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