W curve
A W-curve is a geometric curve in projective space that is invariant under a 1-parametric group of projective transformations. W-curves were first examined by Felix Klein and Sophus Lie in 1871 , they also gave them their names. A ruler is sufficient for the construction of W-curves. Many known curves are W-curves, e.g. B. conics , logarithmic spirals , the graph of power functions such as , logarithms and helices ( helix ). W-curves occur many times in the plant kingdom. The trigonometric functions , for example, are not W-curves .
Naming
The letter 'W' comes from 'throw', which - according to von Staudt - means a series of four points on a straight line. A one-dimensional W-curve (that is, the movement of a point on a projective straight line) is determined by such a series.
"W-curve" sounds very similar to "path curve", and that can be translated as "path curve" in English. Therefore, this term is often found in English literature.
literature
- Felix Klein and Sophus Lie: About those plane curves ... in Mathematische Annalen , Volume 4, 1871; available online at the University of Goettingen
- For an introduction to the W-curves and how to draw them, see Ostheimer and Ziegler: Skalen und Wegkurven (sic!), Verlag am Goetheanum 1996, ISBN 3-7235-0952-5 , or
- Lawrence Edwards: Projective Geometry , Floris Books 2003, ISBN 0-86315-393-3
- For W-curves in nature, see Lawrence Edwards: The vortex of life , Floris Books 1993, ISBN 0-86315-148-5
- For an algebraic classification of the 2- and 3-dimensional W-curves see Classification of pathcurves (PDF; 9.9 MB)
- Georg Scheffers : "Special transcendent curves", in Encyclopedia of Mathematical Sciences , 1903, Volume 3–3, http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN360610161&DMDID=DMDLOG_0117 .