Abstract not found

Abstract not found

These notes are about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using conserved quantities, which can be converted into a discrete theory in a natural way. About referencing: We do not attempt to give a complete reference list, and omit what is already referenced in [59]. We list only articles referenced in the body of the text, or that were written after [59] was published, or were otherwise not included in the reference list in [59], or that were referenced in [59] but need to be updated. About using quaternions: In following with the historical development of the field, we use a model that involves quaternions. However, the use of a more standard model has some advantages, as it can be applied in more general dimensions and settings (see Chapter 10 here, for example), and sometimes gives less cluttered computations. It would be a good exercise to convert this text into one involving a more standard quaternion-free model, but we do not do that here (see [27]), and instead only make occasional comments about this. Acknowledgements: Primary thanks must go to Udo Hertrich-Jeromin, who carefully and patiently taught the author more than half of the material in this text. The author also owes thanks to many others for numerous mathematical tips: Fran

The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creation, manipulation, and interpretation. Lest we get off on the wrong foot, I should caution that this is NOT twodimensional “sphere ” packing: rather than being fixed in size, our circles must adjust their radii in tightly choreographed ways if they hope to fit together in a specified pattern. In posing this as a mathematical tale, I am asking the reader for some latitude. From a tale you expect truth without all the details; you know that the storyteller will be playing with the plot and timing; you let pictures carry part of the story. We all hope for deep insights, but perhaps sometimes a simple story with a few new twists is enough—may you enjoy this tale in that spirit. Readers who wish to dig into the details can consult the “Reader’s Guide ” at the end. Once Upon a Time … From wagon wheel to mythical symbol, predating history, perfect form to ancient geometers, companion to π, the circle is perhaps the most celebrated object in mathematics.

Circular meshes are quadrilateral meshes all of whose faces possess a circumcircle, whereas conical meshes are planar quadrilateral meshes where the faces which meet in a vertex are tangent to a right circular cone. Both are amenable to geometric modeling – recently surface approximation and subdivision-like refinement processes have been studied. In this paper we extend the original defining property of conical meshes, namely the existence of face/face offset meshes at constant distance, to circular meshes. We study the close relation between circular and conical meshes, their vertex/vertex and face/face offsets, as well as their discrete normals and focal meshes. In particular we show how to construct a two-parameter family of circular (resp., conical) meshes from a given conical (resp., circular) mesh. We further discuss meshes which have both properties and their relation to discrete surfaces of negative Gaussian curvature. The offset properties of special quadrilateral meshes and the threedimensional support structures derived from them are highly relevant for computational architectural design of freeform structures. Another aspect important for design is that both circular and conical meshes provide a discretization of the principal curvature lines of a smooth surface, so the mesh polylines represent principal features of the surface described by the mesh.

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy � (H 2 − K)/KdA. They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of � (H 2 − K)dA, which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.

Abstract. The cotangent formula constitutes an intrinsic discretization of the Laplace– Beltrami operator on polyhedral surfaces in a finite element sense. This note gives an overview of approximation and convergence properties of discrete Laplacians and mean curvature vectors for polyhedral surfaces located in the vicinity of a smooth surface in euclidean 3–space. In particular, we show that mean curvature vectors converge in the sense of distributions, but fail to converge in L 2. 1.

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Abstract. We define discrete flat surfaces in hyperbolic 3-space H 3 from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in H 3, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. Along the way, we also examine discrete linear Weingarten surfaces of Bryant type in H 3, and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stoke’s phenomenon. Contents

Figure 1: (Left): A conceptual architectural structure as a P-Hex mesh, computed using the progressive conjugation method. The interior view (top right) shows that the shapes of the P-Hex faces transit smoothly across the parabolic curve. (Bottom right): another P-Hex mesh of free form shape. Free-form meshes with planar hexagonal faces, to be called P-Hex meshes, provide a useful surface representation in discrete differential geometry and are demanded in architectural design for representing surfaces built with planar glass/metal panels. We study the geometry of P-Hex meshes and present an algorithm for computing a free-form P-Hex mesh of a specified shape. Our algorithm first computes a regular triangulation of a given surface and then turns it into a P-Hex mesh approximating the surface. A novel local duality transformation, called Dupin duality, is introduced for studying relationship between triangular meshes and for controlling the face shapes of P-Hex meshes. This report is based on the results presented at Workshop ”Polyhedral Surfaces and Industrial Applications ” held on September 15-