Proceedings ArticleDOI
Average-case ray shooting and minimum weight triangulations
Boris Aronov,Steven Fortune +1 more
- pp 203-211
TLDR
A polynomial-time algorithm is given that computes a triangulations compatible with a set of polyhedral obstacles; the area of the triangulation is within a multiplicative constant of the smallest possible.Abstract:
Consider an environment filled with polyhedral obstacles. The answer to aray-shoottng queyis the first obstacle encountered by the ray. A simple way to answer ray-shooting queries is to triangulate space in a manner compatible with theobstades,and then walk through thetriangulation along therayuntil the first obstacle is encountered. Weshow that the average walk length can be reduced by choosing a triangulation with weight (i.e., length in two dimensions or area in three dimensions) as small as possible. In two dimensions, we observe that the length of the minimum-length triangulation can be estimated by the total length of the obstacles plus the total length of a minimum spanuing tree of the obstacles, up to a logarithmic factor; this gives an a prior-i estimate of the average-cese walk length. Our main result is in three dimensions. We give a polynomial-time algorithm that computes a triangulation compatible with a set of polyhedral obstacles; the area of the triangulation is within a multiplicative constant of the smallest possible.read more
Citations
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Geometric Range Searching and Its Relatives
Pankaj K. Agarwal,Je Erickson +1 more
TL;DR: This volume provides an excellent opportunity to recapitulate the current status of geometric range searching and to summarize the recent progress in this area.
Proceedings ArticleDOI
Fast randomized point location without preprocessing in two- and three-dimensional Delaunay triangulations
TL;DR: The proposed procedure finds the query point by simply “walking through” the triangulation, after selecting a “good starting point” by random sampling, and generalizes and extends a recent result for dD 2 dimensions by proving this procedure takes expected time close to O.
Robust and Efficient Construction of Planar Minkowski Sums.
Eyal Flato,Dan Halperin +1 more
TL;DR: I wish to thank the whole CGAL team and especially CGAL members in Tel-Aviv University for helpful discussions concerning the problems studied in this thesis.
Journal ArticleDOI
Self-customized BSP trees for collision detection
TL;DR: This paper re-explore a classical structure used for collision detection – the binary space partitioning tree and defines self-customized data structures, a concept that is essential in virtual reality environments and their applications.
Journal ArticleDOI
Deferred, Self-Organizing BSP Trees
Sigal Ar,Gil Montag,Ayellet Tal +2 more
TL;DR: This work proposes using information about how the tree is used in order to determine its structure, and demonstrates how this leads to the creation of bsp trees that are small, do not require much preprocessing time, and respond very efficiently to sequences of collision queries.
References
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Book
Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Book
Computational geometry in C
TL;DR: In this paper, the design and implementation of geometry algorithms arising in areas such as computer graphics, robotics, and engineering design are described and a self-contained treatment of the basic techniques used in computational geometry is presented.
Journal ArticleDOI
On range searching with semialgebraic sets
Pankaj K. Agarwal,Jirí Matousek +1 more
TL;DR: A solution with nearly linear space and preprocessing time and withO(n1−1/b+δ) query time is given, whered≤b≤2d−3 and δ>0 is an arbitrarily small constant.
Proceedings ArticleDOI
A pedestrian approach to ray shooting: shoot a ray, take a walk
John Hershberger,Subhash Suri +1 more
TL;DR: In this paper, a simple Steiner triangulation of a simple polygon with the property that a ray can intersect at most O(log n) triangles before reaching the polygon boundary is presented.