1. Area 1: Algorithms Dan Halperin, School of Computer Science, Tel Aviv University

2. Overview  objectives  results  wp 1.1: algorithm design  wp 1.2: integration and evaluation  plans

3. Objectives  extend key geometric data structures and algorithms of computational geometry to curved objects  design algorithms amenable to effective implementations  integrate techniques developed in the project to compare experimentally various approaches  provide complete solutions for some fundamental problems (CGAL)

4. First year results  smallest enclosing ball of balls  improved construction of vertical decompositions of 3D arrangements of surfaces  Voronoi diagrams of circles (additively weighted V.d.)  simple algorithm for visibility graphs for bounded convex sets of constant complexity  Hausdorff distance computation between curves  high-level filtering for arrangements of conic arcs

5. Smallest enclosing ball of balls Fischer, ETH Welzl ’ s point algorithm doesn ’ t generalize LP-type problem key primitive: R-miniball CGAL-based implementation under way

6. Improved output-sensitive constrcution of the vertical decomposition of 3D arrangements of surfaces Shaul-Halperin, Tel Aviv O(n q (n)log n+K log n) time [de Berg-Guibas-H 94] -> O(n log 2 n + K log n ) time n is the # of surfaces and K is the complexity of the decomposition prototye implementation for triangles and for polyhedral surfaces

7. Additively weighted Voronoi diagrams Karavelas-Yvinec, INRIA prototype implementation available CGAL-based version under way

8. A sum of squares theorem for visibility complexes and applications Angelier-Pocchiola, ENS prototype implementation for polygons and for circles available CGAL-based version under way

9. Hausdorff distance for curves, FUBScharf under way: measuring the Hausdorff distance for geometric objects composed of parameterized curves

10. High level filtering for arrangements of conic arcs Wein, TAU prototype implementation available CGAL-based version for segments and circular arcs available (CGAL 2.3) CGAL-based version for conic arcs in progress

12. Summary  new algorithms and data structures for curved objects  emphasis on effective solutions, careful choice of primitives, typically simpler algorithms  implementation, benchmarks  complete, integrated solutions for some fundamental problems (CGAL)

13. Future plans  effective (alternative) decompositions for curved objects  Minkowski sums for curved objects (generalized polygons)  complete CGAL-based solutions for: smallest enclosing ball of balls, arrangements of conic arcs, planar additively weighted V.d., visibility complexes  distance between curved objects  additively weighted V.d. in 3-space  computing pseudo-triangulations efficiently  …