Computational geometry: an introduction
Computational geometry: an introduction
Quadratic bounds for hidden line elimination
SCG '86 Proceedings of the second annual symposium on Computational geometry
Making data structures persistent
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Voronoi diagrams and arrangements
Discrete & Computational Geometry
On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Constructing arrangements of lines and hyperplanes with applications
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Dynamic computational geometry
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
On minima of function, intersection patterns of curves, and davenport-schinzel sequences
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
An efficient algorithm for hidden surface removal, II
Proceedings of the 30th IEEE symposium on Foundations of computer science
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We consider the problem of obtaining sharp (nearly quadratic) bounds for the combinatorial complexity of the lower envelope (i.e. pointwise minimum) of a collection of n bivariate (or generally multi-variate) continuous and "simple" functions, and of designing efficient algorithms for the calculation of this envelope. This problem generalizes the well-studied univariate case (whose analysis is based on the theory of Davenport-Schinzel sequences), but appears to be much more difficult and still largely unsolved. It is a central problem that arises in many areas in computational and combinatorial geometry, and has numerous applications including generalized planar Voronoi diagrams, hidden surface elimination for intersecting surfaces, purely translational motion planning, finding common transversals of polyhedra, and more. In this abstract we provide several partial solutions and generalizations of this problem, and apply them to the problems mentioned above. The most significant of our results is that the lower envelope of n triangles in three dimensions has combinatorial complexity at most O(n2α(n)) (where α(n) is the extremely slowly growing inverse of Ackermann's function), that this bound is tight in the worst case, and that this envelope can be calculated in time O(n2α(n)).