# The power of geometric duality

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DOI: 10.1007/BF01934990

- Cite this article as:
- Chazelle, B., Guibas, L.J. & Lee, D.T. BIT (1985) 25: 76. doi:10.1007/BF01934990

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## Abstract

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 long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen among*n* points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocess*n* points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimal*O*(*k* + log*n*) time algorithm for answering such queries, where*k* is the number of points to be reported. The algorithm requires*O*(*n*) space and*O*(*n* log*n*) 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.