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Unfolding Convex Polyhedra via Quasigeodesics

This work investigates the general unfoldings of convex polyhedra, establishing that every convex polyhedron has a general nonoverlapping unfolding arrangement, or net. It delves into various unfolding techniques, including source unfolding and star unfolding, with connections to quasigeodesics. The study highlights relevant theorems, such as the existence of distinct closed geodesics on closed surfaces homeomorphic to a sphere. It also poses an open conjecture regarding the identification of closed quasigeodesics on convex polyhedra, aiming to highlight efficient polynomial-time algorithms for their discovery.

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Unfolding Convex Polyhedra via Quasigeodesics

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

  2. General Unfoldings of Convex Polyhedra • Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). • 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]

  4. Source Unfolding

  5. Star Unfolding

  6. Star-unfolding of 30-vertex convex polyhedron

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

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

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

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

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

  12. Shortest paths to quasigeodesic do not touch or cross

  13. Insertion of isosceles triangles

  14. Unfolding of Cube

  15. Conjecture

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

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