1. 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

2. 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.

3. 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.

4. 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

5. 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:

6. 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.

7. 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

8. 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.

9. 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

10. 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.

11. 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.

12. 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

13. 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

14. 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.

15. 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

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

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

18. 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

19. 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

20. 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

21. 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.

22. 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.

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

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

25. 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

26. References
Z. Artstein, “Piecewise linear approximation of set valued maps”, Journal of Approximation Theory, vol. 56, pp , 1989.
F. Aurenhammer, R Klein, "Voronoi Diagrams" in Handbook of Computational Geometry, J. R. Sack, J. Urrutia, Eds., Amsterdam: Elsevier, 2000, pp
N. Dyn, E. Farkhi, A. Mokhov, “Approximation of univariate set-valued functions - an overview”, Serdica, vol. 33, pp , 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.

27. 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

28. 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