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

Isoparametric and Dupin Hypersurfaces

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and $H^n$, 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 $S^n$. A hypersurface $M^{n-1}$ in a real space-form is proper Dupin if the number $g$ of distinct principal curvatures is constant on $M^{n-1}$, 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.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

Möbius Geometry and Cyclidic Nets: A Framework for Complex Shape Generation

International audienceFree-form architecture challenges architects, engineers and builders. The geometrical rationalization of complex structures requires sophisticated tools. To this day, two frameworks are commonly used: NURBS modeling and mesh-based approaches. The authors propose an alternative modeling framework called generalized cyclidic nets that automatically yields optimal geometrical properties for the façade and the structure. This framework uses a base circular mesh and Dupin cyclides, which are natural objects of the geometry of circles in space, also known as Möbius geometry. This paper illustrates how new shapes can be generated from generalized cyclidic nets. Finally, it is demonstrated that this framework gives a simple method to generate curved-creases on free-forms. These findings open new perspectives for structural design of complex shells

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

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 $C^1$-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

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