## COMBINATORIAL DECOMPOSITION, GENERIC INDEPENDENCE AND ALGEBRAIC COMPLEXITY OF GEOMETRIC CONSTRAINTS SYSTEMS: APPLICATIONS IN BIOLOGY AND ENGINEERING (2006)

Citations: | 4 - 1 self |

### BibTeX

@MISC{Zhou06combinatorialdecomposition,,

author = {Yong Zhou},

title = {COMBINATORIAL DECOMPOSITION, GENERIC INDEPENDENCE AND ALGEBRAIC COMPLEXITY OF GEOMETRIC CONSTRAINTS SYSTEMS: APPLICATIONS IN BIOLOGY AND ENGINEERING },

year = {2006}

}

### OpenURL

### Abstract

### Citations

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Citation Context ...we discuss how the graph theoretic properties degree of freedom (dof) analysis relate to corresponding properties of the corresponding constraint system. In 2 dimensions, according to Laman’s theorem =-=[18]-=-, if all geometric objects are points and all constraints are distance constraints between these points then any minimal dense cluster represents a generically rigid system. In general, however, while... |

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Citation Context ...neous coordinate matrices obtained from each set of D + 1 point objects. I.e, these constraints are D + 1 degree polynomial inequalities. These correspond to a complete oriented matroid specification =-=[51]-=-. Alternatively, a partial oriented matroid specification could be specified by the user: these assert intersection or separations88 of pairs of convex hulls of subsets of point objects (called respec... |

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Citation Context ...t tractable to count the solution types by solving it. It is desirable to have a combinatorial method to count the solution types and thereafter to get the probabilities of subassemblies. Hendrickson =-=[30]-=- gives a combinatorial method to get conditions for unique graph realization. We propose to extend its method to solve our problem based on the highly overconstrained property of the viral shell.sREFE... |

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Citation Context ...o approaches can be developed which leverages these advantages while incorporating the full generality of the model [25, 29]. While there is a well developed structure theory of complete viral shells =-=[57, 58]-=-, verified by X-ray crystallography and other experimental data, the processes of viral shell assembly are poorly understood. From an experimental point of view, this lack of understanding is due to t... |

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Citation Context ... home cluster to be fixed. Since C is assumed to be wellconstrained, the number of variables and equations equals the total edge weight of the the spanning tree. Most sparse polynomial system solvers =-=[52, 53]-=- (geometric constraint systems are sparse) take time exponential in the number of variables. Hence the overwhelming factor in the algebraic complexity of the system is the number of variables. Degree ... |

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Citation Context ...n as well as several performance measures of DR-planners some of which are relevant to this. It additionally gives a table of comparisons which shows that many of the previous DR-planners for example =-=[3, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]-=-, (or any obvious modifications of them) would inherently fail to incorporate even tree-like input design decompositions orsfeature hierarchies. We explain in Section 3.1 what the difficulty is in the... |

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Citation Context ...n as well as several performance measures of DR-planners some of which are relevant to this. It additionally gives a table of comparisons which shows that many of the previous DR-planners for example =-=[3, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]-=-, (or any obvious modifications of them) would inherently fail to incorporate even tree-like input design decompositions orsfeature hierarchies. We explain in Section 3.1 what the difficulty is in the... |

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Citation Context ...They arise in applications such as mechanical computer aided design, robotics, molecular modeling and teaching geometry. For recent reviews of the extensive literature on geometric constraint solving =-=[2, 3, 4, 6]-=-. A geometric constraint system relates a finite set of primitive geometric objects (points, lines, line segments, conics, and so on) by a finite set of constraints on distance, angle, incidence, tang... |

40 | A brief on constraint solving
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Citation Context ...n as well as several performance measures of DR-planners some of which are relevant to this. It additionally gives a table of comparisons which shows that many of the previous DR-planners for example =-=[3, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]-=-, (or any obvious modifications of them) would inherently fail to incorporate even tree-like input design decompositions orsfeature hierarchies. We explain in Section 3.1 what the difficulty is in the... |

38 |
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37 | Rigidity and scene analysis
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Citation Context ...nge” problem [20, 21]. To date,sthere is no known, tractable, combinatorial characterization of generic rigidity of systems for 3 or higher dimensions even when only points and distances are involved =-=[19, 22]-=-, although several conjectures exist. There are no known general combinatorial characterizations of 2D rigidity, when other constraints besides distances (such as angles) are involved. For constraint ... |

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25 | Finding Solvable Subsets of Constraint Graphs
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Citation Context ... is the parameter associated with the constraint. Most of the constraint solvers so far deal with 2D constraint systems. With the exception of work related to the FRONTIER geometric constraint solver =-=[1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]-=-, to the best of our knowledge, work on stand-alone 3D geometric constraint solvers is relatively sparse [16, 17]. A solution or realization of a geometric constraint system is the (set of) real zero(... |

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Citation Context ...rmediate sub-assemblies are generally unsuccessful. From a modeling point of view, this lack of understanding is due to the fact that prior to the recent model [25, 29], previous computational models =-=[59, 60, 61, 62, 63, 64, 65, 66, 67]-=- generally involve dynamics of (simplified versions) of virus assembly (further description of these approaches and comparison with the approach [25, 29], is given by Sitharam and Mckenna [25]). Dynam... |

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Citation Context ...rmediate sub-assemblies are generally unsuccessful. From a modeling point of view, this lack of understanding is due to the fact that prior to the recent model [25, 29], previous computational models =-=[59, 60, 61, 62, 63, 64, 65, 66, 67]-=- generally involve dynamics of (simplified versions) of virus assembly (further description of these approaches and comparison with the approach [25, 29], is given by Sitharam and Mckenna [25]). Dynam... |

18 |
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Citation Context ... home cluster to be fixed. Since C is assumed to be wellconstrained, the number of variables and equations equals the total edge weight of the the spanning tree. Most sparse polynomial system solvers =-=[52, 53]-=- (geometric constraint systems are sparse) take time exponential in the number of variables. Hence the overwhelming factor in the algebraic complexity of the system is the number of variables. Degree ... |

15 |
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Citation Context ...ic constraint solving can in fact be achieved as a special case, where the order is a complete, total order. More generally, this can be used in the CAD database maintenance of multiple product views =-=[46, 47]-=-, for example the design view and a downstream application client’s view may be somewhat different constraint systems and the two feature hierarchies may not even be refinements of one another, but in... |

15 |
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Citation Context ...rmediate sub-assemblies are generally unsuccessful. From a modeling point of view, this lack of understanding is due to the fact that prior to the recent model [25, 29], previous computational models =-=[59, 60, 61, 62, 63, 64, 65, 66, 67]-=- generally involve dynamics of (simplified versions) of virus assembly (further description of these approaches and comparison with the approach [25, 29], is given by Sitharam and Mckenna [25]). Dynam... |

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Citation Context ...o approaches can be developed which leverages these advantages while incorporating the full generality of the model [25, 29]. While there is a well developed structure theory of complete viral shells =-=[57, 58]-=-, verified by X-ray crystallography and other experimental data, the processes of viral shell assembly are poorly understood. From an experimental point of view, this lack of understanding is due to t... |

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Graph based geometric constraint solving: problems, progress and directions. AMS-DIMACS volume on Computer Aided Design
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Citation Context ...porating the module-rigidity algorithm of this section) is given by Sitharam [11, 15]. The pseudocode has been implemented as part of the downloadable, opensource FRONTIER geometric constraint solver =-=[2, 10, 11, 15]-=-. The basic FA algorithm is based on an extension of the distribute routine for edges (explained above) to vertices and clusters in order for the isolation algorithm to work at an arbitrary stage of t... |

12 |
2001b. Decomposition of geometric constraints systems. Part II: New algorithms
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Citation Context ...vably meets several competing requirements. Specifically, it gives a dof complete DR-plan. The graph transformation performed by the FA cluster simplification is described formally by Hoffmann et al. =-=[6, 7]-=- that provide the vocabulary for proving certain properties of FA that follows45 directly from this simplification. However, other properties of FA require details of the actual DR-planner that ensure... |

12 |
Modeler-independent Feature Recognition in a Distributed Environment
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Citation Context ...ypes. The first type [33, 34], dictates a unified representation language which is an amalgamation of variational constraints with other representation languages such as CSG and Brep. The second type =-=[35]-=- wrestles with a heterogeneous approach, using many servers, one for each representation language, so that the appropriate one can be called when required. Both approaches, while highly general in sco... |

11 |
Frontier: fully enabling geometric constraints for feature based modeling and assembly
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Citation Context ...ree of overlap constraints 1. Solutions 2. Non−tree overlap and original constraints 2. Solved flag Figure 6–4: The structure of the cluster. A cluster object represents a DR-planner’s simplification =-=[2, 6, 9, 10, 11, 15]-=- of a well-constrained or a well-overconstrained subgraph, as well as its original subgraph. Figure 6–4 shows the contents of the cluster. 1. Hierarchy field. These cluster objects are hierarchical an... |

11 |
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Citation Context ...ems are dof-rigid, the converse is not the case in 3D. Standard counterexamples are systems that contain constraint dependences or inexplicit overconstraints hidden in so-called “bananas” or “hinges” =-=[19, 20, 21]-=- (Figure 1–4). In fact, these are the only known types of counterexamples. The module-rigid Frontier vertex algorithm described by Sitharam and Zhou [12] and implemented in FRONTIER [15] is the first ... |

10 |
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Citation Context ...such as intersections; or parametric constraints, while additionally permitting B-rep and other representations of some features. We use the FEMEX and other standard definitions of feature hierarchy, =-=[31, 32]-=- and are concerned primarily with the conceptual design stage. While designers additionally appreciate the expressiveness of variational constraints, today’s CAD systems largely restrict variational c... |

10 |
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Citation Context ...nd unique graph realization into this algorithm. xiisCHAPTER 1 INTRODUCTION Geometric constraint systems have been studied in the context of variational constraint solving in CAD for nearly 2 decades =-=[1, 2, 3, 4]-=-. A geometric constraint system consists of a finite set of primitive geometric objects such as points, lines, planes, conics and so on and a finite set of geometric constraints between them such as d... |

9 | A tractable, approximate, combinatorial 3D rigidity characterization. 5th Automated Deduction in Geometry
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Citation Context ...om of the window. Note. Recall that the dof-rigid FA DR planner (Section 1.2) considered here does not deal with implicit constraint dependences. However, the more general, module-rigid FA DR-planner =-=[12]-=- deals with all known types of constraint dependences such as bananas and hinges. While incorporation of a feature hierarchy into the the module-rigid FA DR-planner [12] has been implemented in FRONTI... |

8 |
Making constraint solvers more useable: The overconstraint problem
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Citation Context ... is the parameter associated with the constraint. Most of the constraint solvers so far deal with 2D constraint systems. With the exception of work related to the FRONTIER geometric constraint solver =-=[1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]-=-, to the best of our knowledge, work on stand-alone 3D geometric constraint solvers is relatively sparse [16, 17]. A solution or realization of a geometric constraint system is the (set of) real zero(... |

8 |
Pattern formation in icosahedral virus capsids: the papova viruses and Nudaurelia capensis b virus
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Citation Context |

7 | Solving minimal, wellconstrained, 3D geometric constraint systems: Combinatorial optimization of algebraic complexity. 5th Automated Deduction in Geometry
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Citation Context ... is the parameter associated with the constraint. Most of the constraint solvers so far deal with 2D constraint systems. With the exception of work related to the FRONTIER geometric constraint solver =-=[1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]-=-, to the best of our knowledge, work on stand-alone 3D geometric constraint solvers is relatively sparse [16, 17]. A solution or realization of a geometric constraint system is the (set of) real zero(... |

7 | Distributed maintenance of multiple product views, Computer-Aided Design 2000;32(7):421–31
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(Show Context)
Citation Context ...ic constraint solving can in fact be achieved as a special case, where the order is a complete, total order. More generally, this can be used in the CAD database maintenance of multiple product views =-=[46, 47]-=-, for example the design view and a downstream application client’s view may be somewhat different constraint systems and the two feature hierarchies may not even be refinements of one another, but in... |

7 | Representing all Minimum Spanning Trees with Applications to Counting and Generation
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(Show Context)
Citation Context ... rooted trees VDC, we find those with the minimum number of equations of maximum degree. Keeping track of the spanning trees in VC and VDC can be achieved in polynomial time using the data structures =-=[54]-=-. However, in practice, the bigs106 O complexity for creation and maintenance of these datastructures hides large constants. Thus for the small sizes of clusters and overlap graphs that are typical in... |

7 | Local rules switching mechanism for viral shell geometry
- Berger, Shor
- 1995
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Citation Context |

6 |
The tetrahedral-octahedral truss
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Citation Context ...f generic constraint dependences. One example (Figure 1–4) with 3D points and distance constraints illustrates the so-called “bananas” problem [19], which generalizes to the so-called “hinge” problem =-=[20, 21]-=-. To date,sthere is no known, tractable, combinatorial characterization of generic rigidity of systems for 3 or higher dimensions even when only points and distances are involved [19, 22], although se... |

6 |
Graph and Combinatorial Analysis for Geometric Constraint Graphs
- Lomonosov
- 2004
(Show Context)
Citation Context ...tial properties of DR-plans and DR-planners. This section is important since we require our feature incorporation algorithm to preserve these properties. Please refer to Hoffmann et al. and Lomonosov =-=[6, 24]-=- for a detailed discussion of these properties. Formally, a dof-DR-plan of a constraint graph G is a directed acyclic graph (dag) whose nodes represent dof-clusters in G, and edges represent containme... |

6 |
Geometry and feature representation for an integration with knowledge based systems
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- 1996
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Citation Context ... constraints. In this chapter, we denote such representations as mixed representations (Figure 3–2). Previous work on such mixed representations can be classified into two broad types. The first type =-=[33, 34]-=-, dictates a unified representation language which is an amalgamation of variational constraints with other representation languages such as CSG and Brep. The second type [35] wrestles with a heteroge... |

5 | Planning geometric constraint decompositions via graph transformations - Hoffmann, Lomonosov, et al. - 1999 |

5 |
Solution management and navigation for 3d geometric constraint systems
- Sitharam, Arbree, et al.
(Show Context)
Citation Context ...sp. 2). 7.2.2 Removing Assumptions Made in Section 1.2 We turn to two assumptions made in Section 1.2. The first is to ensure that the system of equations constructed by the above algorithm is stable =-=[13]-=-. In the leftmost tree (Figure 7–3), although the cluster C is wellconstrained and the dof count is six, there are two non-tree edges representing overlaps. Choosing the overlap constraints between C2... |

5 |
www.d-cubed.co.uk/. In D-cubed commercial geometric constraint solving software
- Owen
(Show Context)
Citation Context ... related to the FRONTIER geometric constraint solver [1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], to the best of our knowledge, work on stand-alone 3D geometric constraint solvers is relatively sparse =-=[16, 17]-=-. A solution or realization of a geometric constraint system is the (set of) real zero(es) of the corresponding algebraic system. In other words, the solution is a class of valid instantiations of (th... |

4 |
Graph algorithms for geometric constraint solving
- Lomonosov
(Show Context)
Citation Context ...nslates to a useful property of dof DR-plans called dof completeness. (We omit proofs). Then we sketch relevant properties of Frontier vertex DR-plans and the corresponding DR-planner (FA DR-planner) =-=[6, 9]-=- which follows this characterization. Let C be a geometric constraint graph. Then Q = {C1, . . .,Cm}, a set of dof rigid proper subgraphs of C, is a complete, maximal, dof rigid decomposition of C if ... |