Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
The complexity and construction of many faces in arrangements of lines and of segments
Discrete & Computational Geometry - Special issue on the complexity of arrangements
The complexity of many cells in arrangements of planes and related problems
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
On the general motion-planning problem with two degrees of freedom
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
On the two-dimensional davenport-schinzel problem
Journal of Symbolic Computation
The upper envelope of Voronoi surfaces and its applications
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
On the sum of squares of cell complexities in hyperplane arrangements
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Arrangements of curves in the plane—topology, combinatorics, and algorithms
Theoretical Computer Science
Counting facets and incidences
Discrete & Computational Geometry
On the zone theorem for hyperplane arrangements
SIAM Journal on Computing
On lines missing polyhedral sets in 3-space
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Combinatorial complexity of translating a box in polyhedral 3-space
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Implementation of a motion planning system in three dimensions
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Vertical decompositions for triangles in 3-space
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Almost tight upper bounds for the single cell and zone problems in three dimensions
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
On translational motion planning in 3-space
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Bounds on the size of tetrahedralizations
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Computational geometry: a retrospective
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
On point location and motion planning among simplices
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
On lazy randomized incremental construction
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
ACM Computing Surveys (CSUR)
Strategies for polyhedral surface decomposition: an experimental study
Proceedings of the eleventh annual symposium on Computational geometry
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We show that the total number of faces bounding any single cell in an arrangement of n (d–1)-simplices in IRd is O(nd–1 log n), thus almost settling a conjecture of Pach and Sharir. We present several applications of this result, mainly to translational motion planning in polyhedral environments. We then extend our analysis technique to derive other results on complexity in simplex arrangements. For example, we show that the number of vertices in such an arrangement, which are incident to the same cell on more than one “side,” is O(nd-1 log n). We also show that the number of repetitons of a “k-flap,” formed by intersecting d–k simplices, along the boundary of the same cell, summed over all cells and all k-flaps, is O(nd-1 log n). We use this quantity, which we call the excess of the arrangement, to derive bounds on the complexity of m distinct cells of such an arrangement.