Dupin Cyclides Research Articles

Bounded domain, bi-quadratic rational parametrizations of Dupin cyclides

Dupin cyclides, their applications in geometric modelling and their parametrization using bi-quadratic patches bounded by lines of curvature, have been investigated in recent years by a number of authors such as Martin, de Pont and Sharrock in 1986, Boehm in 1990, Pratt in 1990, and Degen in 1994. However, no completely reliable and general algorithm for the determination of bi-quadratic cyclide patches has appeared in the literature. This paper presents a new approach that produces any required bi-quadratic patch, bounded by lines of curvature, without non-intrinsic geometric constraints or restrictions. Specifically, if a bi-quadratic parametrization exists for the specified region of the cyclide, then it is correctly determined. Explicit formulae are given for the Bernstein weights and vectors of the parametrizations. The method is neither cyclide specific nor specific to the construction of bi-quadratic rational parametrizations – it may therefore be applied to other surfaces and to higher degree rational constructions.

Branching blend of natural quadrics based on surfaces with rational offsets

A new branching blend between two natural quadrics (circular cylinders/cones or spheres) in many positions is proposed. The blend is a ring shaped patch of a PN surface (surface with rational offset) parametrized by rational bivariant functions of degree ( 6 , 3 ) . The general theory of PN surfaces is developed using Laguerre geometry and a universal rational parametrization of the Blaschke cylinder. The construction is extended via inversion to a PN branching blend of degree ( 8 , 4 ) between Dupin cyclide and a natural quadric.

The Lamé equation in shell membrane theory

A classical nonlinear shell membrane system has recently been demonstrated to have underlying integrable structure. Here, a wide class of corresponding parallel membranes is shown to be generated via a Schrödinger equation of Lamé type. This class is characterized by the existence of a multiplicity of stress distributions for a given membrane geometry. A variety of viable membrane geometries such as generalized Dupin cyclides are constructed explicitly together with the associated one-parameter families of stress resultants. A Lax pair which encapsulates both the Gauss-Mainardi-Codazzi and constrained equilibrium equations is also recorded.

Conversion d'un carreau de Bézier rationnel biquadratique en un carreau de cyclide de Dupin quartique

Dupin cyclides were introduced in 1822 by the French mathematician C-P. Dupin. They are algebraic surfaces of degree 3 or 4. The set of geometric properties of these surfaces has encouraged an increasing interest in using them for geometric modeling. A couple of algorithmes is already developed to convert a Dupin cyclide patch into a rational biquadratic Bezier patch. In this paper, we consider the inverse problem: we investigate the conditions of convertibility of a Bezier patch into a Dupin cyclide one, and we present a conversion algorithm to compute the parameters of a Dupin cyclide with the boundary of the patch that corresponds to the given Bezier patch.

Imperfections in focal conic domains: the role of dislocations

It is usual to think of focal conic domains (FCDs) as perfect geometric constructions in which the layers are folded into Dupin cyclides about an ellipse and a hyperbola that are conjugate. This ideal picture is often far from reality. We have investigated in detail the FCDs in several materials that have a transition from a smectic A (SmA) to a nematic phase (N). The ellipse and the hyperbola are seldom perfect, and the FCD textures also suffer large transformations (in shape and/or in nature) when approaching the transition to the nematic phase, or appear imperfect on cooling from the nematic phase. We interpret these imperfections as due to the interaction of FCDs with dislocations. We analyze theoretically the general principles subtending the interaction mechanisms between FCDs and finite Burgers vector dislocations, namely the formation of kinks on disclinations, to which dislocations are attached, and we present models relating to some experimental results. Whereas the principles of the interactions are very general, their realizations can differ widely in function of the boundary conditions.

Boolean Operations with Implicit and Parametric Representation of Primitives Using R-Functions

We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit and a parametric representation, such as superquadrics, supershapes, and Dupin cyclides. The input is a Constructive Solid Geometry tree (CSG tree) that contains the Boolean operations, the parameters of the primitives, and the global deformations. At each node of the CSG tree, the implicit formulations of the subtrees are used to quickly determine the parts to be transmitted to the parent node, while the primitives' parametric definition are used to refine an intermediary mesh around the intersection curves. The output is both an implicit equation and a mesh representing its solution. For the resulting object, an implicit equation with guaranteed differential properties is obtained by simple combinations of the primitives' implicit equations using R-functions. Depending on the chosen R-function, this equation is continuous and can be differentiable everywhere. The primitives' parametric representations are used to directly polygonize the resulting surface by generating vertices that belong exactly to the zero-set of the resulting implicit equation. The proposed approach has many potential applications, ranging from mechanical engineering to shape recognition and data compression. Examples of complex objects are presented and commented on to show the potential of our approach for shape modeling.

On a nonlinear elastic shell system in liquid crystal theory: generalized Willmore surfaces and Dupin cyclides

An elastic membrane model of smectic A liquid crystal deformation is derivedab initiovia a variational approach. The well-determined nature of the resulting nonlinear model equations reveals that the deformed states of the liquid crystal lamellae can only adopt privileged geometries. These are shown to generalize classical and novel ‘integrable’ geometries associated with Willmore, linear Weingarten and ‘membrane’ O surfaces. The main result establishes that, remarkably, the membrane model admits layered parallel Dupin cyclide structures of the kind originally observed by Friedel and Grandjean in their pioneering experiments of 1910 and subsequently elaborated upon by Friedel in 1922 and later by Bragg.

Energy Considerations for Parabolic Cyclides in SmA Liquid Crystals

ABSTRACT For Dupin cyclides in the SmA liquid crystal phase, Kleman and Lavrentovich [1] found that increasing the saddle-splay constant K¯ caused the minimum of the total elastic energy to occur at a decreased value of the eccentricity e, close to unity. We extend this work to the parabolic cyclides and present an analytical expression for the total elastic energy that is finite over a suitable range of values for the latus rectum, conventionally denoted in this context by −4ℓ (>0). The length of the latus rectum characterises the parabolic focal conic structure analogously to the eccentricity of the Dupin cyclide focal conics. We demonstrate that the total energy is minimised at a particular value of ℓ. It is further observed that the usual saddle-splay elastic term acts independently of ℓ and that varying the value of K¯ does not affect the actual value of ℓ at which the minimum of the elastic energy occurs.

Non-regular and non-Stäckel R-separation for 3-dimensional Helmholtz equation and cyclidic solitons of wave equation

We discuss a class of orthogonal coordinates in E 3 which are non-trivially R-separable for 3-dimensional Euclidean Helmholtz equation. The separation is both non-regular (in the sense of Kalnins and Miller) and non-Stäckel. One family of parametric surfaces consists of parallel Dupin cyclides, the other two consist of circular cones. These coordinates are used to simplify derivation of Friedlander's formulae for modulated progressive wave solutions to the wave equation and to correct some errors of his paper.

Solitons of Wave Equation

Modulated progressive wave solutions (solitons) to (3+1)–dimensional wave equation are discussed within a general geometrical framework. The role of geodesic coordinates defined by hypersurfaces of Riemannian spaces is pointed out in this context. In particular in E 3 orthogonal geodesic coordinates defined by Dupin cyclides are used to simplify derivation of the most nontrivial results of Friedlander on solitons of (3+1)–dimensional wave equations and to correct some of them. The essence of this novel approach is use of the technique of separation of variables in the Kalnins–Miller formulation.

Dupin Cyclide Blends Between Quadric Surfaces for Shape Modeling

Abstract We introduce a novel method to define Dupin cyclide blends between quadric primitives. Dupin cyclides are non‐spherical algebraic surfaces discovered by French mathematician Pierre‐Charles Dupin at the beginning of the 19th century. As a Dupin cyclide can be fully characterized by its principal circles, we have focussed our study on how to determine principal circles tangent to both quadrics being blended. This ensures that the Dupin cyclide we are constructing constitutes a G 1 blend. We use the Rational Quadratic Bézier Curve (RQBC) representation of circular arcs to model the principal circles, so the construction of each circle is reduced to the determination of the three control points of the RQBC representing the circle. In this work, we regard the blending of two quadric primitives A and B as two complementary blending operations: primitive A‐cylinder and cylinder‐primitive B; two Dupin cyclides and a cylinder are then defined for each blending operation. In general the cylinder is not useful and may be reduced to a simple circle. A complete shape design example is presented to illustrate the modeling of Eurographics'04 Hugo using a limited number of quadrics combined using Dupin cyclide blends.Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling

Exploiting curvatures to compute the medial axis for domains with smooth boundary

In the paper, strong proofs of some basics facts about the medial axis transform of a planar region Ω with smooth boundary curve(s) are given. In particular the decomposition of Choi et al. (1997) is derived from the set of regular disks. Two special algorithms to compute the medial axis are presented. Due to the decomposition, they can be applied to each part of Ω separately. Emphasis is given to the local analysis of the boundary's curvatures. This leads to an interesting connection to the theory of Dupin cyclides. With this mean, a predictor/corrector algorithm (as in the numerical analysis of ODE's) is developed. The examples show that the predictor yields already an excellent approximation; so only a few refining steps of the corrector are necessary.

On the equilibrium of shell membranes under normal loading. Hidden integrability

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Rogers C. and Schief W. K. 2003On the equilibrium of shell membranes under normal loading. Hidden integrabilityProc. R. Soc. Lond. A.4592449–2462http://doi.org/10.1098/rspa.2003.1135SectionRestricted accessOn the equilibrium of shell membranes under normal loading. Hidden integrability C. Rogers C. Rogers School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Google Scholar Find this author on PubMed Search for more papers by this author and W. K. Schief W. K. Schief School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Google Scholar Find this author on PubMed Search for more papers by this author C. Rogers C. Rogers School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Google Scholar Find this author on PubMed Search for more papers by this author and W. K. Schief W. K. Schief School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Google Scholar Find this author on PubMed Search for more papers by this author Published:08 October 2003https://doi.org/10.1098/rspa.2003.1135AbstractIt is established that a well–known system of classical shell theory descriptive of membranes in equilibrium is, in fact, integrable. The membranes are shown to have geometries within the integrable class of so–called O surfaces. The membrane O surfaces include inter alia minimal, constant mean curvature, constant Gaussian curvature and, more generally, linear Weingarten surfaces, as well as canal surfaces and Dupin cyclides. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Tellier X, Douthe C, Baverel O and Hauswirth L (2023) Designing funicular grids with planar quads using isotropic Linear-Weingarten surfaces, International Journal of Solids and Structures, 10.1016/j.ijsolstr.2022.112028, 264, (112028), Online publication date: 1-Mar-2023. Hayashi K, Jikumaru Y, Ohsaki M, Kagaya T and Yokosuka Y (2021) Discrete Gaussian Curvature Flow for Piecewise Constant Gaussian Curvature Surface, Computer-Aided Design, 10.1016/j.cad.2021.102992, 134, (102992), Online publication date: 1-May-2021. Tellier X, Douthe C, Hauswirth L and Baverel O (2020) Linear‐Weingarten membranes with funicular boundaries, Structural Concrete, 10.1002/suco.202000030, 21:6, (2293-2306), Online publication date: 1-Dec-2020. Mesnil R, Douthe C, Baverel O and Léger B (2018) Morphogenesis of surfaces with planar lines of curvature and application to architectural design, Automation in Construction, 10.1016/j.autcon.2018.08.007, 95, (129-141), Online publication date: 1-Nov-2018. Douthe C, Mesnil R, Orts H and Baverel O (2017) Isoradial meshes: Covering elastic gridshells with planar facets, Automation in Construction, 10.1016/j.autcon.2017.08.015, 83, (222-236), Online publication date: 1-Nov-2017. Bobenko A, Hertrich-Jeromin U and Lukyanenko I (2014) Discrete constant mean curvature nets in space forms: Steiner’s formula and Christoffel duality, Discrete & Computational Geometry, 10.1007/s00454-014-9622-5, 52:4, (612-629), Online publication date: 1-Dec-2014. Schief W, Szereszewski A and Rogers C (2009) On shell membranes of Enneper type: generalized Dupin cyclides, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8113/42/40/404016, 42:40, (404016), Online publication date: 9-Oct-2009. Rogers C and Szereszewski A (2009) A Bäcklund transformation for L-isothermic surfaces, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8113/42/40/404015, 42:40, (404015), Online publication date: 9-Oct-2009. Guha P and Shipman P (2009) Polarized Hessian covariant: Contribution to pattern formation in the Föppl–von Kármán shell equations, Chaos, Solitons & Fractals, 10.1016/j.chaos.2008.10.025, 41:5, (2828-2837), Online publication date: 1-Sep-2009. Szereszewski A (2009) L-isothermic and L-minimal surfaces, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8113/42/11/115203, 42:11, (115203), Online publication date: 20-Mar-2009. Schief W, Szereszewski A and Rogers C (2007) The Lamé equation in shell membrane theory, Journal of Mathematical Physics, 10.1063/1.2747721, 48:7, (073510), Online publication date: 1-Jul-2007. Schief W (2005) Integrable discrete differential geometry of ‘plated’ membranes in equilibrium, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461:2062, (3213-3229), Online publication date: 8-Oct-2005.Schief W, Kléman M and Rogers C (2005) On a nonlinear elastic shell system in liquid crystal theory: generalized Willmore surfaces and Dupin cyclides, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461:2061, (2817-2837), Online publication date: 8-Sep-2005. This Issue08 October 2003Volume 459Issue 2038 Article InformationDOI:https://doi.org/10.1098/rspa.2003.1135Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/10/2003Published in print08/10/2003 License: Citations and impact KeywordsDupin cyclidesshell membraneintegrabilityO surfacesBäcklund transformation

Isothermic Surfaces Generated via Bäcklund and Moutard Transformations: Boomeron and Zoomeron Connections

Backlund and Moutard transformations are applied to seed classical isothermic surfaces to generate 'solitonic' isothermic surfaces such as Dupin cyclides. Corresponding localized solutions of the zoomeron equation are shown to be related to the dromion solutions of the Davey-Stewartson III equation via a simple Lie point symmetry.

Three views of Dupin cyclides and blending of cones

Three views of Dupin cyclides and blending of cones

Do blending and offsetting commute for Dupin cyclides?

A common method for constructing blending Dupin cyclides for two cones having a common inscribed sphere of radius r>0 involves three steps: (1) computing the (− r) -offsets of the cones so that they share a common vertex, (2) constructing a blending cyclide for the offset cones, and (3) computing the r -offset of the cyclide. Unfortunately, this process does not always work properly. Worse, for some half-cones cases, none of the blending cyclides can be constructed this way. This paper studies this problem and presents two major contributions. First, it is shown that the offset construction is correct for the case of ε≠− r , where ε is the signed offset value; otherwise, a procedure must be followed for properly selecting a pair of principal circles of the blending cyclide. Second, based on Shene's construction in “Blending two cones with Dupin cyclides”, CAGD, 15 (1998) 643–673, a new algorithm is available for constructing all possible blending cyclides for two half-cones. This paper also examines Allen and Dutta's theory of pure blends, which uses the offset construction. To help overcome the difficulties of Allen and Dutta's method, this paper suggests a new algorithm for constructing all possible pure blends. Thus, Shene's diagonal construction is better and more reliable than the offset construction.

Double canal hypersurfaces in the Euclidean space En

A multidimensional analogue of the cyclides of Dupin is considered: double canal hypersurfaces. A generalization is proved of the theorem about the set of centres C1 and C2 of the generating hyperspheres.

Studying cyclides with Laguerre geometry

We develop a theory of Dupin cyclides (including natural quadrics) using associated isotropic hypersurfaces and their control points in the 4-dimensional cyclographic model of Laguerre geometry. In the framework of this approach the classification of cyclides and their parametric and implicit equations are easily re-derived. These results enable us to detect in a constructive way all cases of G 1-continuous compositions of cyclide patches along rational quadratic, cubic and quartic curves.

Blending two cones with Dupin cyclides

This paper presents a complete theory of blending cones with Dupin cyclides and consists of four major contributions. First, a necessary and sufficient condition for two cones to have a blending Dupin cyclide is established. Second, based on the intersection structure of the cones, finer characterization results are obtained. Third, a new construction algorithm that establishes a correspondence between points on one or two coplanar lines and all constructed blending Dupin cyclides makes the construction easy and well-organized. Fourth, the completeness of the construction algorithm is proved. Consequently, all blending Dupin cyclides are organized into one to four one-parameter families, each of which is “parameterized” by points on a special line. It is also shown that each family contains an infinite number of ring cyclides, ensuring the existence of singularity free blending surfaces.

General cyclides

In this paper we consider the problem of joining circles with prescribed tangencies, i.e., along given cones or spheres. The joining surfaces are envelopes of quadratic 1-parameter families of spheres which generalize the family of Dupin cyclides, but still their algebraic degree is four. They are represented with conics in 4-space. Given a conic in Bézier form the implicit equation of the corresponding surface is exhibited.