1. By Dor Lahav

2. Overview Straight Skeletons Convex Polygons Constrained Voronoi diagrams and Delauney triangulations

3. Straight Skeletons

4. Straight Skeleton- Motivation The occurrence of curved edges in the line segment Voronoi diagram V(G) is a disadvantage in the computer representation. a different alternative to V(G)- the Straight Skeleton.

5. Straight Skeleton

6. A few definitions: - A connected component of G is called a figure. - Every simple polygon which rises from a figure, is called a wavefront.

7. Straight Skeleton There are two types of changes: - Edge event: a wavefront edge collapses to length zero. - Split event: a wavefront edge splits due to interference or self interference. - After each type of event, we have a new set of wavefronts.

8. Straight Skeleton The edges of S(G) are pieces of angular bisectors traced out by wavefront vertices. Each vertex of S(G) corresponds to event. So we get a unique structure defining a polygonal partition of a plane.

9. Straight Skeleton Lets see an example…

11. Straight Skeleton How many faces in the diagram? Each segment of G gives rise to two wavefront edges and thus to two faces, one on each side of the segment. Each terminal of G gives rise to one face. This gives a total of 2m+t = O(n) faces(m-edges, t- terminals). There is also an exact bound on the number of vertices…

12. Straight Skeleton Lemma: Let G be a planar straight graph on n points, t of which are terminals. The number of(finite and infinite) vertices of S(G) is exactly 2n+t-2.

13. Straight Skeleton The Straight Skeleton has a 3- dimensional interpretation obtained by defining the height of a point x in the plane as the unique time when x is reached by the wavefront. That’s how S(G) lifts out a polygonal surface. Points of G have height zero.

14. Straight Skeleton

15. Convex Polygons

16. Let C be a convex n-gon in the plane. The medial axis M(C) of C is a tree whose edges are pieces of angular bisectors of C’s sides. There is a simple randomized incremental algorithm that computes M(C) in O(n) time.

17. Convex Polygons

28. Done!

29. Running time? M(C) has an upper bound of 2n-3 edges. Each edge belongs to two faces. Hence, the average number of edges of a face is That’s a constant time per face. So, a total running time of O(n).

30. Convex Polygons- Voronoi diagram

38. Constrained Voronoi diagrams and Delauney triangulations

39. Constrained Voronoi diagrams Let S be a set of n point sites in the plane. Let L be a set of non-crossing line segments spanned by S. We define the bounded distance between two points x and y in the plane as:

40. Constrained Voronoi diagrams For each segment l, the regions clipped by l from the right are extended to the left of l, as if only the sites of these regions were present. The regions clipped by l from the left are extended similarly.

41. Constrained Delauney triangulations If we dualize now by connecting sites of regions that share an edge, a full triangulation that includes L is obtained.