1. Geometric Modeling for Shape Classes Amitabha Mukerjee Dept of Computer Science IIT Kanpur http://www.cse.iitk.ac.in/~amit/

2. Representations 2 from [Requicha ACM Surveys 1980]

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

4. Modeling Fixed Geometries 4

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

6. 6

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

8. Boolean operations 8

9. Intersection of solids  not a solid 9

10. 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

11. Curves and Surfaces 11

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

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

14. 3D Solids : B-rep 14

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

16. Variational Shape Classes 16

17. Familiar Shapes 17

18. Familiar Shapes 18

19. Generating Variational Shapes 19

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

21. Shape Classes for Conceptual Design 21

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

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

24. Constraints on Shape  A Complete Faucet  Driving Parameter Set : { W o, H o, L o,  1,  2 }  Sub-parts:  Inlet  Outlet  Cock

25. 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

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

27. Final Population of Faucets  Names of instances of faucets shown are given as,  [ (A, B); (B, C); (C, D) ]  User Assigned Fitness Table ABCDEF 344444

28. 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