FINDTRIA provides efficient d-dimensional point-in-simplex queries in MATLAB / OCTAVE.
FINDTRIA computes spatial queries for collections of simplexes (triangles, tetrahedrons, etc) in d-dimensional space. Unlike MATLAB's existing point-location facilities, FINDTRIA supports general collections of simplexes, including non-Delaunay, non-convex, and even overlapping configurations, and can compute multiple intersections per query point.
FINDTRIA is based on an efficient d-dimensional AABB-tree, which is used to speed-up the computation of spatial queries.
FINDTRIA is relatively efficient. It's typically many orders of magnitude faster than brute-force searches, and is often faster than MATLAB's inbuilt routine TSEARCHN, especially when the number of query points is large. MATLAB's inbuilt POINTLOCATION routine is usually faster than FINDTRIA when the underlying triangulation is Delaunay, but is often slower -- sometimes by a large factor -- for non-Delaunay triangulations. It is also restricted to two- and three-dimensional problems.
FINDTRIA was not specifically designed to outperform MATLAB's existing point-location routines (though it sometimes does a good job), it's main purpose is to facilitate efficient queries on non-Delaunay triangulations in arbitrary dimensions -- capabilities that are currently unsupported by existing inbuilt routines.
After downloading and unzipping the current repository, navigate to the installation directory within MATLAB / OCTAVE and run the set of examples contained in triademo.m:
triademo(1); % point-location for a 2-dimensional non-Delaunay triangulation.
triademo(2); % point-location for a 3-dimensional non-Delaunay triangulation.
triademo(3); % compare FINDTRIA and TSEARCHN for higher-dimensional problems.
This program may be freely redistributed under the condition that the copyright notices (including this entire header) are not removed, and no compensation is received through use of the software. Private, research, and institutional use is free. You may distribute modified versions of this code UNDER THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL AUTHOR, BOTH SOURCE AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution of this code as part of a commercial system is permissible ONLY BY DIRECT ARRANGEMENT WITH THE AUTHOR. (If you are not directly supplying this code to a customer, and you are instead telling them how they can obtain it for free, then you are not required to make any arrangement with me.)
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FINDTRIA makes use of the AABBTREE package, and is used extensively in the grid-generator MESH2D.The underlying AABB-tree construction and search methods are described in further detail here:
[1] - Darren Engwirda, Locally-optimal Delaunay-refinement and optimisation-based mesh generation, Ph.D. Thesis, School of Mathematics and Statistics, The University of Sydney, September 2014.