In geometry, a W-curve is a curve in
projective ''n''-space that is
invariant under a
1-parameter group of
projective transformations
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
. W-curves were first investigated by
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
and
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius Soph ...
in 1871, who also named them. W-curves in the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
can be constructed with
straightedge alone. Many well-known curves are W-curves, among them
conics,
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
s, powers (like ''y'' = ''x''
3),
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s and the
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helic ...
, but not e.g. the
sine. W-curves occur widely in the realm of plants.
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 th
University of Goettingen* For an introduction on W-curves and how to draw them, see Lawrence Edwards ''Projective Geometry'', Floris Books 2003,
* 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
Classification of pathcurves'
*
Georg Scheffers (1903) "Besondere transzendente Kurven",
Klein's encyclopedia
Felix Klein's ''Encyclopedia of Mathematical Sciences'' is a German mathematical encyclopedia published in six volumes from 1898 to 1933. Klein and Wilhelm Franz Meyer were organizers of the encyclopedia. Its full title in English is ''Encycloped ...
Band 3–3.
Curves
Projective geometry