# W-curve > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/W-curve > Markdown URL: https://mediated.wiki/source/W-curve.md > Source: https://en.wikipedia.org/wiki/W-curve > Source revision: 1285226695 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In geometry, a '''W-curve''' is a curve in [projective ''n''-space](/source/projective_space) that is [invariant](/source/invariant_(mathematics)) under a [1-parameter group](/source/1-parameter_group) of [projective transformations](/source/projective_transformations). W-curves were first investigated by [Felix Klein](/source/Felix_Klein) and [Sophus Lie](/source/Sophus_Lie) in 1871, who also named them. W-curves in the [real projective plane](/source/real_projective_plane) can be constructed with [straightedge](/source/straightedge) alone. Many well-known curves are W-curves, among them [conics](/source/conic_section), [logarithmic spiral](/source/logarithmic_spiral)s, powers (like ''y'' = ''x''3), [logarithm](/source/logarithm)s and the [helix](/source/helix), but not e.g. the [sine](/source/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](/source/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](/source/Georg_Scheffers) (1903) "Besondere transzendente Kurven", [Klein's encyclopedia](/source/Klein's_encyclopedia) Band 3–3. Category:Curves Category:Projective geometry --- Adapted from the Wikipedia article [W-curve](https://en.wikipedia.org/wiki/W-curve) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/W-curve?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.