Unfolding convex polyhedra via quasigeodesics
This presentation is the property of its rightful owner.
Sponsored Links
1 / 21

Unfolding Convex Polyhedra via Quasigeodesics PowerPoint PPT Presentation


  • 57 Views
  • Uploaded on
  • Presentation posted in: General

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).

Download Presentation

Unfolding Convex Polyhedra via Quasigeodesics

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Unfolding convex polyhedra via quasigeodesics

Unfolding Convex Polyhedravia Quasigeodesics

Jin-ichi Ito (Kumamoto Univ.)

Joseph O’Rourke (Smith College)

Costin Vîlcu (S.-S. Romanian Acad.)


Unfolding convex polyhedra via quasigeodesics

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?]


Shortest paths from x to all vertices

Shortest paths from x to all vertices

[Xu, Kineva, O’Rourke 1996, 2000]


Source unfolding

Source Unfolding


Star unfolding

Star Unfolding


Star unfolding of 30 vertex convex polyhedron

Star-unfolding of 30-vertex convex polyhedron


Unfolding convex polyhedra via quasigeodesics

General Unfoldings of Convex Polyhedra

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

  • Source unfolding

  • Star unfolding

  • Quasigeodesic unfolding


Geodesics closed geodesics

Geodesics & Closed Geodesics

  • 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)


Lyusternick schnirelmann theorem

Lyusternick-Schnirelmann Theorem

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


Quasigeodesic

Quasigeodesic

  • Aleksandrov 1948

  • left(p) = total incident face angle from left

  • quasigeodesic: curve s.t.

    • left(p) ≤ 

    • right(p) ≤ 

      at each point p of curve.


Closed quasigeodesic

Closed Quasigeodesic

[Lysyanskaya, O’Rourke 1996]


Shortest paths to quasigeodesic do not touch or cross

Shortest paths to quasigeodesic do not touch or cross


Insertion of isosceles triangles

Insertion of isosceles triangles


Unfolding of cube

Unfolding of Cube


Conjecture

Conjecture


Open find a closed quasigeodesic

Open: Find a Closed Quasigeodesic

Is there an algorithm

polynomial time

or efficient numerical algorithm

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


  • Login