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Smooth spline surface generation over meshes of irregular topology

Smooth spline surface generation over meshes of irregular topology. J.J. Zheng, J.J. Zhang, H.J.Zhou, L.G. Shen The Visual Computer(2005) 21:858-864 Pacific Graphics 2005 Reporter: Chen Wenyu Thursday, Mar 2, 2006. About the author Introduction Zheng-Ball surface patch

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Smooth spline surface generation over meshes of irregular topology

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  1. Smooth spline surface generation over meshes of irregular topology J.J. Zheng, J.J. Zhang, H.J.Zhou, L.G. Shen The Visual Computer(2005) 21:858-864 Pacific Graphics 2005 Reporter: Chen Wenyu Thursday, Mar 2, 2006

  2. About the author • Introduction • Zheng-Ball surface patch • Irregular closed mesh • Irregular open mesh • Conclusions

  3. About the author • 郑津津, professor • 中国科学技术大学精密机械与精密仪器系. • He received his Ph.D. in computer aided geometric modelling from the University of Birmingham, UK, in 1998. • His research interests include CAGD,computer-aided engineering design, microelectro-mechanical systems and computer simulation.

  4. About the author • 张建军, professor • Bournemouth Media School,Bournemouth University. • Ph.D. 1987, 重庆大学. • His research interests include computer graphics, computer-aided design and computer animation..

  5. About the author • H.J. Zhang, 高级工程师 • 中国科大国家同步辐射实验室. • She received her M.Sci. from the University of Central England Birmingham, UK.. • Her research interests include mechanical design, micro-electro-mechanical systems and vacuum technology.

  6. About the author • 沈连婠, professor • 中国科学技术大学精密机械与精密仪器系. • Her research interests include e-design, e-manufacturing, e-education and micro-electromechanical systems

  7. About the author • Introduction • Zheng-Ball surface patch • Irregular closed mesh • Irregular open mesh • Conclusions

  8. Introduction • Regular mesh: each of the mesh points is surrounded by four quadrilaterals

  9. Introduction • generate surfaces over regular meshes: B-spline surfaces…. • generate surfaces over irregular meshes:final surface be ---subdivision surfaces ---spline surface

  10. Introduction • subdivision surfaces C-C subdivision C2Doo-sabin subdivision C1

  11. Spline surface Original mesh M subdivided mesh M1 spline surface

  12. Spline surfaces • Peter(CAGD 93); Loop(sig94) 1. Doo-Sabin subdivision 2. a patch for a pointregular mesh : bi-quadratic B-splineirregular area : bi-cubic surface or triangular patch

  13. Spline surfaces • Loop,DeRose(sig90) 1. subdivision once 2. a patch for a pointregular mesh : bi-quadratic B-splineirregular area : S-patch

  14. Spline surfaces • Peters(sig2000) 1. C-C subdivision 2. a bi-cubic scheme • resulting patches agree with the C-C limit surface except around the irregular vertices

  15. This paper • C-C subdivision: (one face : four edges) • A patch for each vertex regular area: bi-quadratic Bezierirregular area: Zheng-Ball patch

  16. This paper Original mesh M C-C subdivision subdivided mesh M1 Zheng-Ball surface patch spline surface

  17. Compare • Peters’ methods require control point adjustment near extraordinary vertices. But the proposed method needn’t. • Takes fewer steps to process compared with Peters’ methods. • Loops’ methods go through the complicated conversion of control points. But the proposed method is much simpler.

  18. About the author • Introduction • Zheng-Ball surface patch • Irregular closed mesh • Irregular open mesh • Conclusions

  19. Zheng-Ball surface patch • Zheng, J.J., Ball, A.A.: Control point surfaces over non- four-sided areas.CAGD.1997

  20. Definition of the surface Control mesh Zheng-Ball surface patch

  21. domain An n-sided control point surface of degree m is defined by: parameters u = (u1,u2, . . . ,un) must satisfy:

  22. Definition of the basis Zheng-Ball surface patch 条件 • 边界条件: 边界上是多项式曲线 • 边界上对 导数的条件 • 归一性 The patch can be connect to the surrounding patches with C1 continuity

  23. Zheng-Ball surface patch • In this paper, the control mesh

  24. Zheng-Ball surface patch

  25. Zheng-Ball surface patch

  26. Zheng-Ball surface patch in which di are auxiliary variables satisfying

  27. Zheng-Ball surface patch

  28. About the author • Introduction • Zheng-Ball surface patch • Irregular closed mesh • Irregular open mesh • Conclusions

  29. Irregular closed mesh • C-C subdivision • Create patches Control point generation corresponding to a vertex of valence 5

  30. Irregular closed mesh • Two adjacent patches joined with C1 continuity. • They share common boundary points (◦). • control vectors (−→) and(··· →)

  31. Irregular closed mesh • Closed irregular mesh and the resulting geometric model. • Patch structure: Patches on the corners are non-quadrilateral Zheng–Ball patches; • the others are bi-quadratic Bezier patches

  32. About the author • Introduction • Zheng-Ball surface patch • Irregular closed mesh • Irregular open mesh • Conclusions

  33. Irregular open mesh • Boundary vertex • Intermediate vertex • Inner vertex

  34. Irregular open mesh • Examples

  35. About the author • Introduction • Zheng-Ball surface patch • Irregular closed mesh • Irregular open mesh • Conclusions

  36. Conclusions • Original mesh M subdivided mesh M1 C1 spline surface

  37. Thanks

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