By Dor Lahav

Overview Straight Skeletons Convex Polygons Constrained Voronoi diagrams and Delauney triangulations

Straight Skeletons

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.

Straight Skeleton

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

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.

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.

Straight Skeleton Lets see an example…

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…

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.

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.

Straight Skeleton

Convex Polygons

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.

Convex Polygons

Done!

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).

Convex Polygons- Voronoi diagram

Constrained Voronoi diagrams and Delauney triangulations

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:

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.

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