29 research outputs found
Isoparametric and Dupin Hypersurfaces
A hypersurface in a real space-form , or is
isoparametric if it has constant principal curvatures. For and
, the classification of isoparametric hypersurfaces is complete and
relatively simple, but as Elie Cartan showed in a series of four papers in
1938-1940, the subject is much deeper and more complex for hypersurfaces in the
sphere . A hypersurface in a real space-form is proper Dupin if
the number of distinct principal curvatures is constant on , and
each principal curvature function is constant along each leaf of its
corresponding principal foliation. This is an important generalization of the
isoparametric property that has its roots in nineteenth century differential
geometry and has been studied effectively in the context of Lie sphere
geometry. This paper is a survey of the known results in these fields with
emphasis on results that have been obtained in more recent years and discussion
of important open problems in the field.Comment: This is a contribution to the Special Issue "Elie Cartan and
Differential Geometry", published in SIGMA (Symmetry, Integrability and
Geometry: Methods and Applications) at http://www.emis.de/journals/SIGM
Dupin Cyclides as a Subspace of Darboux Cyclides
Dupin cyclides are interesting algebraic surfaces used in geometric design
and architecture to join canal surfaces smoothly and construct model surfaces.
Dupin cyclides are special cases of Darboux cyclides, which in turn are rather
general surfaces in R^3 of degree 3 or 4. This article derives the algebraic
conditions (on the coefficients of the implicit equation) for recognition of
Dupin cyclides among the general implicit form of Darboux cyclides. We aim at
practicable sets of algebraic equations describing complete intersections
inside the parameter space.Comment: 20 pages, 1 figur
A unified approach to blending of constant and varying parametric surfaces with curvature continuity
In this paper, we develop a new approach to blending of
constant and varying parametric surfaces with curvature
continuity. We propose a new mathematical model consisting of a
vector-valued sixth-order partial differential equation (PDE) and
time-dependent blending boundary constraints, and develop an
approximate analytical solution of the mathematical model. The
good accuracy and high computational efficiency are
demonstrated by comparing the new approximate analytical
solution with the corresponding accurate closed form solution. We also investigate the influence of the second partial derivatives on
the continuity at trimlines, and apply the new approximate
analytical solution in blending of constant and varying parametric
surfaces with curvature continuit
The implicit equation of a canal surface
A canal surface is an envelope of a one parameter family of spheres. In this
paper we present an efficient algorithm for computing the implicit equation of
a canal surface generated by a rational family of spheres. By using Laguerre
and Lie geometries, we relate the equation of the canal surface to the equation
of a dual variety of a certain curve in 5-dimensional projective space. We
define the \mu-basis for arbitrary dimension and give a simple algorithm for
its computation. This is then applied to the dual variety, which allows us to
deduce the implicit equations of the the dual variety, the canal surface and
any offset to the canal surface.Comment: 26 pages, to be published in Journal of Symbolic Computatio
Intuitive procedure for constructing geometrically complex objects using cyclides
In the past, cyclide surfaces have been used effectively for the variable radius blending of natural quadric intersections. However, attempts to use cyclides for constructing realistic, freeform composite surfaces met with rather limited success. The paper presents a simple procedure for creating and manipulating geometrically complex objects using tubular cyclide pieces. The method described is intuitive from the designer's point of view, and it is based on the fundamental definitions and properties of the cyclide. Various practical issues involved in this design procedure are discussed, several extensions of the basic technique are described, and implemented examples are provided.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31678/1/0000614.pd
Enumerating the morphologies of non-degenerate Darboux cyclides
International audienceWe provide an enumeration of all possible morphologies of non-degenerate Darboux cyclides. Based on the fact that every Darboux cyclide in R 3 is the stereographic projection of the intersection surface of a sphere and a quadric in R 4 , we transform the enumeration problem of morphologies of Darboux cyclides to the enumeration of the algebraic sequences that characterize the intersection of a sphere and a quadric in R 4
Isoparametric and Dupin Hypersurfaces
A hypersurface Mn−1 in a real space-form Rn, Sn or Hn is isoparametric if it has constant principal curvatures. For Rn and Hn, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Élie Cartan showed in a series of four papers in 1938–1940, the subject is much deeper and more complex for hypersurfaces in the sphere Sn. A hypersurface Mn−1 in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on Mn−1, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field