Unfolding Convex Polyhedra via Quasigeodesics

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Unfolding Convex Polyhedravia Quasigeodesics

• Jin-ichi Ito (Kumamoto Univ.)
• Joseph ORourke (Smith College)
• Costin Vîlcu (S.-S. Romanian Acad.)

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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?
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Shortest paths from x to all vertices
Xu, Kineva, ORourke 1996, 2000
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Source Unfolding
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Star Unfolding
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Star-unfolding of 30-vertex convex polyhedron
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General Unfoldings of Convex Polyhedra

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

• Source unfolding
• Star unfolding
• Quasigeodesic unfolding

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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)

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Lyusternick-Schnirelmann Theorem

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

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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.

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Closed Quasigeodesic
Lysyanskaya, ORourke 1996
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Shortest paths to quasigeodesic do not touch or
cross
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Insertion of isosceles triangles
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Unfolding of Cube
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Conjecture
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Conjectures
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Open Find a Closed Quasigeodesic

• Is there an algorithm
• polynomial time
• or efficient numerical algorithm
• for finding a closed quasigeodesic on a (convex)
  polyhedron?