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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). - PowerPoint PPT Presentation
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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?