional W-curves see ''[https://www.mathart.nl/Pathcurves1010.pdf Classification of pathcurves]'' * Georg Scheffers (1903) "Besondere transzendente Kurven", Klein's encyclopedia Band 3–3.
Category:Curves Category:Projective geometry