[Archimedes Talks Series] (Geometric) Constraints Solving Using Multivariate Spline Functions
Dates
Venue
Artemidos 1 - Amphitheater
Title: (Geometric) Constraints Solving Using Multivariate Spline Functions
Presenter: Professor Gershon Elber, CS, Technion*
Abstract:
We present a subdivision based paradigm to solve a set of (piecewise) rational constraints represented by multivariate (piecewise) rational spline functions. The constraints can be formulated as either inequalities or equalities (semi-algebraic sets). The presented solver is not only capable of finding all solutions, globally, but is also capable of handling solution spaces of dimensions larger than zero. A special care is taken to guarantee all roots are isolated via a single solution geometric test.
This solver was applied to a large variety of geometric problems above the (piecewise) rationals domain that will be discussed as part of the talk. This set 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 (circle tangent to given three circles), bounding circles, bisectors and voronoi regions, mold design, visibility and accessibility, sweeps and envelopes, and self-intersection trimming in offset approximations of curves and surfaces.
Abstract:
We present a subdivision based paradigm to solve a set of (piecewise) rational constraints represented by multivariate (piecewise) rational spline functions. The constraints can be formulated as either inequalities or equalities (semi-algebraic sets). The presented solver is not only capable of finding all solutions, globally, but is also capable of handling solution spaces of dimensions larger than zero. A special care is taken to guarantee all roots are isolated via a single solution geometric test.
This solver was applied to a large variety of geometric problems above the (piecewise) rationals domain that will be discussed as part of the talk. This set 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 (circle tangent to given three circles), bounding circles, bisectors and voronoi regions, mold design, visibility and accessibility, sweeps and envelopes, and self-intersection trimming in offset approximations of curves and surfaces.
Bio:
Gershon Elber is a professor in the Computer Science Department, Technion, Israel. His research interests span computer aided geometric design and computer graphics. Prof. Elber received a BSc in computer engineering and an MSc in computer science from the Technion, Israel in 1986 and 1987, respectively, and a PhD in computer science from the University of Utah, USA, in 1992. Prof. Elber had several academic and Technion appointments, including the deputy vice president for computing, Technion, 2010-2013, and a member and head of the Solid Modeling Association (the parent organization of the solid and physical modeling, SPM, conferences), 2008-2010. He is now the head of the Technion Additive Manufacturing Center. Prof. Elber has served on the editorial board of the Computer Aided Design, Computer Graphics Forum, The Visual Computer, Graphical Models, and the International Journal of Computational Geometry & Applications and has served in many conferences' program committees including Solid Modeling, Shape Modeling, Geometric Modeling and Processing, Pacific Graphics, Computer Graphics International, and Siggraph. Prof. Elber was one of the paper chairs of Solid Modeling 2003 and Solid Modeling 2004, one of the conference chairs of Solid and Physical Modeling 2010, the chair of GDM 2014, the conference co-chair of SIAM GD/SPM 2015, and the conference co-chair of SPM 2018. He has published over 250 papers in international conferences and journals, received 15 best paper awards, presented a dozen keynote plenary talks, and is one of the authors of a book titled "Geometric Modeling with Splines - An Introduction". Prof. Elber received the John Gregory Memorial Award, 2011, in "Appreciation for Outstanding Contributions in Geometric Modeling", the Solid Modeling Association pioneers award in 2016, the Bezier award in 2019, and the Kunii award in 2024.
* 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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Meeting ID: 359 737 944 764
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