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Published online by Cambridge University Press: 05 June 2012
To compute the convex hull of a finite set of points is a classical problem in computational geometry. In two dimensions, there are several algorithms that solve this problem in an optimal way. In three dimensions, the problem is considerably more difficult. As for the general case of any dimension, it was not until 1991 that a deterministic optimal algorithm was designed. In dimensions higher than 3, the method most commonly used is the incremental method. The algorithms described in this chapter are also incremental and work in any dimension. Methods specific to two or three dimensions will be given in the next chapter.
Before presenting the algorithms, section 8.1 details the representation of polytopes as data structures. Section 8.2 shows a lower bound of Ω(n log n + n⌊d/2⌋) for computing the convex hull of n points in d dimensions. The basic operation used by an incremental algorithm is: given a polytope C and a point P, derive the representation of the polytope conv(C ∪ {P}} assuming the representation of C has already been computed. Section 8.3 studies the geometric part of this problem. Section 8.4 shows a deterministic algorithm to compute the convex hull of n points in d dimensions. This algorithm requires preliminary knowledge of all the points: it is an off-line algorithm. Its complexity is O(n log n + n⌊(d+1)/2⌋), which is optimal only in even dimensions.
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