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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?]

[Xu, Kineva, O’Rourke 1996, 2000]

General Unfoldings of Convex Polyhedra

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

- Source unfolding
- Star unfolding
- Quasigeodesic unfolding

- 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)

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

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

[Lysyanskaya, O’Rourke 1996]

Is there an algorithm

polynomial time

or efficient numerical algorithm

for finding a closed quasigeodesic on a (convex) polyhedron?