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Lower Envelopes of Surface Patches in 3-Space

Authors Pankaj K. Agarwal , Esther Ezra , Micha Sharir

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Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Esther Ezra
  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Tel Aviv, Israel

Funding

  • Agarwal, Pankaj K.: Partially supported by NSF grants IIS-18-14493, CCF-20-07556 and CCF-22-23870 and by a US-Israel Binational Science Foundation Grant 2022131.
  • Ezra, Esther: Partially supported by Israel Science Foundation Grant 800/22 and US-Israel Binational Science Foundation Grant 2022131.
  • Sharir, Micha: Partially supported by Israel Science Foundation Grant 495/23.

Cite AsGet BibTex

Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Lower Envelopes of Surface Patches in 3-Space. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.6

Abstract

Let Σ be a collection of n surface patches, each being the graph of a partially defined semi-algebraic function of constant description complexity, and assume that any triple of them intersect in at most s = 2 points. We show that the complexity of the lower envelope of the surfaces in Σ is O(n² log^{6+ε} n), for any ε > 0. This almost settles a long-standing open problem posed by Halperin and Sharir, thirty years ago, who showed the nearly-optimal albeit weaker bound of O(n²⋅ 2^{c√{log n}}) on the complexity of the lower envelope, where c > 0 is some constant. Our approach is fairly simple and is based on hierarchical cuttings and gradations, as well as a simple charging scheme. We extend our analysis to the case s > 2, under a "favorable cross section" assumption, in which case we show that the bound on the complexity of the lower envelope is O(n² log^{11+ε} n), for any ε > 0. Incorporating these bounds with the randomized incremental construction algorithms of Boissonnat and Dobrindt, we obtain efficient constructions of lower envelopes of surface patches with the above properties, whose overall expected running time is O(n² polylog), as well as efficient data structures that support point location queries in their minimization diagrams in O(log²n) expected time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Hierarchical cuttings
  • surface patches in 3-space
  • lower envelopes
  • charging scheme
  • gradation

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