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Unfolding Convex Polyhedra via Quasigeodesics

Unfolding Convex Polyhedra via Quasigeodesics. Jin-ichi Ito (Kumamoto Univ.) Joseph O’Rourke (Smith College) Costin V î lcu (S.-S. Romanian Acad.). General Unfoldings of Convex Polyhedra. Theorem : Every convex polyhedron has a general nonoverlapping unfolding (a net).

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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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