1. Unfolding Convex Polyhedra via Quasigeodesics Jin-ichi Ito (Kumamoto Univ.) Joseph O’Rourke (Smith College) Costin Vîlcu (S.-S. Romanian Acad.)

2. Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). General Unfoldings of Convex Polyhedra  Source unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87]  Star unfolding [Aronov & JOR ’92] [Poincare 1905?]

3. Shortest paths from x to all vertices [Xu, Kineva, O’Rourke 1996, 2000]

5. Source Unfolding

6. Star Unfolding

7. Star-unfolding of 30-vertex convex polyhedron

8. Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). General Unfoldings of Convex Polyhedra  Source unfolding  Star unfolding  Quasigeodesic unfolding

9. Geodesics & Closed Geodesics Geodesic: locally shortest path; straightest lines on surface Geodesic: locally shortest path; straightest lines on surface Simple geodesic: non-self-intersecting Simple geodesic: non-self-intersecting Simple, closed geodesic: Simple, closed geodesic: Closed geodesic: returns to start w/o corner Closed geodesic: returns to start w/o corner (Geodesic loop: returns to start at corner) (Geodesic loop: returns to start at corner)

10. Lyusternick-Schnirelmann Theorem Theorem: Every closed surface homeomorphic to a sphere has at least three, distinct closed geodesics.

11. Quasigeodesic Aleksandrov 1948 Aleksandrov 1948 left(p) = total incident face angle from left left(p) = total incident face angle from left quasigeodesic: curve s.t. quasigeodesic: curve s.t. left(p) ≤  left(p) ≤  right(p) ≤  right(p) ≤  at each point p of curve. at each point p of curve.

12. Closed Quasigeodesic [Lysyanskaya, O’Rourke 1996]

14. Shortest paths to quasigeodesic do not touch or cross

16. Insertion of isosceles triangles

17. Unfolding of Cube

20. Conjecture Base Source Unfolding Star Unfolding pointtheoremtheorem quasigeodesic?theorem

21. Conjectures Base Source Unfolding Star Unfolding pointtheoremtheorem Quasigeodesic?theorem Face??

22. Open: Find a Closed Quasigeodesic Is there an algorithm polynomial time or efficient numerical algorithm for finding a closed quasigeodesic on a (convex) polyhedron?