A decision-support method for information inconsistency resolution in direct modeling of CAD models

Direct modeling is a very recent CAD paradigm that can provide unprecedented modeling flexibility. It, however, lacks the parametric capability, which is indispensable to modern CAD systems. For direct modeling to have this capability, an additional associativity information layer in the form of geometric constraint systems needs to be incorporated into direct modeling. This is no trivial matter due to the possible inconsistencies between the associativity information and geometry information in a model after direct edits. The major issue of resolving such inconsistencies is that there often exist many resolution options. The challenge lies in avoiding invalid resolution options and prioritizing valid ones. This paper presents an effective method to support the user in making decisions among the resolution options. In particular, the method can provide automatic information inconsistency reasoning, avoid invalid resolution options completely, and guide the choice among valid resolution options. Case studies and comparisons have been conducted to demonstrate the effectiveness of the method.Comment: 20 pages, 14 figure

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

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

Cayley configuration spaces of 2D mechanisms, Part I: extreme points, continuous motion paths and minimal representations

We consider longstanding questions concerning configuration spaces of 1-dof tree-decomposable linkages in 2D. By employing the notion Cayley configuration space, i.e., a set of intervals of realizable distance-values for an independent non-edge, we answer the following. (1) How to measure the complexity of the configuration space and efficiently compute that of low algebraic complexity? (2) How to restrict the Cayley configuration space to be a single interval? (3) How to efficiently obtain continuous motion paths between realizations? (4) How to bijectively represent of the Cartesian realization space as a curve in an ambient space of minimum dimension? (5) How robust is the complexity measure (1) and how to efficiently classify linkages according to it? In Part I of this paper, we deal with problems (1)-(4) by introducing the notions of (a) Cayley size, the number of intervals in the Cayley configuration space, (b) Cayley computational complexity of computing the interval endpoints, and (c) Cayley (algebraic) complexity of describing the interval endpoints. Specifically (i) We give an algorithm to find the interval endpoints of a Cayley configuration spac. For graphs with low Cayley complexity, we give the following. (ii) A natural, minimal set of local orientations, whose specification guarantees Cayley size of 1 and $O(|V|^2)$ Cayley computational complexity. Specifying fewer local orientations results in a superpolynomial blow-up of both Cayley size and computational complexity, provided P is different from NP. (iii) An algorithm--for generic linkages--to find a path of continuous motion (provided exists) between two given realizations, in time linear in a natural measure of path length. (iv) A canonical bijective representation of the Cartesian realization space in minimal ambient dimension, also for generic linkages