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) 21:858-864Pacific Graphics 2005

Reporter: Chen WenyuThursday, 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 1998.

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

About the author

张建军 , professor Bournemouth Media Schoo

l, Bournemouth University. Ph.D. 1987, 重庆大学 . His research interests inclu

de 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 th

e 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 includ

e 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 C2

Doo-sabin subdivision C1

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 pointregular mesh : bi-quadratic B-splineirregular area : bi-cubic surface or triangular patch

Spline surfaces Loop,DeRose(sig90)

1. subdivision once 2. a patch for a pointregular mesh : bi-quadratic B-splineirregular 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 Bezierirregular 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 su

rfaces over non- four-sided areas.CAGD.1997

Definition of the surface

Control mesh

Zheng-Ball surface patch

domainAn n-sided control point surface of degree m is defined by:

parameters u = (u1,u2, . . . ,un) 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

iu

( )B u 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

Zheng-Ball surface patch

Zheng-Ball surface patch

in which di are auxiliary variables satisfying

Zheng-Ball surface patch

11

1

1

1 2 3 4 5 6

6

2 2( ) 1

4 ( ) (1,1,1,1,1,1)

( )

min

( , , , , , )

j

j

j

n

n jji i

n

j nj

n

jj

n

S u

u S

B

u

u u u u u u

u

u

u

u =

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 C1 continuity.

They share common boundary points (◦).

control vectors (−→) and(· · · →)

Irregular closed mesh Closed irregular mesh and t

he 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

Conclusions

Original mesh M subdivided mesh M1

C1 spline surface

Thanks