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Edge-Unfolding Medial Axis Polyhedra

Edge-Unfolding Medial Axis Polyhedra. Joseph O’Rourke , Smith College. Unfolding Convex Polyhedra: Albrecht D ü rer, 1425. Snub Cube. Unfolding Polyhedra. Two types of unfoldings: Edge unfoldings : Cut only along edges General unfoldings : Cut through faces too.

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Edge-Unfolding Medial Axis Polyhedra

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

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

  3. Unfolding Polyhedra • Two types of unfoldings: • Edge unfoldings: Cut only along edges • General unfoldings: Cut through faces too

  4. Cube with truncated corner Overlap

  5. General Unfoldings of Convex Polyhedra • Theorem: Every convex polyhedron has a general nonoverlapping unfolding • Source unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87] • Star 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 • [IOV07]: Jin-ichi Ito, JOR, Costin Vîlcu, “Unfolding Convex Polyhedra via Quasigeodesics,” 2007. • Conjecture: Cutting the cut locus of a simple, closed quasigeodesic (plus one additional cut) unfolds without overlap. • Special case: Medial Axis Polyhedra

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

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

  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. Main Theorem • Unfolding U. • Closed, convex region U*. • Could be unbounded. • M(P) = medial axis of P. • Theorem: • Each face fi of U nests inside a cell of M(U*).

  17. Medial Axis & M.A. Polyhedron

  18. Unfolding: U*

  19. Unfolding: Overlay with M(U*)

  20. Partial Construction of Medial Axis

  21. Eight Unfoldings

  22. Un : Un-1

  23. Un Un-1 Bisector rotation

  24. Conclusion Theorem: • Each face fi of U nests inside a cell of M(U*). Corollary: • U does not overlap. • Source unfolding of MAT polyhedron w.r.t. quasigeodesic base does not overlap. Questions: • Does this hold for “convex caps”? • Does this hold more generally? The End

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