Local polyhedra and geometric graphs

Research output: Contribution to journalArticlepeer-review


We introduce a new realistic input model for straight-line geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor among the other vertices of G and (2) the longest and shortest edges of G differ in length by at most a polynomial factor. A polyhedron is local if all its faces are simplices and its edges form a local geometric graph. We show that any boolean combination of two local polyhedra in Rd, each with n vertices, can be computed in O(nlogn) time using a standard hierarchy of axis-aligned bounding boxes. Using results of de Berg, we also show that any local polyhedron in Rd has a binary space partition tree of size O(nlogd- 2n) and depth O(logn); these bounds are tight in the worst case when d≤3. Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions.

Original languageEnglish (US)
Pages (from-to)101-125
Number of pages25
JournalComputational Geometry: Theory and Applications
Issue number1-2
StatePublished - May 2005


  • Binary space partition
  • Bounding volume hierarchy
  • Intersection detection
  • Minkowski sum
  • Realistic input model

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics


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