Dipartimento di Matematica ''F. Casorati''

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Multivariate (Geometric) Constraints Solving using Subdivision based Solvers

Prof. Gershon Elber, Technion - Israel Institute of Technology

Sala conferenze IMATI-CNR, Pavia - Thursday, March 15, 2018 h.15:00

Abstract. We explore a subdivision based paradigm to solve a set of (piecewise) rational constraints represented by (piecewise) rational multivariate spline functions. While the basic approach is robust, it can also be slow. In this talk, we will examine and survey several recently presented schemes to alleviate these computational costs. The addition of inequality filters, single solution isolation techniques, orthogonalization and domain reduction, and the minimization of the memory explosion that results from the exponential dependency on the number of variables will be discussed.

With this machinery, we will demonstrate that this type of solvers can be successfully used to solve a large variety of (geometric) problems above the (piecewise) rationals domain. This set of problems includes point-curve and curve-curve bi-tangents, convex hulls and kernels of planar curves and 3-space surfaces, ray-traps (bouncing billiard balls) between planar curves, the 10th Apollonius problem (a circle tangent to given three circles) and its generalization, bounding circles and spheres, bisectors and Voronoi regions, mold design,
visibility and accessibility, sweeps and envelopes, toolpath design in robotics and self-intersection computation and trimming in offset approximations of curves and surfaces.

Some of this work has been conducted in collaboration with many others including Myung-Soo Kim, SNU, Korea, Elaine Cohen, University of Utah, USA, Tom Grandine, Boeing, USA, Ralph Martin, Cardiff, Michael Barton and Yong Joon Kim, Technion.

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Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602