Geometric inversion of a torus, cylinder or double cone
A Dupin cyclide
In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered c. 1802 by (and named after) Charles Dupin, while he was still a student at the École polytechnique following Gaspard Monge's lectures.[1] The key property of a Dupin cyclide is that it is a channel surface (envelope of a one-parameter family of spheres) in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.
Dupin cyclides are often simply known as cyclides, but the latter term is also used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the Laplace equation in three dimensions.
Dupin cyclides were investigated not only by Dupin, but also by A. Cayley, J.C. Maxwell and Mabel M. Young.
Dupin cyclides are used in computer-aided design because cyclide patches have rational representations and are suitable for blending canal surfaces (cylinder, cones, tori, and others).
In mathematics, a Dupincyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter...
particularly known for work in the field of mathematics, where the Dupincyclide and Dupin indicatrix are named after him; and for his work in the field of...
Clebsch cubic Monkey saddle (saddle-like surface for 3 legs.) Torus Dupincyclide (inversion of a torus) Whitney umbrella Boy's surface Cantor tree surface...
curvature spheres of a surface. It also allows for a natural treatment of Dupincyclides and a conceptual solution of the problem of Apollonius. Lie sphere geometry...
Clebsch cubic Monkey saddle (saddle-like surface for 3 legs.) Torus Dupincyclide (inversion of a torus) Whitney umbrella Right conoid (a ruled surface)...
of circles. The inversion of a cylinder, cone, or torus results in a Dupincyclide. A spheroid is a surface of revolution and contains a pencil of circles...
the Cretaceous and Tertiary Charles Dupin (1784–1873) – mathematician who discovered the Dupincyclide and the Dupin indicatrix Lennis Echterling – clinical...
knot. Its directrix is a curve on a torus e) The 5. picture shows a Dupincyclide (canal surface). Geometry and Algorithms for COMPUTER AIDED DESIGN,...
axis of rotation. The focal surface of a Dupincyclide consists of a pair of focal conics. The Dupincyclides are the only surfaces, whose focal surfaces...
directrices for generating Dupincyclides as canal surfaces in two ways. Focal conics can be seen as degenerate focal surfaces: Dupincyclides are the only surfaces...
Parabolic conoid Plücker's conoid Whitney umbrella Châtelet surfaces Dupincyclides, inversions of a cylinder, torus, or double cone in a sphere Gabriel's...
corresponding Steiner chain. The envelope of the hexlet spheres is a Dupincyclide, the inversion of a torus. The six spheres are not only tangent to the...
field is finite, then it is said to be an arithmetic quartic surface. Dupincyclides The Fermat quartic, given by x4 + y4 + z4 + w4 =0 (an example of a K3...
analogues of Möbius transformations for oriented projective geometry Dupincyclides, shapes obtained from cylinders and tori by inversion At the time of...
with Frank Morley at Johns Hopkins University. Her thesis was titled "Dupin'scyclide as a self-dual surface". With her doctoral degree, Young was eventually...
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