TEL-AVIV UNIVERSITY FACULTY OF EXACT SCIENCES SCHOOL OF MATHEMATICAL SCIENCES An Algorithm for the Computation of the Metric Average of Two Simple Polygons Shay Kels Nira Dyn Evgeny Lipovetsky

Approximation of set-valued functions (N. Dyn, E. Farkhi, A. Mokhov) Approximation of a set-valued function from a finite number of its samples Given a binary operation between sets, many tools from approximation theory can be adapted to set-valued functions Spline subdivision schemes Bernstein operators via de Casteljau algorithm Schoenberg operators via de Boor algorithm The applied operation is termed the metric average (Z. Artstein ) Repeated computations of the metric average are required.

Outline An algorithm that applies tools from computational geometry: segment Voronoi diagrams and planar arrangements to the computation of the metric average of two simple polygons Implementation of the algorithm as a C++ software using CGAL – Computational Geometry Algorithms Library Connectedness and complexity of the metric average. Extension to compact sets that are collections of simple polygons with holes.

Preliminary definitions Let be the collection of nonempty compact subsets of . The Euclidean distance from a point p to a set is The Hausdorff distance between two sets is The set of all projections of a point p on a set is The linear Minkowksi combination of two sets is

The metric average Example in : Let and the one-sided t-weighted metric average of A and B is The metric average of A and B is The metric property:

Conic polygons with holes A conic segment is defined by: its base conic: its beginning point its end point A simple conic polygon is a region of the plane bounded by a single finite chain of conic segments, that intersect only at their endpoints. A simple conic polygon with holes is a conic polygon that contains holes, which are simple conic polygons.

Planar arrangements Given a collection C of curves in the plane, the arrangement of C is the subdivision of the plane into vertices, edges and faces induced by the curves in C. The overlay of two arrangements and is the arrangement produced by edges from and .

Segment Voronoi diagrams For a set S of n simple geometric objects (called sites) , the Voronoi diagram of S is the subdivision of the plane into regions (called faces), each region being associated with some site , and containing all points of the plane for which is closest among all the sites in S. A segment is represented as three objects: an open segment and the endpoints. The diagram is bounded by a frame. All edges are conic segments. The diagram constitutes of an arrangement of conic segments.

Computation of the metric average - the algorithm The metric average can be written as: Computation of where A, B are simple polygons 1. Compute the sets 2. Compute 3. Compute 4. Compute

Computation of the metric average - the metric faces Let A, B be simple polygons and let F be a Voronoi face of VDB , we call a connected component of as a metric face originating from F. The metric faces are faces of the overlay of the arrangements representing VDB and A \ B, which are intersection of the bounded faces of the two original arrangements. Each metric face “inherits” the Voronoi site of the face of VDB that contains it.

Computation of the metric average - the metric faces(1) By definition of the Voronoi diagram, for thus The transform is a continuous and one-to-one function from F to . The region bounded by is We can compute the metric average only for boundaries of the metric faces and only relative to the corresponding Voronoi sites.

Computation of the metric average - the algorithm (1) Computation of the one-sided metric average 1. Compute the segment Voronoi diagram induced by 2 . Overlay with and obtain the metric faces with their corresponding sites 3. For each metric face in the collection found in 2, compute

Computation of the metric average - the algorithm (2) for a metric face F 1. For each conic segment in a. compute b. add the result of (a) to the collection of conic segments already computed 2. Return the resulting collection of conic segments as boundary of a conic polygon ( i.e. we computed ) S

Computation of the metric average - the algorithm (3) Computation of for a conic segment and the corresponding point Voronoi site S is the set of points satisfying where and are collinear. 1. Express in terms of p and S 2. Substitute into the conic equation of and by collecting the terms obtain the conic equation of For a segment Voronoi site S, compute and the computation is similar.

Complexity bounds Proposition: Let A,B be simple polygons and let n be the sum of the number of vertices in A and the number of vertices in B. Let k be the combinatorial complexity (the sum of the number of vertices, the number of edges, and the number of faces) of the overlay of the arrangements representing the sets and . Then: k is . The combinatorial complexity of with is Then the run-time complexity of the computation of the metric average is .

The metric average of two simple convex polygons with Examples The metric average of two simple convex polygons with

Examples (1) The metric average of two simple polygons with

Connectedness of the metric average The metric average of two intersecting simple polygons can be a union of several disjoin conic polygons. The connectedness problem is model by a graph. Vertices: connected components of - metric faces of - metric faces of

Connectedness of the metric average (1) are called metric connected if and only if the set is connected. There is an edge on the graph between each two vertices corresponding to metric connected elements. is connected iff the metric connectivity graph is connected

Connectedness of the metric average (2) Several propositions considering metric connectedness, for example: Proposition: Let be simple polygons and be metric faces , . are metric connected if and only if there are points and , satisfying: and . In terms of metric faces and the corresponding Voronoi sites: condition 1 condition 2 condition 3

The metric average of two simple polygonal sets A set consisting of pairwise disjoint polygons with holes is termed a simple polygonal set. The segment Voronoi diagram induced by the boundary of a simple polygonal set is well defined. Let be simple polygonal sets and F a face of , a connected component of is termed a metric face originating from F. The metric faces are conic polygons with holes.

The metric average of two simple polygonal sets (1) Let the metric face F be a conic polygon P with holes . The operation is a continuous and one-to-one function from F to is a conic polygon with holes The computation is similar to the computation of the metric average and the modified metric average of two simple polygons. The implementation is supported by CGAL.

Examples , where A is a polygon and B consists of two polygons contained in A.

Examples (1) ,where A, B are simple polygonal sets.

Future work An algorithm for the computation of the metric average of two-dimensional compact sets with boundaries consisting of spline curves An algorithm for the computation of the metric average of two polyhedra. Research for new set averaging operations with the ‘metric property’ relative to some distance Work in progress: new set averaging operation with superior geometric features and the ‘metric property’ relative to the measure of the symmetric difference distance

References Z. Artstein, “Piecewise linear approximation of set valued maps”, Journal of Approximation Theory, vol. 56, pp. 41-47, 1989. F. Aurenhammer, R Klein, "Voronoi Diagrams" in Handbook of Computational Geometry, J. R. Sack, J. Urrutia, Eds., Amsterdam: Elsevier, 2000, pp. 201-290.   N. Dyn, E. Farkhi, A. Mokhov, “Approximation of univariate set-valued functions - an overview”, Serdica, vol. 33, pp. 495-514, 2007. D. Halperin, "Arrangements", in Handbook of Discrete and Computational Geometry, J. E. Goodman, J. O’Rourke, Eds., Chapman & Hall/CRC, 2nd edition, 2004, pp 529–562. The CGAL project homepage. http://www.cgal.org/.

Appendix A: Computation of the metric average with Voronoi diagrams – the mathematics Let A, B be simple polygons, the set A\ B can be written as and therefore For a point p on the interior of a Voronoi face F therefore (*) or in terms of metric faces (**) (*) and (**) can be extended to any two compact sets A, B in

Appendix A: Computation of the metric average with Voronoi diagrams – the mathematics (1) For a site S(F) of the segment Voronoi diagram and a point p in R2 the set is a singleton. can be regarded as a function which is continuous and one-to-one. The boundary of a metric face is a simple closed curve, so is its mapping under G, and therefore stands for the region bounded by The metric average can be computed as where