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55
On the convergence of metric and geometric properties of polyhedral surfaces
 GEOMETRIAE DEDICATA
, 2005
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A discrete Laplace–Beltrami operator for simplicial surfaces
 Discrete Comput. Geom
"... Abstract We define a discrete LaplaceBeltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finiteelements Laplacian (the so called "cotan formula ..."
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Cited by 59 (3 self)
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Abstract We define a discrete LaplaceBeltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finiteelements Laplacian (the so called "cotan formula") except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete LaplaceBeltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Keywords: Laplace operator, Delaunay triangulation, Dirichlet energy, simplicial surfaces, discrete differential geometry 1 Dirichlet energy of piecewise linear functions Let S be a simplicial surface in 3dimensional Euclidean space, i.e. a geometric simplicial complex in Ê 3 whose carrier S is a 2dimensional submanifold,
Geometry of Multilayer Freeform Structures for Architecture
, 2007
"... The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes. We approach them via the concept of parallel meshes, and present methods of computation and optimization. We discuss planar faces, beams of controlled height, node g ..."
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Cited by 49 (17 self)
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The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes. We approach them via the concept of parallel meshes, and present methods of computation and optimization. We discuss planar faces, beams of controlled height, node geometry, and multilayer constructions. Beams of constant height are achieved with the new type of edge offset meshes. Mesh parallelism is also the main ingredient in a novel discrete theory of curvatures. These methods are applied to the construction of quadrilateral, pentagonal and hexagonal meshes, discrete minimal surfaces, discrete constant mean curvature surfaces, and their geometric transforms. We show how to design geometrically optimal shapes, and how to find a meaningful meshing and beam layout for existing shapes.
The focal geometry of circular and conical meshes
, 2006
"... 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 subdivi ..."
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Cited by 29 (10 self)
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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 subdivisionlike 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 twoparameter 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.
Circle packing: a mathematical tale
 Notices Amer. Math. Soc
, 2003
"... 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 creat ..."
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Cited by 27 (2 self)
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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.
Packing circles and spheres on surfaces
 TO APPEAR IN THE ACM SIGGRAPH CONFERENCE PROCEEDINGS
, 2009
"... Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces' incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate ci ..."
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Cited by 11 (6 self)
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Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces' incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings, hexagonal meshes which carry a torsionfree support structure, hybrid trihex meshes, and others. We show how triangle meshes can be optimized so as to have the incircle packing property. We explain their relation to conformal geometry and implications on solvability of optimization. The examples we give confirm that this kind of meshes is a rich source of geometric structures relevant to architectural geometry.
Hexagonal Meshes with Planar Faces
"... Figure 1: (Left): A conceptual architectural structure as a PHex mesh, computed using the progressive conjugation method. The interior view (top right) shows that the shapes of the PHex faces transit smoothly across the parabolic curve. (Bottom right): another PHex mesh of free form shape. Freef ..."
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Cited by 8 (1 self)
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Figure 1: (Left): A conceptual architectural structure as a PHex mesh, computed using the progressive conjugation method. The interior view (top right) shows that the shapes of the PHex faces transit smoothly across the parabolic curve. (Bottom right): another PHex mesh of free form shape. Freeform meshes with planar hexagonal faces, to be called PHex 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 PHex meshes and present an algorithm for computing a freeform PHex mesh of a specified shape. Our algorithm first computes a regular triangulation of a given surface and then turns it into a PHex 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 PHex meshes. This report is based on the results presented at Workshop ”Polyhedral Surfaces and Industrial Applications ” held on September 15