Let P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct place- ments of Q with respect to P under translations is O(n 2 + m + rain(rim + nm)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n + m) log(n + m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q.

We present a near-quadratic time algorithm that computes a point inside a simple polygon P having approximately the largest visibility polygon inside P , and near-linear time algorithm for nding the point that will have approximately the largest Voronoi region when added to an n-point set. We apply the same technique to nd the translation that approximately maximizes the area of intersection of two polygonal regions in near-quadratic time.

We propose the ShareCam Problem: controlling a single robotic pan, tilt, zoom camera based on simultaneous frame requests from n online users. To solve it, we propose a new piecewise linear metric, Intersection Over Maximum (IOM), for the degree of satisfaction for each users. To maximize overall satisfaction, we present several algorithms. For a discrete set of m distinct zoom levels, we give an exact algorithm that runs in O(n m) time. The algorithm can be distributed to run in O(nm) time at each client and in O(n log n + mn) time at the server.

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

Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕP of P that minimizes the convex hull of ϕP ∪ Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕP and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact nearlinear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1 + ε)-approximation in time O(ε −1/2 log n + ε −3/2 log a (1/ε)) if the two sets are convex polygons with n vertices in total, where a ∈ {0, 1, 2} depending on the version of the problem. 1

was preceded by a one-day workshop entitled “CGAL Innovations and Applications: Robust Geometric

We analyze a probabilistic algorithm for matching plane compact sets with sufficiently nice boundaries under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. We give a time bound that does not depend on the number of vertices in the case of polygons. 1

We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices. 1