We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of non-negative, constant descriptioncomplexity algebraic functions. We first give an FPTAS for optimizing such a sum of algebraic functions, and then we apply it to several geometric optimization problems. We obtain the first FPTAS for two fundamental geometric shape matching problems in fixed dimension: maximizing the volume of overlap of two polyhedra under rigid motions, and minimizing their symmetric difference. We obtain the first FPTAS for other problems in fixed dimension, such as computing an optimal ray in a weighted subdivision, finding the largest axially symmetric subset of a polyhedron, and computing minimum-area hulls. 1

We present upper and lower bounds for the number of iterations performed by the Iterative Closest Point (ICP) algorithm. This algorithm has been proposed by Besl and McKay [4] as a successful heuristics for pattern matching under translation, where the input consists of two point sets in d-space, for d ≥ 1, but so far it seems not to have been rigorously analyzed. The considered (standard) measure of resemblance that the algorithm attempts to optimize is the RMS (root mean squared distance). We show that the number of iterations performed by the algorithm is polynomial in the number of input points. In particular, this bound is quadratic in the one-dimensional problem, for which we present a lower bound construction of Ω(n log n) iterations, where n is the overall size of the input. We also present several structural geometric properties of the algorithm. We show that at each iteration of the algorithm the cost function monotonically and strictly decreases along the vector ∆t of the relative translation. As a result, we conclude that the polygonal path π, obtained by concatenating all the relative translations that are computed during the execution of the algorithm, does not intersect itself. In particular, in the one-dimensional problem all the relative translations of the ICP algorithm are in the same (left or right) direction. 1