1. Edge-Unfolding Medial Axis Polyhedra Joseph O’Rourke, Smith College

2. Unfolding Convex Polyhedra: Albrecht Dürer, 1425 Snub Cube

3. Unfolding Polyhedra zTwo types of unfoldings: yEdge unfoldings: Cut only along edges yGeneral unfoldings: Cut through faces too

4. Cube with truncated corner Overlap

5. General Unfoldings of Convex Polyhedra zTheorem: Every convex polyhedron has a general nonoverlapping unfolding ØSource unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87] ØStar unfolding [Aronov & JOR ’92] [Poincare 1905?]

6. Shortest paths from x to all vertices [Xu, Kineva, JOR 1996, 2000]

7. Cut locus from x a.k.a., the ridge tree [SS86]

8. Source Unfolding: cut the cut locus

9. Quasigeodesic Source Unfolding z[IOV07]: Jin-ichi Ito, JOR, Costin Vîlcu, “Unfolding Convex Polyhedra via Quasigeodesics,” 2007. zConjecture: Cutting the cut locus of a simple, closed quasigeodesic (plus one additional cut) unfolds without overlap. zSpecial case: Medial Axis Polyhedra

10. Quasigeodesic Source Unfolding z[IOV07]: Jin-ichi Ito, JOR, Costin Vîlcu, “Unfolding Convex Polyhedra via Quasigeodesics,” 2007. zConjecture: Cutting the cut locus of a simple, closed quasigeodesic (plus one additional cut) unfolds without overlap. zSpecial case: Medial Axis Polyhedra point

11. Simple, Closed Quasigeodesic [Lysyanskaya, JOR 1996] Lyusternick-Schnirelmann Theorem:   3

12. A Medial Axis Polyhedron

13. Medial axis of a convex polygon

14. Medial axis = cut locus of ∂P

15. Medial Axis & M.A. Polyhedron

16. Example (in Mathematica) MAT.Polyhedra.nb

17. Main Theorem zUnfolding U. zClosed, convex region U *. yCould be unbounded. zM(P) = medial axis of P. zTheorem: yEach face f i of U nests inside a cell of M(U * ).

18. Medial Axis & M.A. Polyhedron

19. Unfolding: U *

20. Unfolding: Overlay with M(U * )

21. Partial Construction of Medial Axis

22. Eight Unfoldings

24. U n : U n-1

25. U n  U n-1 Bisector rotation

26. Induction Base

27. Sidedness

28. Conclusion Theorem: yEach face f i of U nests inside a cell of M(U * ). Corollary: y U does not overlap. y Source unfolding of MAT polyhedron w.r.t. quasigeodesic base does not overlap. Questions: yDoes this hold for “convex caps”? yDoes this hold more generally? The End