1. Exponential Quasi-interpolatory Subdivision Scheme Yeon Ju Lee and Jungho Yoon Department of Mathematics, Ewha W. University Seoul, Korea

2. Exponential quasi-interpolatory s.s. Contents  Subdivision scheme – several type of s.s.  Quasi-interpolatory subdivision scheme  Construction  Smoothness & accuracy  Example  Exponential quasi-interpolatory subdivision scheme  Construction  Smoothness  Example

3. Exponential quasi-interpolatory s.s. Subdivision scheme  Useful method to construct smooth curves and surfaces in CAGD  The rule :

4. Exponential quasi-interpolatory s.s. Subdivision scheme  Rule :  Interpolatory s.s. & Non-interpolatory s.s  Stationary s.s. & Non-stationary s.s

5. Exponential quasi-interpolatory s.s. B-spline subdivision scheme   It has maximal smoothness C m-1 with minimal support.  It has approximation order only 2 for all m.  Cubic-spline :

6. Exponential quasi-interpolatory s.s. Interpolatory subdivision scheme   4-point interpolatory s.s. :  The Smoothness is C 1 in some range of w.  The Approximation order is 4 with w=1/16.

7. Exponential quasi-interpolatory s.s.  Goal We want to construct a new scheme which has good smoothness and approximation order.

8. Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  Construction

9. Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  Advantage  L : odd (L+1,L+2)-scheme. So in even pts case, it has tension.  L : even (L+2,L+2)-scheme. It has tension in both case.  This scheme has good smoothness.  It has approximation order L+1.

10. Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  The mask set of cubic case In cubic case, the mask can reproduce polynomials up to degree 3. odd case : use 4-pts even case : use 5-pts with tension v

11. Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  Various basic limit function which start with 

12. Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme

13. Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme

14. Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  Comparison of schemes which use cubic Cubic-spline4-pts interp. s.s.S a Where L=3 Support of limit ftn[-2, 2][-3, 3][-4, 4] Maximal SmoothnessC2C2 C1C1 C3C3 Approximatio n Order 244

15. Exponential quasi-interpolatory s.s. Example

16. Comparison with some example  Example E=0.8169 E=0.1428

17. Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  General case LMask setSm.Range of tension 3 O=[-1/16,9/16,9/16,-1/16] E= [-v, 4v,1-6v,4v,-v] C3C3 0.0208<v<0.0404 4 O=[-v, – 77/2048+5v,385/512-10v, 385/1024+10v,-55/512-5v,35/2048+v] E(i)=O(7-i) for i=1:6 C3C3 -0.0106<v<-0.0012 5 O=[3, – 25,150,150, – 25,3]/256] E=[-v,6v, – 15v,1+20v,-15v,6v,-v] C4C4 -0.0084<v<-0.0046 6 O=[-v,385/65536+7v, – 2079/32768-21v, 51975/65536+35v,5775/16384-35v, -7245/65536+21v,945/32768-7v,-231/65536+v] E(i)=O(9-i) for i=1:8 C4C4 0.0007<v<0.0017 7 O=[-5,49, – 245,1225,1225, – 245,49, – 5]/2048 E=[-v, 8v, – 28v,56v,1-70v,56v, – 28v,8v, – v] C5C5 0.0012<v<0.0015

18. Exponential quasi-interpolatory s.s.  Construction

19. Exponential quasi-interpolatory s.s. Analysis of non-stationary s.s.

20. Exponential quasi-interpolatory s.s.

21.  Example E=7.7716e-016 E=0.1434

22. Exponential quasi-interpolatory s.s. Next Study