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My current big unsolved problem is to use Nvidia GPUs to compute the volumes of all the nonempty intersections between the tetrahedra in two different 3D triangulations of the same polyhedron.
Generally, I develop and implement fast parallel algorithms on very large geometric datasets in CAD and GIS. I've also modeled and processed large terrain databases, e.g., to compress, to compute hydrography and visibility, and to site observers, and compressed 5D environmental data sets. The algorithms are the fastest in their class; awards are listed below.

Latest paper:
Salles V. G. de Magalhães, W. Randolph Franklin, and Marcus V. A. Andrade. An efficient and exact parallel algorithm for intersecting large 3d triangular meshes using arithmetic filters. Computer Aided Design, 2019. https://doi.org/10.1016/j.cad.2019.102801
We present 3DEPUGOVERLAY, a fast, exact, parallel, memoryefficient, algorithm for computing the intersection between two large 3D triangular meshes with geometric degeneracies. Applications include CAD/CAM, CFD, GIS, and additive manufacturing. 3DEPUGOVERLAY combines 5 techniques: multiple precision rational numbers to eliminate roundoff errors during the computations; Simulation of Simplicity to properly handle geometric degeneracies; simple data representations and only local topological information to simplify the correct processing of the data and make the algorithm more parallelizable; a uniform grid to efficiently index the data, and accelerate testing pairs of triangles for intersection or locating points in the mesh; and parallel programming to exploit current hardware. 3DEPUGOVERLAY is up to 101 times faster than LibiGL, and comparable to QuickCSG, a parallel inexact algorithm. 3DEPUGOVERLAY is also more memory efficient. In all test cases, 3DEPUGOVERLAY’s result matched the reference solution. It is freely available for nonprofit research and education at https://github.com/sallesviana/MeshIntersection.

Recent papers and talks:
2020
 Salles V. G. de Magalhães, W. Randolph Franklin, and Marcus V. A. Andrade.
An efficient and exact parallel algorithm for intersecting large 3d triangular meshes using arithmetic filters.
J. Computer Aided Design, March 2020.
online 20191219.
doi:https://doi.org/10.1016/j.cad.2019.102801.
[abstract▼] [details] [full text] [BibTeX▼]
2019
 W. Randolph Franklin and Salles Viana Gomes de Magalhães.
Minimal representations of polygons and polyhedra.
In John Krumm, editor, 1st ACM SIGSPATIAL International Workshop on Spatial Gems (SpatialGems 2019).
ACM, nov 2019.
URL: https://www.spatialgems.net/.
[abstract▼] [details] [full text] [BibTeX▼]  Marcelo de Matos Menezes, Salles Viana Gomes Magalhães, W. Randolph Franklin, Matheus Aguilar de Oliveira, and Rodrigo E. O. Bauer Chichorro.
Accelerating the exact evaluation of geometric predicates with GPUs.
In 28th International Meshing Roundtable. Buffalo, NY, USA, 16 Oct 2019.
[abstract▼] [details] [full text] [slides] [BibTeX▼]  W. Randolph Franklin and Salles Viana Gomes de Magalhães.
Implementing simulation of simplicity for geometric degeneracies.
unpublished, 2019.
[details] [full text] [BibTeX▼]  W. Randolph Franklin and Salles V. G. de Magalhães.
Computing intersection areas of overlaid 2d meshes.
In IGS2019 International Geometry Summit Posters' proceedings. Vancouver, Canada, 17–21 June 2019.
[abstract▼] [details] [full text] [poster] [fastforward] [BibTeX▼]
2018
 Salles Viana Gomes de Magalhães, W. Randolph Franklin, and Ricardo dos Santos Ferreira.
Fast analysis of upstream features on spatial networks (GIS Cup).
In Proceedings of the 26th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, SIGSPATIAL '18, 622–625. New York, NY, USA, 2018. ACM.
Winner (1st place).
doi:https://doi.org/10.1145/3274895.3276474.
[abstract▼] [details] [full text] [slides] [BibTeX▼]  W. Randolph Franklin, Salles V. G. de Magalhães, and Marcus V. A. Andrade.
Data structures for parallel spatial algorithms on large datasets (vision paper).
In Proceedings of BigSpatial’18: 7th ACM SIGSPATIAL Workshop on Analytics for Big Geospatial Data. Seattle, USA, 6 Nov 2018.
[abstract▼] [details] [full text] [slides] [BibTeX▼]  W. Randolph Franklin, Salles V. G. de Magalhães, and Marcus V. A. Andrade.
Exact fast parallel intersection of large 3D triangular meshes (extended abstract).
In 28th Annual Fall Workshop on Computational Geometry. Queens College, CUNY, New York City, 26–27 Oct 2018.
[abstract▼] [details] [full text] [BibTeX▼]  W. Randolph Franklin, Salles V. G. de Magalhães, and Marcus V. A. Andrade.
Exact fast parallel intersection of large 3D triangular meshes.
In 27th International Meshing Roundtable. Alberqueque, New Mexico, 2 Oct 2018.
[abstract▼] [details] [full text] [slides] [BibTeX▼]
2017
 W. Randolph Franklin.
Applications of geometry.
In Kenneth H Rosen, editor, Handbook of Discrete and Combinatorial Mathematics, Discrete Mathematics and Its Applications, chapter 13.8, pages 998–1022.
CRC Press, 2nd edition, 1 Dec 2017.
[abstract▼] [details] [full text] [BibTeX▼]
 Salles V. G. de Magalhães, W. Randolph Franklin, and Marcus V. A. Andrade.
An efficient and exact parallel algorithm for intersecting large 3d triangular meshes using arithmetic filters.
J. Computer Aided Design, March 2020.
online 20191219.
doi:https://doi.org/10.1016/j.cad.2019.102801.

Professional summary
 Professor, ECSE Dept, RPI
 BSc (Toronto)
 AM, PhD, Applied Math (Harvard)
 Program Director, Numeric, Symbolic, and Geometric Computation Program, CISE, National Science Foundation, 2000—2002
 Visiting Professor, UC Berkeley, 1985—1986
 Visiting positions at Genoa, Laval, CSIRO Canberra, National University of Singapore, 1992—1993.
 Visitor at Georgia Tech, 2016.
 GPG key
 Vcard
 Email: frankwrATSIGNrpiONEDOTedu

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