1. Part II: Paper c: Skeletons, Roofs, and the Medial Axis Joseph ORourke Smith College

2. Outline zVoronoi Diagram zMedial Axis yGrassfire Transformation zStraight Skeleton yConstant-sloped roofs (cf. David Bélanger notes) yProperties (cf. Kevin Danaher notes)

3. Voronoi Applet (Paul Chew, Cornell) zhttp://www.cs.cornell.edu/Info/People/chew/Delaunay.htmlhttp://www.cs.cornell.edu/Info/People/chew/Delaunay.html

4. Skeletons & Roofs zDavid Bélanger, McGill Univ. zroofs.html (local)roofs.html zhttp://www.sable.mcgill.ca/~dbelan2/roofs/roofs.html (remote)http://www.sable.mcgill.ca/~dbelan2/roofs/roofs.html

5. Straight Skeleton in 1-Cut Thm zShrink boundary yHandle nonconvex polygons new event when vertex hits opposite edge yHandle nonpolygons butt vertices of degree 0 and 1

6. Straight Skeletons An alternative to the medial axis Kevin Danaher Computer Geometry Fall 2002 http://figment.csee.usf.edu/~aparasha/cgeom/StraightSkeletons.ppt

7. Straight Skeleton

8. Shrinking Process (contd) zPolygon hierarchy during shrinking

9. Events zTwo events can occur: yEdge event: an edge shrinks to zero, making its neighboring edges adjacent. ySplit event: A reflex vertex runs to an edge and splits it, thus splitting the whole polygon. New adjacencies occur between the split edge and each of the two edges incident to the reflex vertex.

10. Events (contd)

11. Formal Definitions zThe straight skeleton, S(P), of polygon, P, is the union of the pieces of the angular bisectors traced out by the polygon vertices during the shrinking process. zEach edge, e, sweeps out a certain area called the face of e. zBisector pieces are called arcs, and their endpoints which are not vertices of P are called nodes of S(P).

15. Properties zIf P is an n-gon, then S(P): yrealizes 2n -3 arcs yrealizes n -2 nodes yDivides P into n monotone polygons

16. Why straight skeleton? zThe straight skeleton has a lower combinatorial complexity than the medial axis for non-convex polygons. yMedial axis has 2n+r –3 arcs (with r parabolically curved) -vs- 2n –3 for straight skeleton

17. Bibliography zO. Aichholzer, F. Aurenhammer, D. Alberts, and B. Gartner. A novel type of skeleton for polygons. Journal of universal computer science, www.iicm.edu/jucs_1_12, Institute for Image Processing and Computer Supported New Media, 1(12):752-761, 1995 www.iicm.edu/jucs_1_12 zO. Aichholzer and F. Aurenhammer, Straight skeletons for general polygonal figures in the plane, Proc.2nd COCOON, Lecture Notes in Computer Science, 1090, Springer-Verlag, Berlin, 1996, pp. 117--126. zP. Felkel, S. Obdrzalek, Straight Skeleton Implementaion, 14th Spring Conference on Computer Graphics (SCCG'98), 210-218, 1998.