|
|
|
| Multi-degree B-spline curves |
| SHU Beng, HONG Guo-Zhao |
| (Institute of Computer Graphics and Image Processing, Zhejiang University, Hangzhou 310027, China) |
|
|
|
Abstract Multi-degree B-spline (MD-spline) curves are special B-spline curves with various degrees on different intervals, thus adapted to the development of CAD modeling system. MD-spline curves whose maximal variational degree was lower than three were investigated. This kind of MD-splines inherit most properties of polynomial B-splines, such as variation diminishing property, convexity preserving property, etc, and enjoy other advantageous properties for modeling, such as degeneration property, knot insertion property. Also the whole MD-spline curve is at least Cn-1, where n is the smallest degree of whole curve segments. In addition, the relation between MD-spline and B-spline was presented. MD-spline can be transformed into B-spline through knot insertion, simultaneously the degree elevation of B-spline can be interpreted as corner cutting process through MD-spline. MD-splines can effectively reduce the numbers of spline curves control points and knot vectors while keeping the desired accuracy, which are very good for geometric design and data transmission of CAD system．
|
|
Published: 18 November 2009
|
|
|
变次数B-样条曲线
变次数B-样条（MD-样条）曲线是在不同区间有不同次数的特殊B-样条曲线.为了适应CAD造型系统的发展，研究了最大变化次数小于3的MD-样条曲线.这类MD-样条继承了多项式B-样条的变差缩减性、保凸性等大多数性质,并具有退化性、嵌入节点等独特性质.整个MD-样条曲线至少是Cn-1连续的,这里n为整个曲线段的最小次数.研究了MD-样条与B-样条的关系,可以通过嵌入节点将MD-样条转化为B-样条,同时通过MD-样条能够将B-样条的升阶看成是几何割角的过程.变次数B-样条能够在保持理想精度的条件下,有效地减少控制顶点和节点向量的数目,有利于几何设计和CAD系统的数据传输.
|
|
| ［1］ BARRY J. Discrete beta-splines ［J］. ACM SIGGRAPH Computer Graphics, 1987, 21(4): 137144．
［2］ 王文涛,汪国昭. 带形状参数的双曲多项式均匀B-样条［J］. 软件学报, 2005, 16(4): 625634．
WANG Wen-tao, WANG Guo-zhao. Hyperbolic polynomial uniform B-spline with shape parameter ［J］. Journal of Software, 2005, 16(4): 625634．
［3］ WANG Guo-zhao, CHEN Qin-yu, ZHOU Ming-hua. NUAT B-spline curves ［J］. Computer Aided Geometric Design, 2004, 21(2): 193205．
［4］ LU Yong-gang, WANG Guo-zhao, YANG Xun-nian. Uniform hyperbolic polynomial B-spline curves ［J］. Computer Aided Geometric Design, 2002, 19(6): 379393．
［5］ KAKLIS P D, PANDELIS D G. Convexity-preserving polynomial splines of non-uniform degree ［J］. IMA Journal of Numerical Analysis, 1990, 10(2): 223234．
［6］ KAKLIS P D, SAPIDIS N S. Convexity preserving interpolatory parametric splines of non-uniform polynomial degree ［J］. Computer Aided Geometric Design, 1995, 12(1): 126．
［7］ COSTANTINI P. Variable degree polynomial splines ［M］. LEMhAUT A, RABUT C, SCHUMAKER L L. Curves and surfaces with application in CAGD. Nashville: Vanderbilt
University Press, 1997: 8594．
［8］ COSTANTINI P. Curve and surface construction using variable degree polynomial splines ［J］. Computer Aided Geometric Design, 2000, 17(5): 419446．
［9］ MAZURE M L. Quasi-Chebyshev splines with connection matrics: application to variable degree polynomial splines ［J］. Computer Aided Geometric Design, 2001, 18(3): 287298
．
［10］ THOMAS W S, ZHENG Jian-min, SONG Xiao-wen. Knot intervals and multi-degree splines ［J］. Computer Aided Geometric Design, 2003, 20(7): 455468．
［11］ PIEGL L, TILLER W. The NURBS book ［M］. 2nd ed. Berlin: Springer, 1997: 8182．
［12］ COHEN E, RIESENFELD R F, ELBER G. Geometric modeling with splines: an introduction ［M］. Wellesley: A K Peters, 2001: 194195．
［13］ HUANG Qi-xing, HU Shi-min, RALPH R M. Fast degree elevation and knot insertion for B-spline curves ［J］. Computer Aided Geometric Design, 2005, 22(2): 183197．
［14］ DE BOOR C. Cutting corners always works ［J］. Computer Aided Geometric Design, 1987, 4(2): 125131．
［15］ PALUSZNY M, PRAUTZSCH H, SCHFER M. A geometric look at corner cutting ［J］. Computer Aided Geometric Design, 1997, 14(5): 421447．
［16］ 王国瑾,汪国昭,郑建民. 计算机辅助几何设计［M］. 北京：高等教育出版社, 2001: 89． |
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
| |
Shared |
|
|
|
|
| |
Discussed |
|
|
|
|
