1. Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

4. Surface patches can be modeled mathematically in parametric form as A closed, connected composite surface represents the shape of a solid. This surface, in turn, is composed of surface patches, aesthetics, aerodynamics, fluid flow etc. may influence surface design Surfaces of aircraft wings and fuselage, car body and its doors, seats, and windshields are all designed by combining surface patches at their boundaries. scalar polynomials in parameters (u, v)

5. Tensor product surface patches Boundary interpolating patches Sweep surfaces Quadric (Analytic) surface patches

6. Let and be univariate functions such that u U and v V is called a tensor product surface with domain U V C ij 3 The surface is bi-quadratic for m = n = 2 and bi-cubic for m = n = 3e.g.

7. v = constant u = constant

8. Generalization m and n are user-chosen degrees in parameters u and v For a bi-cubic surface patch, one needs to specify 16 sets of data as control points and/or slopes One for each D ij patches with degrees in u and v greater than 3 can be modeled one can as well choose the degrees unequal in parameters for most applications, use of bi-cubic surface patches seems adequate

9. Hermite functions In matrix form

10. (v =0) (u =0) (v =1) (u =1)

11. Fergusons patch = UMGM T V T Geometric matrix Ferguson coefficient matrix

12. A simple Ferguson Bicubic Patch Specifying twist vectors is not easy; we assign them 0 values r(u, v) = UMGM T V T =

13. A simple Ferguson Bicubic Patch