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