Geometry of Orthogonal Surfaces CanaDAM 2007 Banff May 28, 2007 - - PowerPoint PPT Presentation

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Geometry of Orthogonal Surfaces CanaDAM 2007 Banff May 28, 2007 - - PowerPoint PPT Presentation

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Geometry of Orthogonal Surfaces CanaDAM 2007 Banff May 28, 2007 Stefan Felsner Technische Universit at Berlin contains joint work with Sarah Kappes and Florian Zickfeld Orthogonal Surfaces The dominance order on IR d : x y x i y

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Geometry of Orthogonal Surfaces

CanaDAM 2007 Banff May 28, 2007 Stefan Felsner Technische Universit¨ at Berlin contains joint work with Sarah Kappes and Florian Zickfeld

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Orthogonal Surfaces

The dominance order on IRd: x ≤ y ⇐ ⇒ xi ≤ yi for i = 1, .., d The orthogonal surface SX generated by a finite X ⊂ IRd is the boundary of the filter X =

  • y ∈ IRd : ∃x ∈ X with y ≥ x
  • .
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An Example in 2-D

A point set X

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An Example in 2-D

The surface SX The filter X

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An Example in 3-D

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Flats and their Features

A Flat is a connected piece of the intersection with an

  • rthogonal hyperplane.

Upper and lower boundary are pieces of orthogonal surfaces

  • f dimension one less.
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Characteristic points

Characteristic points are points incident to flats of all colors. The CP-order is the dominance order on characteristic points.

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More Terminology

Surface SX is generic if every flat has a single minimum. Surface SX is suspended if it has exactly d unbounded flats. Surface SX is rigid if the CP-order is ranked

  • Fact. A generic surface is rigid.
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Connections with Polytopes I

Theorem [Scarf 1979]. The CP-order of a generic suspended orthogonal surface in IRd is isomorphic to the face lattice of a simplicial d-polytope (minus 0, 1 and one facet).

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Connections with Polytopes II

Theorem [Schnyder 1989]. The face lattice of every simplicial 3-polytope (minus 0, 1 and one facet) is the CP-order of a generic suspended

  • rthogonal surfaces in IR3.

Theorem [Felsner 2003]. The face lattice of every 3-polytope (minus 0, 1 and one facet) is the CP-order of a rigid suspended orthogonal surfaces in IR3.

(Implies the Brightwell-Trotter Theorem.)

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Realizability Problems

  • Which orthogonal surfaces in IRd have a corresponding

d-polytope? (Scarf: generic; YES).

  • Which d-polytopes have a corresponding orthogonal

surface in IRd? (Schnyder/F: d = 3; YES).

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Realizability Problems

  • Which orthogonal surfaces in IRd have a corresponding

d-polytope? (Scarf: generic; YES).

  • Which d-polytopes have a corresponding orthogonal

surface in IRd? (Schnyder/F: d = 3; YES).

  • generic suspended orthogonal surface in IRd
  • ←

− simplicial d-polytope.

  • Proof. Neighbourly 4-polytopes have complete graphs

as 2-skeletons, but dim(K13) = 5.

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Bad Surfaces

D = 2, 4, 2, 4 C = 4, 2, 1, 3 B = 3, 1, 3, 2 A = 1, 3, 3, 1 B A

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Good Surfaces

3-D 2-D Theorem [Kappes 06]. If all flats of a surface are generic, cogeneric or parallel, then the extended CP-order is a CW- poset.

  • Conjecture. In this situation the CP-order is polytopal.
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Good Surfaces

Theorem [Kappes 06]. If all flats of a surface are generic, cogeneric or parallel, then the extended CP-order is a CW- poset.

  • Conjecture. In this situation the CP-order is polytopal.
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Good Polytopes

Let a d-polytope P be realizable by an orthogonal surface in IRd

  • If F is a simplicial face and Ps is obtained by stacking

a new vertex above F, then Ps is realizable.

  • If x is a simple vertex and Pc is obtained by cutting x,

then Pc is realizable.

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Good Polytopes

Let a d-polytope P be realizable by an orthogonal surface in IRd

  • If F is a simplicial face and Ps is obtained by stacking

a new vertex above F, then Ps is realizable.

  • If x is a simple vertex and Pc is obtained by cutting x,

then Pc is realizable.

  • If P has a suspended realization, then the pyramid over

P is realizable.

  • If P has a suspended realization, then the product of

P with a path is realizable.

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Part II Planar Graphs and Orthogonal Surfaces in 3-D

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Schnyder Woods

G a 3-connected planar graph with special vertices a1,a2,a3

  • n the outer face.

Axioms for 3-coloring and orientation of edges: (W1 - W2) Rule of edges and half-edges: (W3) Rule of vertices: (W4) No face boundary is a directed cycle in one color.

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Schnyder Woods - Paths and Regions

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Schnyder Woods - Paths and Regions

R1 R2 R3

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Schnyder Woods - Regions

  • If u ∈ Ro

i (v) then Ri(u) ⊂ Ri(v).

  • If u ∈ ∂Ri(v) then Ri(u) ⊆ Ri(v)

(equality, iff there is a bi-directed path between u and v.)

v u v u

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Counting Faces in Schnyder Regions I

φi(v) = # faces in Ri(v). Embed v at (φ1(v), φ2(v)) Theorem. 3-connected planar graphs admit convex drawings on the (f − 1) × (f − 1) grid.

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Counting Faces in Schnyder Regions II

Embed v at (φ1(v), φ2(v), φ3(v))

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Counting Faces in Schnyder Regions II

Embed v at (φ1(v), φ2(v), φ3(v))

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Weighted Count

  • Theorem. Every coplanar orhogonal surface supporting a

Schnyder wood S can be obtained from weighted regions. 1/2 1 1/2 2 2 1/2 1/2

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Non-rigid Surfaces

Counting faces doesn’t yield an order preserving embedding

  • f FG \ F∞ into IR3.
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Relations for Flats

  • Lemma. The arrow-relation on flats of one color is acyclic.
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Shifting Flats

= ⇒ The Brightwell-Trotter Theorem.

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Rigid or Coplanar

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The End

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The End

Thank you.