, Volume 25, Issue 1, pp 7690
First online:
The power of geometric duality
 Bernard ChazelleAffiliated withDepartment of Computer Science, Brown University
 , Leo J. GuibasAffiliated withComputer Science Laboratory, Xerox PARC, Palo Alto Research Center
 , D. T. LeeAffiliated withNorthwestern University
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This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two longstanding problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimumarea triangle with vertices chosen amongn points in the plane; the other is to produce an optimal algorithm for the halfplane range query problem. This problem is to preprocessn points in the plane, so that given a test halfplane, one can efficiently determine all points lying in the halfplane. We describe an optimalO(k + logn) time algorithm for answering such queries, wherek is the number of points to be reported. The algorithm requiresO(n) space andO(n logn) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.
 Title
 The power of geometric duality
 Journal

BIT Numerical Mathematics
Volume 25, Issue 1 , pp 7690
 Cover Date
 198503
 DOI
 10.1007/BF01934990
 Print ISSN
 00063835
 Online ISSN
 15729125
 Publisher
 Kluwer Academic Publishers
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 Authors

 Bernard Chazelle ^{(1)}
 Leo J. Guibas ^{(2)}
 D. T. Lee ^{(3)}
 Author Affiliations

 1. Department of Computer Science, Brown University, Box 1910, Providence, R.I., USA
 2. Computer Science Laboratory, Xerox PARC, Palo Alto Research Center, 94304, Palo Alto, California, USA
 3. Northwestern University, 60201, Evanston, Illinois, USA