Philippe Guigue

Fast and Robust Triangle-Triangle Overlap Test Using Orientation Predicates

Abstract This paper presents an algorithm for determining whether two triangles in three dimensions intersect. The general scheme is identical to the one proposed by Moller [Möller 97]. The main difference is that our algorithm relies exclusively on the sign of 4 × 4 determinants and does not need any intermediate explicit constructions which are the source of numerical errors. Besides the fact that the resulting code is more reliable than existing methods, it is also more efficient. The source code is available online.

Inner and outer rounding of set operations on lattice polygonal regions

Robustness problems due to the substitution of the exact computation on real numbers by the rounded floating pointarithmetic are often an obstacle to obtain practical implementation of geometric algorithms. If the adoption of the exact computation paradigm [13] gives a satisfactory solution to this kind of problemsfor purely combinatorial algorithms this solution does not allow to solvein practice the case of algorithms that cascade the construction of new geometric objects.In this paper we consider the problem of rounding the intersection of two polygonal regionsonto the integer lattice with inclusion properties. Namely given two polygonal regions A and B having their vertices on the integer lattice the inner and outer rounding modesconstruct two polygonal regions with integer vertices such that they respectively are included and containing the exact intersection of A and B. We also prove interesting results on the Hausdorff distance the size and the convexity of these polygonal regions.

THE SHUFFLING BUFFER

The complexity of randomized incremental algorithms is analyzed with the assumption of a random order of the input. To guarantee this hypothesis, the n data have to be known in advance in order to be mixed what contradicts with the on-line nature of the algorithm. We present the shuffling buffer technique to introduce sufficient randomness to guarantee an improvement on the worst case complexity by knowing only k data in advance. Typically, an algorithm with O(n2) worst-case complexity and O(n) or O(nlog n) randomized complexity has an complexity for the shuffling buffer. We illustrate this with binary search trees, the number of Delaunay triangles or the number of trapezoids in a trapezoidal map created during an incremental construction.

Inner and outer rounding of boolean operations on lattice polygonal regions

Robustness problems due to the substitution of the exact computation on real numbers by the rounded floating point arithmetic are often an obstacle to obtain practical implementation of geometric algorithms. If the adoption of the exact computation paradigm [C.K. Yap, T. Dube, The exact computation paradigm, in: D.-Z. Du, F.K. Hwang (Eds.), Computing in Euclidean Geometry, in: Lecture Notes Series on Computing, vol. 4, second ed., World Scientific, Singapore, 1995, pp. 452-492, http://cs.nyu.edu/cs/faculty/yap/papers/paradigm.ps] gives a satisfactory solution to this kind of problems for purely combinatorial algorithms, this solution does not allow to solve in practice the case of algorithms that cascade the construction of new geometric objects. In this report, we consider the problem of rounding the intersection of two polygonal regions onto the integer lattice with inclusion properties. Namely, given two polygonal regions A and B having their vertices on the integer lattice, the inner and outer rounding modes construct two polygonal regions A@?@?B and A@?@?B with integer vertices such that A@?@?B@?A@?B@?A@?@?B. We also prove interesting results on the Hausdorff distance, the size and the convexity of these polygonal regions.