1. 1 Outline What is polar form? B-Spline –Labeling, why (0,1,2), (1,2,3)… ? –Knot Vector, why (-2,-1,0,1,2,…) ? Derivation of B-Spline points –Maybe(!?)

2. 2 What is Polar Form Labeling? The sequences of parameters applied at each level of pyramid for de Casteljau evaluation. P0P0 P1P1 P2P2 1-u 0 u0u0 u0u0 u1u1 1-u 1 u1u1 u0u0 P new = (u 0, u 1 )

3. 3 Connecting Bezier Curves u 0 = 0 u 0 = 1, u 1 = 0 u 1 = 1

4. 4 Subdivision of Curve u = 0 u = 0.5 u = 1

5. 5 Ideal Uniform Parameterization But Hard to Control !!! u = 0 u = 1 u = 2

6. 6 Remember Curve-Subdivision u = 0 u = 1 u = 2 u = 0 u = 1 u = 2 u = 0 u = 1

7. 7 Controllability u = 2 u = 0 u = 1

8. 8 Adding the 3 rd Segment u = 2 u = 0 u = 1 u = 3

9. 9 Reverse Subdivision u = 2 u = 0 u = 1 u = 3

10. 10 Adding New Control u = 2 u = 0 u = 1 u = 3

11. 11 Prepend Segments u=-1~0 u = 2 u = 0 u = 1 u = 3 u = -1

12. 12 Add the New Control u = 2 u = 0 u = 1 u = 3 u = -1

13. 13 Prepend new Segment u = -2~-1 u = 2 u = 0 u = 1 u = 3 u = -1 u = -2

14. 14 Add new control Should be mid point, sorry for bad drawing u = 2 u = 0 u = 1 u = 3 u = -1 u = -2

15. 15 Should be mid point, sorry for bad drawing u = 2 u = 0 u = 1 u = 3 u = -1 u = -2

16. 16

17. 17 (0,0,0) (1,1,1) (0,0,1) (0,1,1)

18. 18 (0,0,0) (1,1,1) (0,0,1) (0,1,1) (2,2,2) (1,1,2) (1,2,2)

19. 19 (0,0,0) (1,1,1) (0,0,1) (0,1,1) (2,2,2) (1,1,2) (1,2,2) (0,1,2) Extrapolate

20. 20 (0,0,0) (1,1,1) (0,0,1) (0,1,1) (2,2,2) (1,1,2) (1,2,2) (0,1,2) (0,0,0) (0,0,1) (0,1,1) (0,0,0) (0,0,1) (0,1,2) (0,1,1) (0,0,0) (0,0,1) (1,1,1) (0,1,2) (0,1,1) (0,0,0) (0,0,1) (1,1,2) (1,1,1) (0,1,2) (0,1,1) (0,0,0) (0,0,1) (1,2,2) (1,1,2) (1,1,1) (0,1,2) (0,1,1) (0,0,0) (0,0,1) (2,2,2) (1,2,2) (1,1,2) (1,1,1) (0,1,2) (0,1,1) (0,0,0) (0,0,1)

21. 21 (2,2,2) (1,2,2) (1,1,2) (1,1,1) (0,1,2) (0,1,1) (0,0,0) (0,0,1) (3,3,3) (2,3,3) (2,2,3) (1,2,3)

22. 22 (2,2,2) (1,2,2) (1,1,2) (1,1,1) (0,1,2) (0,1,1) (0,0,0) (0,0,1) (3,3,3) (2,3,3) (2,2,3) (1,2,3) (-1,-1,-1) (0,-1,-1) (0,0,-1) (1,0,-1)

23. 23 (2,2,2) (1,2,2) (1,1,2) (1,1,1) (0,1,2) (0,1,1) (0,0,0) (0,0,1) (3,3,3) (2,3,3) (2,2,3) (1,2,3) (-1,-1,-1) (0,-1,-1) (0,0,-1) (-1,0,1) (-2,-2,-2) (-1,-1,-2) (-1,-2,-2) (-2,-1,0)

24. 24 (0,1,2) (1,2,3) (-1,0,1) (-2,-1,0) u = 0 u = 1