## Combinatorial Classification of 2D Underconstrained Systems (Abstract) (2005)

Citations: | 2 - 2 self |

### BibTeX

@MISC{Gao05combinatorialclassification,

author = {Heping Gao and Meera Sitharam},

title = {Combinatorial Classification of 2D Underconstrained Systems (Abstract)},

year = {2005}

}

### OpenURL

### Abstract

Approaches for characterizing, classifying, decomposing, solving and navigating the solution set of generically wellconstrained geometric constraint systems have been studied extensively, both in 2D [16, 8, 9, 10, 15, 13] and in 3D [18, 4]. Significant progress has also been made in understanding generically overconstrained systems [14, 7]. However, while the study of underconstrained systems is acknowledged to be important and crucial for

### Citations

195 |
On graphs and rigidity of plane skeletal structures
- Laman
- 1970
(Show Context)
Citation Context ... Approaches for characterizing, classifying, decomposing, solving and navigating the solution set of generically wellconstrained geometric constraint systems have been studied extensively, both in 2D =-=[16, 8, 9, 10, 15, 13]-=- and in 3D [18, 4]. Significant progress has also been made in understanding generically overconstrained systems [14, 7]. However, while the study of underconstrained systems is acknowledged to be imp... |

113 | Conditions for unique graph realizations
- Hendrickson
- 1992
(Show Context)
Citation Context ...ric constraint systems have been studied extensively, both in 2D [16, 8, 9, 10, 15, 13] and in 3D [18, 4]. Significant progress has also been made in understanding generically overconstrained systems =-=[14, 7]-=-. However, while the study of underconstrained systems is acknowledged to be important and crucial for both the classical CAD, Robotics and newer molecular modeling applications of geometric constrain... |

57 | Solving geometric constraint systems
- Kramer
- 1992
(Show Context)
Citation Context ... Approaches for characterizing, classifying, decomposing, solving and navigating the solution set of generically wellconstrained geometric constraint systems have been studied extensively, both in 2D =-=[16, 8, 9, 10, 15, 13]-=- and in 3D [18, 4]. Significant progress has also been made in understanding generically overconstrained systems [14, 7]. However, while the study of underconstrained systems is acknowledged to be imp... |

31 | Correctness proof of a geometric constraint solver
- Fudos, CM
- 1993
(Show Context)
Citation Context ... G that have a a DR-plan of size at most b1(|G|) and relate these classes to each other. Give an efficient algorithm to recognize the graphs in such a class. For example, triangle decomposable graphs =-=[5]-=-, Henneberg 1 graphs [16], and quadratically solvable graphs [10, 11, 12] are such classes. In general, choosing a particular algorithm or method A for finding DR-plans, characterize the class CA of w... |

22 | Solving Geometric Constraint Systems II: A Symbolic Approach and Decision of Reconstructability
- Gao, Chou
- 1998
(Show Context)
Citation Context ... Approaches for characterizing, classifying, decomposing, solving and navigating the solution set of generically wellconstrained geometric constraint systems have been studied extensively, both in 2D =-=[16, 8, 9, 10, 15, 13]-=- and in 3D [18, 4]. Significant progress has also been made in understanding generically overconstrained systems [14, 7]. However, while the study of underconstrained systems is acknowledged to be imp... |

16 |
Solving Geometric Constraint Systems I: A Global Propagation Approach
- Gao, Chou
- 1998
(Show Context)
Citation Context |

13 |
Geometric Constraint Solving
- Fudos
- 1995
(Show Context)
Citation Context |

12 |
2001b. Decomposition of geometric constraints systems. Part II: New algorithms
- Hoffmann, Lomonosov, et al.
(Show Context)
Citation Context ... systems, most algorithms - for both the existence and the navigation problems - deal with two distinct subproblems. The first subproblem (a) is to obtain a so-called Decomposition-Recombination plan =-=[1]-=-, which is a purely combinatorial object that involves only the constraint graph G. A DR-plan DG for a well-constrained graph G can be viewed as an efficient combinatorial description or roadmap of a ... |

12 |
Geometric Constraint Solving and Applications
- Brüderlin, Roller
- 1998
(Show Context)
Citation Context |

12 |
Graph based geometric constraint solving: problems, progress and directions. AMS-DIMACS volume on Computer Aided Design
- Sitharam
- 2005
(Show Context)
Citation Context ...lassifying, decomposing, solving and navigating the solution set of generically wellconstrained geometric constraint systems have been studied extensively, both in 2D [16, 8, 9, 10, 15, 13] and in 3D =-=[18, 4]-=-. Significant progress has also been made in understanding generically overconstrained systems [14, 7]. However, while the study of underconstrained systems is acknowledged to be important and crucial... |

8 |
Making constraint solvers more useable: The overconstraint problem
- Hoffman, Sitharam, et al.
- 2004
(Show Context)
Citation Context ...ric constraint systems have been studied extensively, both in 2D [16, 8, 9, 10, 15, 13] and in 3D [18, 4]. Significant progress has also been made in understanding generically overconstrained systems =-=[14, 7]-=-. However, while the study of underconstrained systems is acknowledged to be important and crucial for both the classical CAD, Robotics and newer molecular modeling applications of geometric constrain... |

8 | The nonsolvability by radicals of generic 3-connected planar graphs
- Owen, Power
(Show Context)
Citation Context ...e classes to each other. Give an efficient algorithm to recognize the graphs in such a class. For example, triangle decomposable graphs [5], Henneberg 1 graphs [16], and quadratically solvable graphs =-=[10, 11, 12]-=- are such classes. In general, choosing a particular algorithm or method A for finding DR-plans, characterize the class CA of well-constrained graphs G for which A finds a DR-plan of size at most b1(|... |

6 |
Graph and Combinatorial Analysis for Geometric Constraint Graphs
- Lomonosov
- 2004
(Show Context)
Citation Context ...for finding the realization, given the DR-plan DG: the definition of the DR-plan’s size would vary accordingly. However, for most common notions of size, finding an optimally sized DR-plan is NP-hard =-=[17]-=- even for the 2D distance constraint systems being considered here. In other words, the efficiency of description of the candidate solution set is directly related to the complexity of the subproblem ... |

3 |
Vila-Marta and Josep Vilaplana-Pasto. Transforming an under-constrained geometric constraint problem into a well-constrained one
- Joan-Arinyo, Soto-Riera, et al.
- 2003
(Show Context)
Citation Context ...te classes of underconstrained graphs G for which there is a set of edges U for which G ∪ U has a small DR-plan, is triangle decomposable, Henneberg 1, quadratically solvable etc. For one such class, =-=[3]-=- gives an algorithm for finding such edges. However, we observe that, without further qualification, the picture is not rosy for general 2D distance constraint graphs. We can find underconstrained gra... |

2 |
JianHua Fan and Meera Sitharam and Yong Zhou. Elimination in generically rigid 3d geometric constraint systems
- Peters
- 2004
(Show Context)
Citation Context ... finds real solutions. This is a combinatorial measure of size of the DR-plan that captures the complexity of subproblem (b). Finer such combinatorial measures of algebraic complexity can be found in =-=[6]-=-. In this context, the complexity of subproblem (b) is defined for the worst case tuple d and is hence treated as a combinatorial property dependent only on the constraint graph G. This complexity cou... |

1 |
Are all 3-connected generic constriant configurations of points on a plane non-radical
- Owen, Power
- 2004
(Show Context)
Citation Context ...e classes to each other. Give an efficient algorithm to recognize the graphs in such a class. For example, triangle decomposable graphs [5], Henneberg 1 graphs [16], and quadratically solvable graphs =-=[10, 11, 12]-=- are such classes. In general, choosing a particular algorithm or method A for finding DR-plans, characterize the class CA of well-constrained graphs G for which A finds a DR-plan of size at most b1(|... |