Unfolding convex polyhedra via quasigeodesics
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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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Unfolding Convex Polyhedravia Quasigeodesics

Jin-ichi Ito (Kumamoto Univ.)

Joseph O’Rourke (Smith College)

Costin Vîlcu (S.-S. Romanian Acad.)


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?]


Shortest paths from x to all vertices

[Xu, Kineva, O’Rourke 1996, 2000]


Source Unfolding


Star Unfolding


Star-unfolding of 30-vertex convex polyhedron


General Unfoldings of Convex Polyhedra

  • Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net).

  • Source unfolding

  • Star unfolding

  • Quasigeodesic unfolding


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)


Lyusternick-Schnirelmann Theorem

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


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.


Closed Quasigeodesic

[Lysyanskaya, O’Rourke 1996]


Shortest paths to quasigeodesic do not touch or cross


Insertion of isosceles triangles


Unfolding of Cube


Conjecture


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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