TY - GEN
T1 - On ray shooting for triangles in 3-space and related problems
AU - Ezra, Esther
AU - Sharir, Micha
N1 - Publisher Copyright: © Esther Ezra and Micha Sharir; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).
PY - 2021/6/1
Y1 - 2021/6/1
N2 - We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in ℝ3, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections in a set of lines in ℝ3, and (v) output-sensitive construction of an arrangement of triangles in three dimensions. Our approach is based on the polynomial partitioning technique. For example, our ray-shooting algorithm processes a set of n triangles in R3 into a data structure for answering ray shooting queries amid the given triangles, which uses O(n3/2+ε) storage and preprocessing, and answers a query in O(n1/2+ε) time, for any ε > 0. This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly n5/8. The algorithms for the other problems have similar performance bounds, with similar improvements over previous results. We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer m queries on n objects in max {O(m2/3n5/6+ε + n1+ε), O(m5/6+εn2/3 + m1+ε) time, for any ε > 0, again an improvement over the earlier bounds.
AB - We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in ℝ3, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections in a set of lines in ℝ3, and (v) output-sensitive construction of an arrangement of triangles in three dimensions. Our approach is based on the polynomial partitioning technique. For example, our ray-shooting algorithm processes a set of n triangles in R3 into a data structure for answering ray shooting queries amid the given triangles, which uses O(n3/2+ε) storage and preprocessing, and answers a query in O(n1/2+ε) time, for any ε > 0. This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly n5/8. The algorithms for the other problems have similar performance bounds, with similar improvements over previous results. We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer m queries on n objects in max {O(m2/3n5/6+ε + n1+ε), O(m5/6+εn2/3 + m1+ε) time, for any ε > 0, again an improvement over the earlier bounds.
KW - Polynomial partitioning
KW - Ray shooting
KW - Three dimensions
KW - Tradeoff
UR - http://www.scopus.com/inward/record.url?scp=85108240306&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2021.34
DO - 10.4230/LIPIcs.SoCG.2021.34
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 37th International Symposium on Computational Geometry, SoCG 2021
A2 - Buchin, Kevin
A2 - de Verdiere, Eric Colin
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 37th International Symposium on Computational Geometry, SoCG 2021
Y2 - 7 June 2021 through 11 June 2021
ER -