Computational geometry: an introduction
Computational geometry: an introduction
Constructing higher-dimensional convex hulls at logarithmic cost per face
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Optimal point location in a monotone subdivision
SIAM Journal on Computing
A general approach to d-dimensional geometric queries
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
A probabilistic algorithm for the post office problem
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
The shortest watchtower and related problems for polyhedral terrains
Information Processing Letters
Visibility problems for polyhedral terrains
Journal of Symbolic Computation
The complexity of many faces in arrangements of lines of segments
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Arrangements of lines in 3-space: a data structure with applications
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
A deterministic algorithm for partitioning arrangements of lines and its application
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
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
An algorithm for generalized point location and its applications
Journal of Symbolic Computation
Multidimensional divide-and-conquer
Communications of the ACM
Intersecting two polyhedra one of which is convex
FCT '85 Fundamentals of Computation Theory
Efficiently Computing and Representing Aspect Graphs of Polyhedral
Efficiently Computing and Representing Aspect Graphs of Polyhedral
Primitives for computational geometry
Primitives for computational geometry
ACM SIGACT News
Cutting hyperplane arrangements
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Stabbing and ray shooting in 3 dimensional space
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Finding stabbing lines in 3-dimensional space
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
Ray shooting and parametric search
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Finding a line transversal of axial objects in three dimensions
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Diameter, width, closest line pair, and parametric searching
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Incidence and nearest-neighbor problems for lines in 3-space
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Generalized hidden surface removal
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
New bounds for lower envelopes in three dimensions, with applications to visibility in terrains
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
On lines missing polyhedral sets in 3-space
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
Computational geometry: a retrospective
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Height distributional distance transform methods for height field ray tracing
ACM Transactions on Graphics (TOG)
ACM Computing Surveys (CSUR)
Efficient generation of k-directional assembly sequences
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
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We study combinatorial and algorithmic problems involving arrangements of n lines in 3-dimensional space, and then present applications of our results to a variety of problems on polyhedral terrains. Our main results include:A tight &THgr;(n2) bound on the complexity of the space of all lines passing above all the n given lines (their “upper envelope”) and satisfying a certain orientation consistency constraint.A preprocessing procedure using near-quadratic time and storage that builds a structure supporting &Ogr;(log n) time queries for testing if a line lies above all the given lines.An &Ogr;(n4/3+&egr;) randomized expected time algorithm, for any fixed &egr; 0, that tests the “towering property”: do n given red lines lie all above n given blue lines?A preprocessing procedure for a polyhedral terrain &Sgr; with n edges, that uses near-quadratic time and storage and builds a structure supporting &Ogr;(log2 n) time rayshooting queries for computing the first intersection of an arbitrary query ray with &Sgr;.Finding the smallest vertical distance between two disjoint polyhedral terrains with a total of n edges, in time &Ogr;(n4/3+&egr;), for any &egr; 0.Computing the upper envelope (pointwise maximum) of two polyhedral terrains with a total of n edges, in time &Ogr;(n1.5+&egr; + klog2 n), for any &egr; 0, where &kgr; is the size of the output envelope.The tools used to obtain these results include Plücker coordinates for lines in space, random sampling in geometric problems, and a new variant of segment trees.