{"title":"Almost Tight Bound for the Union of Fat Tetrahedra in Three Dimensions","authors":"Esther Ezra, M. Sharir","doi":"10.1109/FOCS.2007.9","DOIUrl":null,"url":null,"abstract":"We show that the combinatorial complexity of the. union of n \"fat\" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least .some fixed constant) of arbitrary sizes, is O(n2+epsiv),for any epsiv > 0: the bound is almost tight in the worst case, thus almost settling a conjecture of Pach el al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-K'irkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Delta behave as fat dihedral wedges in Delta. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R3 having arbitrary side lengths, is O(n2+epsiv), for any epsiv > 0 again, significantly extending the result of [24]. Our analysis can easily he extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have, arbitrary sizes) in R3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"21","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/FOCS.2007.9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 21
Abstract
We show that the combinatorial complexity of the. union of n "fat" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least .some fixed constant) of arbitrary sizes, is O(n2+epsiv),for any epsiv > 0: the bound is almost tight in the worst case, thus almost settling a conjecture of Pach el al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-K'irkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Delta behave as fat dihedral wedges in Delta. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R3 having arbitrary side lengths, is O(n2+epsiv), for any epsiv > 0 again, significantly extending the result of [24]. Our analysis can easily he extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have, arbitrary sizes) in R3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.