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Functionals that penalize bending or stretching of a surface play a key role in geometric and scientific computing, but to date have ignored a very basic requirement: in many situations, surfaces must not pass through themselves or each other. This paper develops a numerical framework for optimization of surface geometry while avoiding (self-)collision. The starting point is thetangent-point energy, which effectively pushes apart pairs of points that are close in space but distant along the surface. We develop a discretization of this energy for triangle meshes, and introduce a novel acceleration scheme based on a fractional Sobolev inner product. In contrast to similar schemes developed for curves, we avoid the complexity of building a multiresolution mesh hierarchy by decomposing our preconditioner into two ordinary Poisson equations, plus forward application of a fractional differential operator. We further accelerate this scheme via hierarchical approximation, and describe how to incorporate a variety of constraints (on area, volume,etc.). Finally, we explore how this machinery might be applied to problems in mathematical visualization, geometric modeling, and geometry processing.
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Sum-of-squares geometry processing
Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due to the expansive list ofgeometric primitives, e.g., splines, triangles, and hexahedra, joined with an ever-expanding variety ofobjectivesone might want to achieve with them. With the recent increase in attention towardhigher-order surfaces, we can expect a variety of challenges porting existing solutions that work on triangle meshes to work on these more complex geometry types. In this paper, we present a framework for solving many core geometry processing problems on higher-order surfaces. We achieve this goal through sum-of-squares optimization, which transforms nonlinear polynomial optimization problems into sequences of convex problems whose complexity is captured by a singledegreeparameter. This allows us to solve a suite of problems on higher-order surfaces, such as continuous collision detection and closest point queries on curved patches, with only minor changes between formulations and geometries.
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- Award ID(s):
- 1838071
- PAR ID:
- 10601942
- Publisher / Repository:
- Association for Computing Machinery (ACM)
- Date Published:
- Journal Name:
- ACM Transactions on Graphics
- Volume:
- 40
- Issue:
- 6
- ISSN:
- 0730-0301
- Format(s):
- Medium: X Size: p. 1-13
- Size(s):
- p. 1-13
- Sponsoring Org:
- National Science Foundation
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