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Circular arc structures

Authors:

Johannes WallnerAuthors Info & Claims

SIGGRAPH '11: ACM SIGGRAPH 2011 papers

Article No.: 101, Pages 1 - 12

https://doi.org/10.1145/1964921.1964996

Published: 25 July 2011 Publication History

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Abstract

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 so-called circular arc structures as a means to faithfully realize freeform designs without giving up smooth appearance. In contrast to non-smooth 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 three-dimensional designs.

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Cited By

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Zahri M(2019)Mathematical Modeling of a Class of Symmetrical Islamic DesignSymmetry10.3390/sym1104051711:4(517)Online publication date: 10-Apr-2019
https://doi.org/10.3390/sym11040517
Jiang CWang JWallner JPottmann HLai YMartin R(2014)Freeform honeycomb structuresProceedings of the Symposium on Geometry Processing10.1111/cgf.12444(185-194)Online publication date: 9-Jul-2014
https://dl.acm.org/doi/10.1111/cgf.12444
Bartoň MPottmann HWallner J(2014)Detection and reconstruction of freeform sweepsComputer Graphics Forum10.1111/cgf.1228733:2(23-32)Online publication date: 1-May-2014
https://dl.acm.org/doi/10.1111/cgf.12287
Show More Cited By

Index Terms

Circular arc structures
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models
2. Theory of computation
  1. Randomness, geometry and discrete structures

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Reviews

Reviewer: Joseph J. O'Rourke

A circular arc structure (CAS) is a mesh of a surface with edges realized by circular arcs, such that all edges incident to a vertex are tangent to a common plane, and form a repeatable pattern of angles about interior vertices so that all vertices are congruent. The goal is to approximate a free-form architectural design with repetitions of the costly parts, primarily the node and arc structures. The CAS is computed by a nonlinear optimization of a "functional" that penalizes deviations from the multiple goals. A good initialization is crucial; a conformal mapping from a regular network (say, hexagonal) with the prescribed vertex angles provides it. We can avoid the torsion at nodes by following principal lines of curvature. We can augment the functional so that circular-arc edges all have the same radius; this is another cost savings. A M?bius transformation preserves smoothness, angles, and circular arc-ness, and so provides geometric guidance for CASs. In particular, applying M?bius transformations to tori results in the so-called Dupin cyclides, and leads to cyclidic CASs with attractive properties. For example, the offset of a cyclidic patch is again cyclidic, which enables construction of structural elements with parallel glass panels. A cyclidic CAS model may also be naturally converted to a developable strip model, which has manufacturing implications: we can form such strips by bending flat strips. This paper is rich with ideas, emerging from the fruitful interaction of classical differential geometry, modern discrete geometry, numerical optimization, and architectural pragmatics. The 25 color figures are nothing short of stunning. Online Computing Reviews Service

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Published In

SIGGRAPH '11: ACM SIGGRAPH 2011 papers

August 2011

869 pages

ISBN:9781450309431

DOI:10.1145/1964921

Editor:
Hugues Hoppe

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 25 July 2011

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Austrian Science Fund

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SIGGRAPH '11

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SIGGRAPH

SIGGRAPH '11: Special Interest Group on Computer Graphics and Interactive Techniques Conference

August 7 - 11, 2011

British Columbia, Vancouver, Canada

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SIGGRAPH '11 Paper Acceptance Rate 82 of 432 submissions, 19%;

Overall Acceptance Rate 1,822 of 8,601 submissions, 21%

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Cited By

View all

Zahri M(2019)Mathematical Modeling of a Class of Symmetrical Islamic DesignSymmetry10.3390/sym1104051711:4(517)Online publication date: 10-Apr-2019
https://doi.org/10.3390/sym11040517
Jiang CWang JWallner JPottmann HLai YMartin R(2014)Freeform honeycomb structuresProceedings of the Symposium on Geometry Processing10.1111/cgf.12444(185-194)Online publication date: 9-Jul-2014
https://dl.acm.org/doi/10.1111/cgf.12444
Bartoň MPottmann HWallner J(2014)Detection and reconstruction of freeform sweepsComputer Graphics Forum10.1111/cgf.1228733:2(23-32)Online publication date: 1-May-2014
https://dl.acm.org/doi/10.1111/cgf.12287
Rörig TSechelmann SKycia AFleischmann M(2014)Surface Panelization Using Periodic Conformal MapsAdvances in Architectural Geometry 201410.1007/978-3-319-11418-7_13(199-214)Online publication date: 3-Dec-2014
https://doi.org/10.1007/978-3-319-11418-7_13

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Freeform surfaces from single curved panels