Unfolding Convex Unfolding Convex PolyhedraPolyhedra

via Quasigeodesicsvia Quasigeodesics

Jin-ichi Ito (Kumamoto Univ.)Jin-ichi Ito (Kumamoto Univ.)

Joseph O’Rourke (Smith Joseph O’Rourke (Smith College)College)

Costin VCostin Vîîlcu (S.-S. Romanian lcu (S.-S. Romanian Acad.)Acad.)

TheoremTheorem: Every : Every convexconvex polyhedron polyhedron has a has a generalgeneral nonoverlapping nonoverlapping unfolding (a net).unfolding (a net).

General Unfoldings of General Unfoldings of Convex PolyhedraConvex Polyhedra

Source unfolding [Sharir & Schorr ’86, Source unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87]Mitchell, Mount, Papadimitrou ’87]

Star unfolding [Aronov & JOR ’92]Star unfolding [Aronov & JOR ’92]

[Poincare 1905?]

Shortest paths from Shortest paths from xx to all to all verticesvertices

[Xu, Kineva, O’Rourke 1996, 2000]

Source UnfoldingSource Unfolding

Star UnfoldingStar Unfolding

Star-unfolding of 30-vertex Star-unfolding of 30-vertex convex polyhedronconvex polyhedron

TheoremTheorem: Every : Every convexconvex polyhedron polyhedron has a has a generalgeneral nonoverlapping nonoverlapping unfolding (a net).unfolding (a net).

General Unfoldings of General Unfoldings of Convex PolyhedraConvex Polyhedra

Source unfolding Source unfolding Star unfolding Star unfolding Quasigeodesic unfoldingQuasigeodesic unfolding

Geodesics & Closed Geodesics & Closed GeodesicsGeodesics

GeodesicGeodesic: locally shortest path; : locally shortest path; straightest lines on surfacestraightest lines on surface

Simple geodesicSimple geodesic: non-self-: non-self-intersectingintersecting

Simple, Simple, closed geodesicclosed geodesic:: Closed geodesic: returns to start w/o Closed geodesic: returns to start w/o

cornercorner (Geodesic loop: returns to start at (Geodesic loop: returns to start at

corner)corner)

Lyusternick-Schnirelmann Lyusternick-Schnirelmann TheoremTheorem

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

QuasigeodesicQuasigeodesic

Aleksandrov 1948Aleksandrov 1948 left(p) = total incident face angle left(p) = total incident face angle

from leftfrom left quasigeodesic: curve s.t. quasigeodesic: curve s.t.

left(p) ≤ left(p) ≤ right(p) ≤ right(p) ≤

at each point p of curve.at each point p of curve.

Closed QuasigeodesicClosed Quasigeodesic

[Lysyanskaya, O’Rourke 1996]

Shortest paths to Shortest paths to quasigeodesic do not touch quasigeodesic do not touch

or crossor cross

Insertion of isosceles Insertion of isosceles trianglestriangles

Unfolding of CubeUnfolding of Cube

ConjectureConjecture

BaseBase Source Source UnfoldingUnfolding

Star Star UnfoldingUnfolding

pointpoint theoremtheorem theoremtheorem

quasigeodesicquasigeodesic ?? theoremtheorem

Open: Find a Closed Open: Find a Closed QuasigeodesicQuasigeodesic

Is there an algorithmIs there an algorithmpolynomial timepolynomial time

or efficient numerical algorithmor efficient numerical algorithm

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