Geometrical system decomposition using a multigroup approach

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Geometrical system decompositionusing a
multi-group approach

• P. Mathis, P. Schreck
• LSIIT Université Louis Pasteur
• UMR CNRS 7005
• Strasbourg - France

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2D dimensioned sketch a constraint system
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Decomposition

• Goal cut the constraint system in smaller ones
• Because
• With resolution approach time is shorter
• With construction approach more problems are
  solved
• Rely on invariance under displacements
• Idea considering others groups

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First example
decomposable
graph not decomposable
but not well constrained modulo the displacement
group
(well constrained modulo the similarity group)
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Second example
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Constraint typology
translation
displacement
similarity
transformation
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Similarity invariant part
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Sub-figure 1similarities invariance
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Sub-figure 1similarities invariance
Solution 1 orbit of sub-figure 1 under
similarities
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Displacement invariant part
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Sub-figure 2Displacements invariance
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Assembling subf1 and subf2
Similarity computation
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New constraints extracted border
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Remaining problemTranslation invariance
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Sub-figure 3Translation invariance
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Assembling
Displacement computation
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Key points

• Hierarchy of groups
• G-reference entities to be fixed
• Border
• Assembling of figures inv. under different groups

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Conclusion

• Decomposition using transformation groups is more
  powerful
• Bottom-up algorithm computing remaining system,
  G-reference, border
• Principles valid in 3D

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Construction vs Resolution

• Construction
• yield several solutions
• solution space is scanable
• Resolution
• deal with all types of constraint
• numerical solving

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Second example
fixed direction