Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1112.6008

Cayley Configuration Spaces of a Common Class of Mechanisms in Two Dimensions

Meera Sitharam·Menghan Wang·William Sims·Heping Gao
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study Cayley configuration spaces of a class of 1 degree-of-freedom linkages (graphs with specified edge lengths), obtained by dropping an edge from a tree-decomposable graph. The class includes well-known mechanisms based on the four-bar, as well as strandbeest, cardioid, limacon etc. The Cayley configuration space is the set of intervals of attainable lengths for a \emph{base} nonedge (e.g. the dropped edge) over the linkage's 2 dimensional realizations. We require \emph{quadratic radical solvability (QRS)} (an extension of ruler-and-compass-realizability) of the interval endpoints, and tree-decomposability guarantees efficient, ruler-and-compass construction of the linkage realization, given the Cayley configuration. Due to these restrictions of Kempe universality, this class of \emph{low Cayley complexity (LCC)} graphs is common in mechanical computer aided design and kinematics. Our main contributions are the following. (1) We show that the definition of LCC is robust, and depends only on the graph, no matter the choice of base nonedge whose addition ensures tree-decomposability. (2) We give an efficient algorithmic characterization of LCC graphs (3) We show (graph) planarity is equivalent to LCC for a natural subclass of 1-degree-of-freedom tree-decomposable graphs. Counterexamples show impossibility of such finite forbidden minor characterizations when the above subclass is enlarged. (4) We give an easily testable definition of genericity of LCC linkages (i.e. with underlying LCC graphs) based on their edge lengths. (5) For generic LCC linkages, we give an algorithm to find both paths of continuous motion (provided they exist) between two distinct realizations, in time linear in a discrete measure of the length of the path. Nontrivial generalizations of these results to non-LCC, 1-degree-of-freedom tree-decomposable linkages. Several accessible open problems are posed.

Explore related subjects

cs.CG

Keep this discovery

BibTeXRIS

Meera Sitharam, Menghan Wang, William Sims, Heping Gao. 2011-12-27. Cayley Configuration Spaces of a Common Class of Mechanisms in Two Dimensions. https://arxiv.org/abs/1112.6008

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Almost Linear Universal Point Sets for Planar Graphs

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

cs.CG↗
arXiv2026

Some results on Archdeacon's conjecture for rotation systems

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

cs.CG↗
arXiv2026

The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

cs.CG↗