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40
Discrete Laplace operators: No free lunch
, 2007
"... Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set ..."
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Cited by 48 (1 self)
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Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends wellknown and widelyused operators.
Infinitesimally rigid polyhedra. I. Statics of frameworks
 Transactionsofthe American Mathematical Society
, 1984
"... Abstract. From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation ..."
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Cited by 26 (4 self)
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Abstract. From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra. The static techniques introduced provide a new simplified proof of Alexandrov's theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in 3space, detailed analogues of Alexandrov's theorem are presented for convex 4polytopes built as bar and joint frameworks in 4space. 1. Introduction. Over
PseudoTriangulations  a Survey
 CONTEMPORARY MATHEMATICS
"... A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory an ..."
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Cited by 16 (4 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straightline drawings from abstract versions called combinatorial pseudotriangulations, algorithms and applications of pseudotriangulations.
Overcoming Superstrictness in Line Drawing Interpretation
 IEEE Trans. on ?attern Analysis and Machine Intelli#ence
, 2002
"... AbstractÐThis paper presents a new algorithm for correcting incorrect line drawingsÐincorrect projections of a polyhedral scene. Such incorrect drawings arise, e.g., when an image of a polyhedral world is taken, the edges and vertices are extracted, and a drawing is synthesized. Along the way, the t ..."
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Cited by 14 (3 self)
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AbstractÐThis paper presents a new algorithm for correcting incorrect line drawingsÐincorrect projections of a polyhedral scene. Such incorrect drawings arise, e.g., when an image of a polyhedral world is taken, the edges and vertices are extracted, and a drawing is synthesized. Along the way, the true positions of the vertices in the 2D projection are perturbed due to digitization errors and the preprocessing. As most available algorithms for interpreting line drawings are ªsuperstrict,º theyjudge these noisyinputs as incorrect and fail to reconstruct a threedimensional scene from them. The presented method overcomes this problem bymoving the positions of all vertices until a veryclose correct drawing is found. The closeness criterion is to minimize the sum of squared distances from each vertex in the input drawing to its corrected position. With this tool, anysuperstrict method for line drawing interpretation is now practical, as it can be applied to the corrected version of the input drawing. Index TermsÐLine drawing interpretation, superstrictness, scene understanding, correction algorithms. æ 1
Interactive Vault Design
 International Journal of Space Structures
"... In the last two decades the rise of computeraided design and modelling techniques enabled a new language of doubly curved surfaces in architecture. The relation between architectural design and ..."
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Cited by 10 (3 self)
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In the last two decades the rise of computeraided design and modelling techniques enabled a new language of doubly curved surfaces in architecture. The relation between architectural design and
A NOVEL FORM FINDING METHOD FOR FABRIC FORMWORK FOR CONCRETE SHELLS
"... This paper introduces a new form finding approach for the design of flexible formwork, consisting of prestressed structural membranes, for the construction of thin, anticlastic, concrete shells. The approach is based on the Force Density Method and an adaptation of the nonlinear extension of Thrust ..."
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Cited by 9 (2 self)
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This paper introduces a new form finding approach for the design of flexible formwork, consisting of prestressed structural membranes, for the construction of thin, anticlastic, concrete shells. The approach is based on the Force Density Method and an adaptation of the nonlinear extension of Thrust Network Analysis (TNA). First, the adaptation of the basic version of TNA to tensiononly structures is discussed, and fundamental assumptions are formulated. Key aspects herein are the use of reciprocal diagrams and the control of forces in indeterminate networks. Then, the strategy using an overall optimization process, for finding the closest possible tensiononly, equilibrium surface to a given target surface under the appropriate loading (i.e. weight of the concrete) is presented. Finally, the strategy is applied to a chosen target surface and loads, and the equilibrium solutions for different starting points of the optimization process are discussed.
Analysing Spatial Realizability of Line Drawings Through . . .
, 1998
"... This work proves that the realizability of a line drawing without occluding segments can be verfred by checking the concurrence of groups of three lines to a single point. These lines are either those supporting segments in the drawing or new ones added during the test itself. Although this result w ..."
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Cited by 6 (4 self)
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This work proves that the realizability of a line drawing without occluding segments can be verfred by checking the concurrence of groups of three lines to a single point. These lines are either those supporting segments in the drawing or new ones added during the test itself. Although this result was essentially
EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS
"... Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We surv ..."
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Cited by 4 (2 self)
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Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of its most important applications, including molecular conformation, localization of sensor networks and statics. Key words. Matrix completion, barandjoint framework, graph rigidity, inverse problem, protein conformation, sensor network.
Geometrybased Understanding of Structures
 Journal of the International Association for Shell and Spatial Structures
"... The development of structures with complex, curved geometry typically consists of a series of iterative steps in which formal and structural considerations are addressed separately. Since such a sequential process often results in solutions that fail to realise the aesthetic and structural intention ..."
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Cited by 4 (2 self)
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The development of structures with complex, curved geometry typically consists of a series of iterative steps in which formal and structural considerations are addressed separately. Since such a sequential process often results in solutions that fail to realise the aesthetic and structural intentions of the original design, there is clearly a need for an integrated approach providing bidirectional control over both form and forces at every stage of the design exploration. This paper therefore describes and advocates an approach based on elements of graphic statics, in which the design of structures is based on geometrical rather than analytical or numerical representations of the relation between form and forces. The presented approach adopts the key principles of graphic statics and extends them with the latest research on form finding, structural design and optimisation techniques.
Isostaticity and controlled force transmission in the cytoskeleton: a model awaiting experimental evidence
 Biophys. J
, 1970
"... ABSTRACT A new model is proposed for force transmission through the cytoskeleton (CSK). A general discussion is first presented on the physical principles that underlie the modeling of this phenomenon. Some fundamental problems of conventional models—continuous and discrete—are examined. It is argu ..."
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Cited by 4 (0 self)
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ABSTRACT A new model is proposed for force transmission through the cytoskeleton (CSK). A general discussion is first presented on the physical principles that underlie the modeling of this phenomenon. Some fundamental problems of conventional models—continuous and discrete—are examined. It is argued that mediation of focused forces is essential for good control over intracellular mechanical signals. The difficulties of conventional continuous models in describing such mediation are traced to a fundamental assumption rather than to their being continuous. Relevant advantages and disadvantages of continuous and discrete modeling are discussed. It is concluded that favoring discrete models is based on two misconceptions, which are clarified. The model proposed here is based on the idea that focused propagation of mechanical stimuli in frameworks over large distances (compared to the mesh size) can only occur when considerable regions of the CSK are isostatic. The concept of isostaticity is explained and a recently developed continuous isostaticity theory is briefly reviewed. The model enjoys several advantages: it leads to good control over force mediation; it explains nonuniform stresses and action at a distance; it is continuous, making it possible to model force propagation over long distances; and it enables prediction of individual force paths. To be isostatic, or nearly so, CSK networks must possess specific structural characteristics, and these are quantified. Finally, several experimental observations are interpreted using the new model and implications are discussed. It is also suggested that this approach may give insight into the dynamics of reorganization of the CSK. Many of the results are amenable to experimental measurements, providing a testing ground for the proposed picture, and generic experiments are suggested.