73 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
Darboux cyclides and webs from circles
Motivated by potential applications in architecture, we study Darboux
cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin
cyclides and quadrics, and they carry up to six real families of circles.
Revisiting the classical approach to these surfaces based on the spherical
model of 3D Moebius geometry, we provide computational tools for the
identification of circle families on a given cyclide and for the direct design
of those. In particular, we show that certain triples of circle families may be
arranged as so-called hexagonal webs, and we provide a complete classification
of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure
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
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
Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides
Cyclidic nets are introduced as discrete analogs of curvature line
parametrized surfaces and orthogonal coordinate systems. A 2-dimensional
cyclidic net is a piecewise smooth -surface built from surface patches of
Dupin cyclides, each patch being bounded by curvature lines of the supporting
cyclide. An explicit description of cyclidic nets is given and their relation
to the established discretizations of curvature line parametrized surfaces as
circular, conical and principal contact element nets is explained. We introduce
3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinate
systems and investigate them in detail. Our considerations are based on the Lie
geometric description of Dupin cyclides. Explicit formulas are derived and
implemented in a computer program.Comment: 39 pages, 30 figures; Theorem 2.7 has been reformulated, as a
normalization factor in formula (2.4) was missing. The corresponding
formulations have been adjusted and a few typos have been correcte
Geometry and Shape of Minkowski's Space Conformal Infinity
We review and further analyze Penrose's 'light cone at infinity' - the
conformal closure of Minkowski space. Examples of a potential confusion in the
existing literature about it's geometry and shape are pointed out. It is argued
that it is better to think about conformal infinity as of a needle horn
supercyclide (or a limit horn torus) made of a family of circles, all
intersecting at one and only one point, rather than that of a 'cone'. A
parametrization using circular null geodesics is given. Compactified Minkowski
space is represented in three ways: as a group manifold of the unitary group
U(2) a projective quadric in six-dimensional real space of signature (4,2) and
as the Grassmannian of maximal totally isotropic subspaces in complex
four--dimensional twistor space. Explicit relations between these
representations are given, using a concrete representation of antilinear action
of the conformal Clifford algebra Cl(4,2) on twistors. Concepts of space-time
geometry are explicitly linked to those of Lie sphere geometry. In particular
conformal infinity is faithfully represented by planes in 3D real space plus
the infinity point. Closed null geodesics trapped at infinity are represented
by parallel plane fronts (plus infinity point). A version of the projective
quadric in six-dimensional space where the quotient is taken by positive reals
is shown to lead to a symmetric Dupin's type `needle horn cyclide' shape of
conformal infinity.Comment: 19 pages, 8 figures, a dozen of typos fixe
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