Edge-Unfolding Medial Axis Polyhedra Joseph O’Rourke, Smith College.

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Presentation transcript:

Edge-Unfolding Medial Axis Polyhedra Joseph O’Rourke, Smith College

Unfolding Convex Polyhedra: Albrecht Dürer, 1425 Snub Cube

Unfolding Polyhedra zTwo types of unfoldings: yEdge unfoldings: Cut only along edges yGeneral unfoldings: Cut through faces too

Cube with truncated corner Overlap

General Unfoldings of Convex Polyhedra zTheorem: Every convex polyhedron has a general nonoverlapping unfolding ØSource unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87] ØStar unfolding [Aronov & JOR ’92] [Poincare 1905?]

Shortest paths from x to all vertices [Xu, Kineva, JOR 1996, 2000]

Cut locus from x a.k.a., the ridge tree [SS86]

Source Unfolding: cut the cut locus

Quasigeodesic Source Unfolding z[IOV07]: Jin-ichi Ito, JOR, Costin Vîlcu, “Unfolding Convex Polyhedra via Quasigeodesics,” zConjecture: Cutting the cut locus of a simple, closed quasigeodesic (plus one additional cut) unfolds without overlap. zSpecial case: Medial Axis Polyhedra

Quasigeodesic Source Unfolding z[IOV07]: Jin-ichi Ito, JOR, Costin Vîlcu, “Unfolding Convex Polyhedra via Quasigeodesics,” zConjecture: Cutting the cut locus of a simple, closed quasigeodesic (plus one additional cut) unfolds without overlap. zSpecial case: Medial Axis Polyhedra point

Simple, Closed Quasigeodesic [Lysyanskaya, JOR 1996] Lyusternick-Schnirelmann Theorem:   3

A Medial Axis Polyhedron

Medial axis of a convex polygon

Medial axis = cut locus of ∂P

Medial Axis & M.A. Polyhedron

Example (in Mathematica) MAT.Polyhedra.nb

Main Theorem zUnfolding U. zClosed, convex region U *. yCould be unbounded. zM(P) = medial axis of P. zTheorem: yEach face f i of U nests inside a cell of M(U * ).

Medial Axis & M.A. Polyhedron

Unfolding: U *

Unfolding: Overlay with M(U * )

Partial Construction of Medial Axis

Eight Unfoldings

U n : U n-1

U n  U n-1 Bisector rotation

Induction Base

Sidedness

Conclusion Theorem: yEach face f i of U nests inside a cell of M(U * ). Corollary: y U does not overlap. y Source unfolding of MAT polyhedron w.r.t. quasigeodesic base does not overlap. Questions: yDoes this hold for “convex caps”? yDoes this hold more generally? The End