Edge-Unfolding Edge-Unfolding Medial Axis PolyhedraMedial Axis Polyhedra

Joseph O’RourkeJoseph O’Rourke, Smith College, Smith College

Unfolding Convex Polyhedra: Unfolding Convex Polyhedra: Albrecht DAlbrecht Düürer, 1425rer, 1425

Snub Cube

Unfolding PolyhedraUnfolding Polyhedra

Two types of unfoldings: Edge unfoldings: Cut only along edges General unfoldings: Cut through faces too

Cube with truncated cornerCube with truncated corner

Overlap

General Unfoldings of Convex General Unfoldings of Convex PolyhedraPolyhedra

Theorem: Every convex polyhedron has a general nonoverlapping unfolding

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

Ø Star unfolding [Aronov & JOR ’92]

[Poincare 1905?]

Shortest paths from Shortest paths from xx to all vertices to all vertices

[Xu, Kineva, JOR 1996, 2000]

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

Source Unfolding: cut the cut locusSource Unfolding: cut the cut locus

Quasigeodesic Source Unfolding Quasigeodesic Source Unfolding

[IOV07]: Jin-ichi Ito, JOR, Costin Vîlcu, “Unfolding Convex Polyhedra via Quasigeodesics,” 2007.

Conjecture: Cutting the cut locus of a simple, closed quasigeodesic (plus one additional cut) unfolds without overlap.

Special case: Medial Axis Polyhedra

Quasigeodesic Source Unfolding Quasigeodesic Source Unfolding

[IOV07]: Jin-ichi Ito, JOR, Costin Vîlcu, “Unfolding Convex Polyhedra via Quasigeodesics,” 2007.

Conjecture: Cutting the cut locus of a simple, closed quasigeodesic (plus one additional cut) unfolds without overlap.

Special case: Medial Axis Polyhedra

point

Simple, Closed QuasigeodesicSimple, Closed Quasigeodesic

[Lysyanskaya, JOR 1996]

Lyusternick-Schnirelmann Theorem: Lyusternick-Schnirelmann Theorem: 33

A Medial Axis PolyhedronA Medial Axis Polyhedron

Medial axis of a convex polygonMedial axis of a convex polygon

Medial axis = cut locus of ∂PMedial axis = cut locus of ∂P

Medial Axis & M.A. PolyhedronMedial Axis & M.A. Polyhedron

Main TheoremMain Theorem

Unfolding U.Closed, convex region U*.

Could be unbounded.

M(P) = medial axis of P.

Theorem: Each face fi of U nests inside a cell of M(U*).

Medial Axis & M.A. PolyhedronMedial Axis & M.A. Polyhedron

Unfolding: Unfolding: UU**

Unfolding: Overlay with Unfolding: Overlay with M(UM(U**))

Partial Construction of Medial AxisPartial Construction of Medial Axis

Eight UnfoldingsEight Unfoldings

UUn n : U: Un-1n-1

UUnn U Un-1n-1

Bisector rotation

ConclusionConclusion

Theorem: Each face fi of U nests inside a cell of M(U*).

Corollary: U does not overlap. Source unfolding of MAT polyhedron

w.r.t. quasigeodesic base does not overlap.

Questions: Does this hold for “convex caps”?Does this hold more generally? The End