Igamberdiev, Alexander: A duality transform for realizing convex polytopes with small integer coordinates. 2016
Inhalt
- Introduction
- Methods for realizing planar graphs: overview
- Efficient duality transforms in R3
- Preliminaries
- Equilibrium stresses
- Maxwell–Cremona lifting and the canonical equilibrium stress
- Wheel-decomposition for equilibrium stresses
- An efficient reverse of the Maxwell–Cremona lifting
- Braced stresses
- Definition
- Canonical braced stresses
- Braced stresses and equilibrium stresses: flat embeddings
- Braced stresses and stresses of braced graphs
- Scalability of braced stresses
- Projective equivalence with equilibrium stresses
- Free lunch: wheel-decomposition for braced stresses
- Free lunch: projective properties of equilibrium stresses
- Canonical braced stresses as Hessians of volume
- Orthogonal projection of the canonical braced stresses
- Lovász lifting procedure
- Lovász dual transform
- Lovász lifting procedure: convex case
- Lovász lifting procedure: general case
- Geometry of the wheel-decomposition
- Geometry of the wheel-decomposition: equilibrium stresses
- Duality transforms
- Prismatoid
- Efficient duality transforms in Rd
- Preliminaries
- Combinatorics and notation of d-polytopes: overview
- Conventions on orientations
- Orientation on faces of a polytope
- Volumes
- Realizations of polytopes
- Approaches to stresses in higher dimensions
- Equilibrium stress
- Braced stress
- From equilibrium stresses to braced stresses and back
- Scalability of braced stresses
- Canonical braced stress
- Maxwell–Cremona lifting
- Maxwell–Cremona lifting
- Canonical equilibrium stress
- Full version of Maxwell–Cremona lifting theorem
- Creasing formula for canonical equilibrium stress
- Lovász construction in Rd
- Realizations of truncated polytopes
- Truncated and stacked polytopes
- High level description of the algorithm
- Step 1: Realizations of stacked polytopes
- Step 2: Stacking of the missing vertex
- Step 3: Realizing truncated polytopes
- Duality transform for general simplicial polytopes
- Conclusion