Geometric Constraint Solving Based on kconnected Graph Decomposition

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Geometric Constraint Solving Based on
k-connected Graph Decomposition

• Gui-Fang Zhang
• Beijing Forestry University, China
• Xiao-Shan Gao
• Chinese Academy of Sciences, China

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Contents

• Introduction
• D-tree Decomposition
• Merge

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Geometry Constraint Solving (GCS)

• One of the key techniques to parametric CAD

Methods for GCS

• Graph Analysis Methods
• Rule-based Methods
• Numerical Computation Methods
• Symbolic Computation Methods

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Applications of GCS

• Parametric CAD
• Linkage Design and Simulation
• Computer Vision
• Kinematic Analysis of Robotics
• Molecular Design
• Computer Aided
• Instruction(CAI),

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A Basic Idea of GCS Divide Conquer

• Reduce a large problem into smaller ones
• Divide QiDC(Pi)
• Graph analysis approach
• Rule-based approach
• Symbolic computation approach
• Conquer
• Merge Q1 ? Q2 ? ? Qm
• Numerical computation
• Rule-based approach
• Symbolic computation approach

Key Steps Decomposition and Merge
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Triangular DecompositionOwen, Hoffmann,
Joan-Arinyo
The method can only solve problems that can be
decomposed into triangles.
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Contents

• Introduction
• K-tree Decomposition
• Merge

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Graph Representation of Geometric problem
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Deficit of a Constrained Graph

• G(V,E,?) a geometric constraint graph
• deficit(G)(DOF(V)-D)- DOF(E)
• D is 3 in 2D or 6 in 3D.
• If G is a structurally well-constrained,
• deficit(G)0.
• If G is not structurally over-constrained,
• deficit(G) 0.

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Connectivity of a graph

• Two vertices x and y of graph G are said to be
  k-connected if there exist k vertex-disjoint
  paths from x to y in G. The connectivity of x and
  y is denoted by ?(x,y), which is the number of
  vertex disjoint paths from x to y in G.
• Theorem of Whitney. A graph G(V,E) is
• k-connected iff ?(x,y) k
• that is?(G)min?(x,y) x,y? V

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Connectivity of well-constrained graph

• Let G(V,E) be a well-constrained graph for a
  geometric constraint problem. We have
• Theorem
• 3 in 2D
• ?(G) 5 in 3D for points planes
• 7 in 3D for points, planes
  lines.

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Decomposition of k-connected well-constrained
Graph

• Separating graph Gs subgraph of G induced by a
  separating k-tuple Vs v1, v2,,vk
• Split graph

Gs
G1
G2
The relation of split graphs G1,G2 and cut graph
Gs
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A Key Relation

• G k-connected structurally well-constrained
  graph
• Gs separating graph
• G1, G2 the split graphs.
• Theorem. deficit(G1)deficit(G2) deficit (Gs).
• Consequences
• G1, G2 are structurally well-constrained
• Gs is structurally well-constrained
• A structurally well-constrained bi-connected
  constraint graph can always be split into two
  structurally well-constrained graphs in 2D.
  (This is the case of triangular decomposition of
  Owen et al)

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Examples

• deficit(Gs)0 Two split graphs are rigid
• deficit(G_s)1 one split graph is rigid

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• deficit(Gs)gt1

A bad split no rigid is generated
A good split one rigid is generated
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An K-tree decomposition
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A Decomposition Tree

• A D-tree for a structurally well-constrained
  k-connected graph G(V,E,?) is a binary tree.
• The root of the tree is the graph G.
• Left child L and right child R are defined as
  follows.
• For each node N in the tree, its left child L is
  the split graph of N which is either a triangle
  or a structurally well-constrained subgraph of N,
  and the right child R is the(modified) split
  graph of N with L which is either a triangle or a
  structurally well-constrained graph.
• All leaves are either a triangle or a
  structurally well-constrained j-connected (j lt
  8) constraint graph.

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The Algorithm to Find the Decomposition Tree

• Key Idea find the k-connected separating set and
  split the graph with this set.
• Complexity
• K2 O(n)
• K3 O(n2)
• Other cases O(nk)

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Contents

• Introduction
• D-tree Decomposition
• Merge

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Merge of Two Rigids R1 and R2 Sharing a
Separating Set S
Ss
R1
R2
G1
G2

• Computationally Easy Only need to do a
  translation and a rotation
• Key point for what kind of Ss, R1 and R2 can
  merge into a larger rigid?

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Merge bi-connected graphs in 3D

• Ss a,b the separating pair
• R1 and R2 the split subgraphs of G
• Theorem. R is a rigid body iff
• a,b are of the following three cases
• a point and a line which are not incident
• a plane and a line which are not parallel,
  perpendicular or incident to each other
• two lines which are not parallel to each other.

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Cases that cannot be merged

• The Double Banana problem

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Merge tri-connected graphs in 3D

• Theorem. R is a rigid body iff
• Ssa,b,c are of the following cases
• Three points
• Three planes intersect at one point
• Three lines not parallel to each other
• Two points and one plane
• One line and two points not on the line at the
  same time
• One point and two lines
• One point and two planes

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Thanks