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Angular constraints on planar frameworks

Sean Dewar, Georg Grasegger, Anthony Nixon, Zvi Rosen, William Sims, Meera Sitharam, David Urizar

Abstract

Consider a collection of points and the sets of slopes or directions of the lines between pairs of points. It is known that the algebraic matroid on this set of elements is the well studied 2-dimensional rigidity matroid. This article analyzes a construction on top of the set of slopes given by an angle constraint system of incidences and angles. In this setting we provide a matricial rigidity formulation of the problem for colored graphs, an algebro-geometric reformulation, precise necessary conditions and a combinatorial characterization of the generic behaviour for a special case.

Angular constraints on planar frameworks

Abstract

Consider a collection of points and the sets of slopes or directions of the lines between pairs of points. It is known that the algebraic matroid on this set of elements is the well studied 2-dimensional rigidity matroid. This article analyzes a construction on top of the set of slopes given by an angle constraint system of incidences and angles. In this setting we provide a matricial rigidity formulation of the problem for colored graphs, an algebro-geometric reformulation, precise necessary conditions and a combinatorial characterization of the generic behaviour for a special case.
Paper Structure (16 sections, 23 theorems, 52 equations, 5 figures, 3 tables)

This paper contains 16 sections, 23 theorems, 52 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Previous work on angles
  3. Analogous cases in the literature
  4. Mathematical setting
  5. Angle rigidity
  6. The angle-rigidity matrix
  7. Infinitesimally angle-rigid frameworks
  8. Angle-rigid graphs
  9. Algebraic matroid formulation
  10. A necessary condition for minimal angle-rigidity
  11. Extension moves for angle-rigid graphs
  12. Computational results
  13. Conjecture, strategies, and partial results
  14. A color-swap move
  15. The picture variety and partitioning vertices
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $(G,c,p)$ be a line-angle incidence structure. If there exist vertices $v,w \in V$ where $p_v\neq p_w$, then the following vectors form a basis of the trivial infinitesimal flexes of $(G,c,p)$:

Figures (5)

  • Figure 1: $K_4$ is dependent in $\mathcal{R}_2$, i. e. the distance / slope of one of the pairs of points (vertices) is dependent on the distances / slopes of the remaining pairs, however, the shown set of angles is independent.
  • Figure 2: Colored triangles $(K_3,c)$ and $(K_3,c')$.
  • Figure 5: Counterexample graphs.
  • Figure 6: The 5 non-isomorphic bichromatic colorings of $K_4$.
  • Figure 7: Two graphs with 5 vertices that are 2-color-rigid.

Theorems & Definitions (49)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Theorem 2.8
  • ...and 39 more