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Part II: Paper c: Skeletons, Roofs, and the Medial Axis

Part II: Paper c: Skeletons, Roofs, and the Medial Axis. Joseph O’Rourke Smith College. Outline. Voronoi Diagram Medial Axis Grassfire Transformation Straight Skeleton Constant-sloped roofs (cf. David Bélanger notes) Properties (cf. Kevin Danaher notes).

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Part II: Paper c: Skeletons, Roofs, and the Medial Axis

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  1. Part II: Paperc: Skeletons, Roofs, and the Medial Axis Joseph O’Rourke Smith College

  2. Outline • Voronoi Diagram • Medial Axis • Grassfire Transformation • Straight Skeleton • Constant-sloped roofs (cf. David Bélanger notes) • Properties (cf. Kevin Danaher notes)

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

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

  5. Straight Skeleton in 1-Cut Thm • Shrink boundary • Handle nonconvex polygons new event when vertex hits opposite edge • Handle 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 (cont’d) • Polygon hierarchy during shrinking

  9. Events • Two events can occur: • Edge event: an edge shrinks to zero, making its neighboring edges adjacent. • Split 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 (cont’d)

  11. Formal Definitions • The 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. • Each edge, e, sweeps out a certain area called the face of e. • Bisector pieces are called arcs, and their endpoints which are not vertices of P are called nodes of S(P).

  12. Properties • If P is an n-gon, then S(P): • realizes 2n -3 arcs • realizes n -2 nodes • Divides P into n monotone polygons

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

  14. Bibliography • O. 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 • O. 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. • P. Felkel, S. Obdrzalek, Straight Skeleton Implementaion, 14th Spring Conference on Computer Graphics (SCCG'98), 210-218, 1998.

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