Approximating a real number by a rational number with a limited denominator: A geometric approach
Résumé
We consider the problem of determining the rational number which best approximates the real number a and such that its denominator belongs to an interval [b, b']. There is a related geometric problem consisting in finding the integer point lying in the vertical domain D of the form {(x, y) is an element of R(2) vertical bar b <= x <= b'} such that the straight line passing through the origin and through this point best approximates the straight line L of slope a passing through the origin. The computation of this point is interlinked with the computation of both the convex hulls of the integer points located above and below the straight line L respectively and lying in the vertical domain D. In the literature, many general convex hull algorithms exist, as the gift wrapping algorithm for example. However, we focus on two interesting approaches to compute these convex hulls which are especially appropriated in our special configuration. The first one mainly uses number theory and runs in 0(log(b')) time. The other is in line with computational geometry as the method proposed in 1999 by Balza-Gomez er al. [H. Balza-Gomez, J.-M. Moreau, D. Michelucci, Convex hull of grid points below a line or a convex curve, in: DGCI '99: Proceedings of the 8th International Conference on Discrete Geometry for Computer Imagery, Springer, Marne-la-Vallee, France, 1999, pp. 361-374] which runs in O(log(b' - b)) time. We propose a new method for the computation of these convex hulls which combines number theory and computational geometry. Our method preserves the optimal time complexity and is the first being output sensitive. Indeed, we compute the convex hulls in time linear in their vertex number. Moreover, the resulting algorithm is very simple and so is suitable for implementation.