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Interactive Visualization of Rotational Symmetry Fields on Surfaces
 IEEE Trans. Visualization and Computer Graphics
, 2011
"... Abstract—Rotational symmetries (RoSys) have found uses in several computer graphics applications, such as global surface parameterization, geometry remeshing, texture and geometry synthesis, and nonphotorealistic visualization of surfaces. The visualization of Nway rotational symmetry (NRoSy) fiel ..."
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Abstract—Rotational symmetries (RoSys) have found uses in several computer graphics applications, such as global surface parameterization, geometry remeshing, texture and geometry synthesis, and nonphotorealistic visualization of surfaces. The visualization of Nway rotational symmetry (NRoSy) fields is a challenging problem due to the ambiguities in the N directions represented by an Nway symmetry. We provide an algorithm that allows faithful and interactive representation of NRoSy fields in the plane and on surfaces, by adapting the wellknown line integral convolution (LIC) technique from vector and secondorder tensor fields. Our algorithm captures N directions associated with each point in a given field by decomposing the field into multiple different vector fields, generating LIC images of these fields, and then blending the results. To address the loss of contrast caused by the blending of images, we observe that the pixel values in LIC images closely approximate normally distributed random variables. This allows us to use concepts from probability theory to correct the loss of contrast without the need to perform any image analysis at each frame. Index Terms—Rotational symmetry, RoSy, visualization, tensor field visualization, image blending, contrast adjustment. Ç 1
Circular arc structures
"... The most important guiding principle in computational methods for freeform architecture is the balance between cost efficiency on the one hand, and adherence to the design intent on the other. Key issues are the simplicity of supporting and connecting elements as well as repetition of costly parts. ..."
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Cited by 8 (3 self)
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The most important guiding principle in computational methods for freeform architecture is the balance between cost efficiency on the one hand, and adherence to the design intent on the other. Key issues are the simplicity of supporting and connecting elements as well as repetition of costly parts. This paper proposes socalled circular arc structures as a means to faithfully realize freeform designs without giving up smooth appearance. In contrast to nonsmooth meshes with straight edges where geometric complexity is concentrated in the nodes, we stay with smooth surfaces and rather distribute complexity in a uniform way by allowing edges in the shape of circular arcs. We are able to achieve the simplest possible shape of nodes without interfering with known panel optimization algorithms. We study remarkable special cases of circular arc structures which possess simple supporting elements or repetitive edges, we present the first global approximation method for principal patches, and we show an extension to volumetric structures for truly threedimensional designs.
Hexagonal global parameterization of arbitrary surfaces
 In: ACM SIGGRAPH ASIA 2010 Sketches
, 2010
"... Abstract—We introduce hexagonal global parameterization, a new type of surface parameterization in which parameter lines respect sixfold rotational symmetries (6RoSy). Such parameterizations enable the tiling of surfaces with nearly regular hexagonal or triangular patterns, and can be used for tri ..."
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Abstract—We introduce hexagonal global parameterization, a new type of surface parameterization in which parameter lines respect sixfold rotational symmetries (6RoSy). Such parameterizations enable the tiling of surfaces with nearly regular hexagonal or triangular patterns, and can be used for triangular remeshing. Our framework to construct a hexagonal parameterization, referred to as HEXCOVER, extends the QUADCOVER algorithm and formulates necessary conditions for hexagonal parameterization. We also provide an algorithm to automatically generate a 6RoSy field that respects directional and singularity features in the surface. We demonstrate the usefulness of our geometryaware global parameterization with applications such as surface tiling with nearly regular textures and geometry patterns, as well as triangular and hexagonal remeshing. Index Terms—Surface parameterization, rotational symmetry, hexagonal global parameterization, triangular remeshing, pattern synthesis on surfaces, texture synthesis, geometry synthesis, regular patterns. 1
M.: Interactive design exploration for constrained meshes
"... In architectural design, surface shapes are commonly subject to geometric constraints imposed by material, fabrication or assembly. Rationalization algorithms can convert a freeform design into a form feasible for production, but often require design modifications that might not comply with the de ..."
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Cited by 4 (2 self)
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In architectural design, surface shapes are commonly subject to geometric constraints imposed by material, fabrication or assembly. Rationalization algorithms can convert a freeform design into a form feasible for production, but often require design modifications that might not comply with the design intent. In addition, they only offer limited support for exploring alternative feasible shapes, due to the high complexity of the optimization algorithm. We address these shortcomings and present a computational framework for interactive shape exploration of discrete geometric structures in the context of freeform architectural design. Our method is formulated as a mesh optimization subject to shape constraints. Our formulation can enforce soft constraints and hard constraints at the same time, and handles equality constraints and inequality constraints in a unified way. We propose a novel numerical solver that splits the optimization into a sequence of simple subproblems that can be solved efficiently and accurately. Based on this algorithm, we develop a system that allows the user to explore designs satisfying geometric constraints. Our system offers full control over the exploration process, by providing direct access to the specification of the design space. At the same time, the complexity of the underlying optimization is hidden from the user, who communicates with the system through intuitive interfaces.
Weighted triangulations for geometry processing
"... In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary t ..."
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Cited by 3 (0 self)
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In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primaldual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closedform expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of wellcentered meshes, selfsupporting surfaces, and sphere packing.
Darboux Cyclides and Webs from Circles
, 2011
"... Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order ≤ 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 ..."
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Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order ≤ 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 Möbius geometry, we provide computational tools for the identificiation 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 socalled hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.
Case Studies in CostOptimized Paneling of Architectural Freeform Surfaces
"... Paneling an architectural freeform surface refers to an approximation of the design surface by a set of panels that can be manufactured using a selected technology at a reasonable cost, while respecting the design intent and achieving the desired aesthetic quality of panel layout and surface smoothn ..."
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Cited by 2 (1 self)
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Paneling an architectural freeform surface refers to an approximation of the design surface by a set of panels that can be manufactured using a selected technology at a reasonable cost, while respecting the design intent and achieving the desired aesthetic quality of panel layout and surface smoothness. Eigensatz and coworkers [Eigensatz et al. 2010] have recently introduced a computational solution to the paneling problem that allows handling largescale freeform surfaces involving complex arrangements of thousands of panels. We extend this paneling algorithm to facilitate effective design exploration, in particular for local control of tolerance margins and the handling of sharp crease lines. We focus on the practical aspects relevant for the realization of largescale freeform designs and evaluate the performance of the paneling algorithm with a number of case studies. Eigensatz et al. reference surface panelized surface plane cylinder paraboloid torus cubic mold types Figure 1: Given a reference surface (top row), our paneling algorithm produces a rationalization of the the input. The solution (middle row) employs a small set of molds that can be reused for costeffective panel production (bottom row), while preserving surface smoothness and respecting original design intent. The shown paneling solution (using metal) is 40% cheaper than the production alternative of using custom molds for each individual panels. The solution shown in Figure 10 is 60 % cheaper compared to using custom molds for each individual panels.
Geometric Computing for Freeform Architecture
"... Geometric computing has recently found a new field of applications, namely the various geometric problems which lie at the heart of rationalization and constructionaware design processes of freeform architecture. We report on our work in this area, dealing with meshes with planar faces and meshes w ..."
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Geometric computing has recently found a new field of applications, namely the various geometric problems which lie at the heart of rationalization and constructionaware design processes of freeform architecture. We report on our work in this area, dealing with meshes with planar faces and meshes which allow multilayer constructions (which is related to discrete surfaces and their curvatures), triangles meshes with circlepacking properties (which is related to conformal uniformization), and with the paneling problem. We emphasize the combination of numerical optimization and geometric knowledge. 1
Statics aware grid shells
 Computer Graphics Forum
, 2015
"... b c Figure 1: We perform a FEM static analysis of the input surface to obtain a stress field, from which we derive a double orthogonal line field (a), an anisotropy scalar field (b) and a density scalar field (c). Then we build a polygonal tessellation whose elements are sized and aligned according ..."
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b c Figure 1: We perform a FEM static analysis of the input surface to obtain a stress field, from which we derive a double orthogonal line field (a), an anisotropy scalar field (b) and a density scalar field (c). Then we build a polygonal tessellation whose elements are sized and aligned according to the stress tensor field; this tessellation is optimized for symmetry and regularity of faces. A small scale model is fabricated to validate the result with load tests. We introduce a framework for the generation of polygonal gridshell architectural structures, whose topology is designed in order to excel in static performances. We start from the analysis of stress on the input surface and we use the resulting tensor field to induce an anisotropic nonEuclidean metric over it. This metric is derived by studying the relation between the stress tensor over a continuous shell and the optimal shape of polygons in a corresponding gridshell. Polygonal meshes with uniform density and isotropic cells under this metric exhibit variable density and anisotropy in Euclidean space, thus achieving a better distribution of the strain energy over their elements. Meshes are further optimized taking into account symmetry and regularity of cells to improve aesthetics. We experiment with quad meshes and hexdominant meshes, demonstrating that our gridshells achieve better static performances than stateoftheart gridshells.
1Interactive Visualization of Rotational Symmetry Fields on Surfaces
"... Abstract—Rotational symmetries have found uses in several computer graphics applications, such as global surface parameterization, geometry remeshing, texture and geometry synthesis, and nonphotorealistic visualization of surfaces. The visualization of Nway rotational symmetry (NRoSy) fields is a ..."
Abstract
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Abstract—Rotational symmetries have found uses in several computer graphics applications, such as global surface parameterization, geometry remeshing, texture and geometry synthesis, and nonphotorealistic visualization of surfaces. The visualization of Nway rotational symmetry (NRoSy) fields is a challenging problem due to the ambiguities in the N directions represented by an Nway symmetry. We provide an algorithm that allows faithful and interactive representation of NRoSy fields in the plane and on surfaces, by adapting the wellknown Line Integral Convolution (LIC) technique from vector and secondorder tensor fields. Our algorithm captures the N directions associated with each point in a given field by decomposing the field into multiple different vector fields, generating LIC images of these fields, and then blending the results. To address the loss of contrast caused by the blending of images, we observe that the pixel values in LIC images closely approximate normally distributed random variables. This allows us to use concepts from probability theory to correct the loss of contrast without the need to perform any image analysis at each frame.