Geometric Modeling
for Shape Classes
Amitabha Mukerjee
Dept of Computer Science
IIT Kanpur
http://www.cse.iitk.ac.in/~amit/
Representations
from [Requicha ACM Surveys 1980]
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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
kilian-mitra-07 : Geometric-modeling-shape-interpolation,
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Shape Classes for
Conceptual Design
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Design = Search in Ill-structured spaces
From Goel [VSRD 99]
Applications to Conceptual Design
1.Geometric
Parametrization
2.Formulation of
cumulative objective
3.Parameter Search
and optimization
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Constraints on Shape
A Complete Faucet
Driving Parameter
Set :
{ Wo , Ho , Lo , 1 , 2 }
Sub-parts:
Inlet
Outlet
Cock
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
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 “functionality“ - mathematical function
 “aesthetics” - User interaction
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
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4
4
4
4
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Conclusion
•
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
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