Post on 17-Jan-2016
Geometric Modeling for Shape Classes
Amitabha MukerjeeDept of Computer Science
IIT Kanpurhttp://www.cse.iitk.ac.in/~amit/
Representations
2from [Requicha ACM Surveys 1980]
Parametric design vs Conceptual Design
Conceptual Variation approximated using a finite set of parameters
Modeling Fixed Geometries
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Mathematical Structures
• Vectors, orthonormal bases– distances and norms– Angles
• Transformations• Motions, boolean operations
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Representing Geometrical Objects
• As Primitives• Spatial decomposition• Boolean (Constructive) operations
– Continuous constructions: Extrusion / Sweep
• Boundary based modeling
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Boolean operations
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Intersection of solids not a solid
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Boundary is not unique specifier
• Depends on the embedding space– A boundary on a sphere may represent either side
– May need additional neighbourhood information
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Curves and Surfaces
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• Implicit equations– Line: p = u.p1 + (1-u). p2
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• Plane: (p-p0).n = 0
• If n = {a,b,c} and p0.n = -d, we have ax+by+cz+d=0
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3D Solids : B-rep
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Algorithms
• Point membership classification– 2D planar shapes
– 3D ??
• Line – Shape intersection• Solid boolean operations
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Variational Shape Classes
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Familiar Shapes
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Familiar Shapes
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Generating Variational Shapes
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Generating Variational Shapes
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kilian-mitra-07 : Geometric-modeling-shape-interpolation,
Shape Classes for Conceptual Design
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Design = Search in Ill-structured spaces
From Goel [VSRD 99]
Applications to Conceptual Design
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1.Geometric Parametrization
2.Formulation of cumulative objective
3.Parameter Search and optimization
Constraints on Shape
A Complete FaucetDriving Parameter
Set : { Wo , Ho , Lo , 1 , 2 }
Sub-parts: InletOutletCock
Algorithms
• Boolean operations on probabilistic sets– Point membership classification?
• Output also in terms of probability density function
• Boolean operations on objects and classes• Function evaluation
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Generating Variational Shapes
“functionality“ - mathematical function “aesthetics” - User interaction
143143
Final Population of Faucets
Names of instances of faucets shown are given as ,
[ (A , B); (B , C); (C , D) ]
User Assigned Fitness Table
A B C D E F
3 4 4 4 4 4
Conclusion
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• Computational processes are moving from deterministic to probabilistic
• Geometric modeling will also need to move more in this direction, which is also cognitively viable.
• Need structures for modeling ambiguous shapes
• Many algorithmic challenges even for unique shapes, output for shape classes will also be probabilistic