Geometric Modeling for Shape Classes Amitabha Mukerjee Dept of Computer Science IIT Kanpur

Representations 2 from [Requicha ACM Surveys 1980]

Parametric design vs Conceptual Design  Conceptual Variation approximated using a finite set of parameters

Modeling Fixed Geometries 4

Mathematical Structures Vectors, orthonormal bases – distances and norms – Angles Transformations Motions, boolean operations 5

6

Representing Geometrical Objects As Primitives Spatial decomposition Boolean (Constructive) operations – Continuous constructions: Extrusion / Sweep Boundary based modeling 7

Boolean operations 8

Intersection of solids  not a solid 9

Boundary is not unique specifier Depends on the embedding space – A boundary on a sphere may represent either side – May need additional neighbourhood information 10

Curves and Surfaces 11

Implicit equations – Line: p = u.p1 + (1-u). p2 12

Plane: (p-p0).n = 0 If n = {a,b,c} and p0.n = -d, we have ax+by+cz+d=0 13

3D Solids : B-rep 14

Algorithms Point membership classification – 2D planar shapes – 3D ?? Line – Shape intersection Solid boolean operations 15

Variational Shape Classes 16

Familiar Shapes 17

Familiar Shapes 18

Generating Variational Shapes 19

Generating Variational Shapes 20 kilian-mitra-07 : Geometric-modeling-shape-interpolation,

Shape Classes for Conceptual Design 21

Design = Search in Ill-structured spaces From Goel [VSRD 99]

Applications to Conceptual Design 23 1.Geometric Parametrization 2.Formulation of cumulative objective 3.Parameter Search and optimization

Constraints on Shape  A Complete Faucet  Driving Parameter Set : { W o, H o, L o,  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 25

Generating Variational Shapes  “functionality“ - mathematical function  “aesthetics” - User interaction 143

Final Population of Faucets  Names of instances of faucets shown are given as,  [ (A, B); (B, C); (C, D) ]  User Assigned Fitness Table ABCDEF

Conclusion 28 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