arXiv subjects

Heping Gao

Publications and source records attributed to Heping Gao.

4 recordsLinked to original sources

arXiv2012

Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part II: Combinatorial Characterization of Complexity

We continue to study Cayley configuration spaces of 1-dof linkages in 2D begun in Part I of this paper, i.e. the set of attainable lengths for a non-edge. In Part II, we focus on the algebraic complexity of describing endpoints of the intervals in the set, i.e., the Cayley complexity. Specifically, We focus on Cayley configuration spaces of a natural class of 1-dof linkages, called 1-dof tree-decomposable linkages. The underlying graphs G satisfy the following: for some base non-edge f, G \cup f is quadratic-radically solvable (QRS), meaning that G \cup f is minimally rigid, and given lengths \bar{l} of all edges, the corresponding linkage (G \cup f, \bar{l}) can be simply realized by ruler and compass starting from f. It is clear that the Cayley complexity only depends on the graph G and possibly the non-edge f. Here we ask whether the Cayley complexity depends on the choice of a base non-edge f. We answer this question in the negative, thereby showing that low Cayley complexity is a property of the graph G (independent of the non-edge f). Then, we give a simple characterization of graphs with low Cayley complexity, leading to an efficient algorithmic characterization, i.e. an efficient algorithm for recognizing such graphs. Next, we show a surprising result that (graph) planarity is equivalent to low Cayley complexity for a natural subclass of 1-dof triangle-decomposable graphs. While this is a finite forbidden minor graph characterization of low Cayley complexity, we provide counterexamples showing impossibility of such finite forbidden minor characterizations when the above subclass is enlarged.

cs.CG↗

arXiv2011

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

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.

cs.CG↗

arXiv2009

Characterizing graphs with convex and connected configuration spaces

We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS), which includes Linkages and Frameworks. Each representation corresponds to a choice of Cayley parameters and yields a different parametrized configuration space. Significantly, we give purely graph-theoretic, forbidden minor characterizations that capture (i) the class of graphs that always admit efficient configuration spaces and (ii) the possible choices of representation parameters that yield efficient configuration spaces for a given graph. In addition, our results are tight: we show counterexamples to obvious extensions. This is the first step in a systematic and graded program of combinatorial characterizations of efficient configuration spaces. We discuss several future theoretical and applied research directions. Some of our proofs employ an unusual interplay of (a) classical analytic results related to positive semi-definiteness of Euclidean distance matrices, with (b) recent forbidden minor characterizations and algorithms related to the notion of d-realizability of EDCS. We further introduce a novel type of restricted edge contraction or reduction to a graph minor, a "trick" that we anticipate will be useful in other situations.

cs.CG↗

arXiv2008

Characterizing 1-Dof Henneberg-I graphs with efficient configuration spaces

We define and study exact, efficient representations of realization spaces of a natural class of underconstrained 2D Euclidean Distance Constraint Systems(EDCS) or Frameworks based on 1-dof Henneberg-I graphs. Each representation corresponds to a choice of parameters and yields a different parametrized configuration space. Our notion of efficiency is based on the algebraic complexities of sampling the configuration space and of obtaining a realization from the sample (parametrized) configuration. Significantly, we give purely combinatorial characterizations that capture (i) the class of graphs that have efficient configuration spaces and (ii) the possible choices of representation parameters that yield efficient configuration spaces for a given graph. Our results automatically yield an efficient algorithm for sampling realizations, without missing extreme or boundary realizations. In addition, our results formally show that our definition of efficient configuration space is robust and that our characterizations are tight. We choose the class of 1-dof Henneberg-I graphs in order to take the next step in a systematic and graded program of combinatorial characterizations of efficient configuration spaces. In particular, the results presented here are the first characterizations that go beyond graphs that have connected and convex configuration spaces.

cs.CG↗