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: Pacific Graphics 2005 Reporter: Chen Wenyu Thursday, Mar 2, 2006

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

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 His research interests include CAGD,computer-aided engineering design, microelectro-mechanical systems and computer simulation.

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..

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.

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

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

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

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

Introduction subdivision surfaces C-C subdivision C 2 Doo-sabin subdivision C 1

Spline surface Original mesh M subdivided mesh M1 spline surface

Spline surfaces Peter(CAGD 93); Loop(sig94) 1. Doo-Sabin subdivision 2. a patch for a point regular mesh : bi-quadratic B-spline irregular area : bi-cubic surface or triangular patch

Spline surfaces Loop,DeRose(sig90) 1. subdivision once 2. a patch for a point regular mesh : bi-quadratic B-spline irregular area : S-patch

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

This paper C-C subdivision: (one face : four edges) A patch for each vertex regular area: bi-quadratic Bezier irregular area: Zheng-Ball patch

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

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.

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

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

Definition of the surface Control mesh Zheng-Ball surface patch

domain An n-sided control point surface of degree m is defined by: parameters u = (u 1,u 2,...,u n ) must satisfy:

Definition of the basis Zheng-Ball surface patch 1. 边界条件 : 边界上是多项式曲线 2. 边界上对 导数的条件 3. 归一性 条件 The patch can be connect to the surrounding patches with C1 continuity

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

Zheng-Ball surface patch

in which d i are auxiliary variables satisfying

Zheng-Ball surface patch

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

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

Irregular closed mesh Two adjacent patches joined with C 1 continuity. They share common boundary points ( ◦ ). control vectors (− → ) and( · · · → )

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

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

Irregular open mesh Boundary vertex Intermediate vertex Inner vertex

Irregular open mesh Examples

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

Original mesh M  subdivided mesh M 1  C 1 spline surface

Thanks