This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point by simply "walking through" the triangulation, after selecting a "good starting point" by random sampling. The analysis generalizes and extends a recent result for d = 2 dimensions by proving this procedure takes expected time close to O(n 1/(d+1)) for point location in Delaunay triangulations of n random points in d = 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice.

Matching two geometric objects in 2D and 3D spaces is a central problem in computer vision, pattern recognition and protein structure prediction. In particular, the problem of aligning two polygonal chains under translation and rotation to minimize their distance has been studied using various distance measures. It is well known that the Hausdorff distance is useful for matching two point sets, and that the Fréchet distance is a superior measure for matching two polygonal chains. The discrete Fréchet distance closely approximates the (continuous) Fréchet distance, and is a natural measure for the geometric similarity of the folded 3D structures of bio-molecules such as proteins. In this paper, we present new algorithms for matching two polygonal chains in 2D to minimize their discrete Fréchet distance under translation and rotation, and an effective heuristic for matching two polygonal chains in 3D. We also describe our empirical results on the application of the discrete Fréchet distance to the protein structure-structure alignment.

This short note considers the problem of point location in a Delaunay triangulation of n random points, using no additional preprocessing or storage other than a standard data structure representing the triangulation. A simple and easy-to-implement (but, of course, worst-case suboptimal) heuristic is shown to take expected time O(n 1/3 ).

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