第36卷第6期 中国神学技术大学学报 Vol.36,No.6 2006年6月 JOURNAL OF UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA Jun.2006 Article ID:0253-2778(2006)06-0573-09 Dimensions of spline spaces over general T-meshes HUANG Zhang-jin, DENG Jian-song, LI Xin (De partment of Matheratics,Umiversity of Science and Technology of China.Hefei 230026,China) Abstract: The spline spaces over general T-meshes and the periodic spline spaces over regular T- meshes were studied. With the method based on B-nets the dimension formulae were proposed as well. Key words: T-mesh; spline; Bezier ordinate; dimension CLC number: O174. 4; O241. 5 Document code: A AMS Subject Classification(2000): 65D07 一般T网格上样条空间的维数 黄章进，邓建松，李新 (中国科学技术大学数学系，安徽合肥230026) 摘要：研究了一般T网格上的样条空间和规则T网格上的周期样条空间，并用基于B网的方法给出了它们 的维数公式. 关键词：T网格；样条；Bézier纵标；维数 spline over T-meshes, i. e., for every vertex, a 0 Introduction blending function is defined. Each of the blending T-meshes are formed by a set of horizontal functions comes from some tensor-product spline line segments and a set of vertical line segments, space. Though this type of spline supports many where T-junctions are allowed. It could be thought valuable operations within a consistent framework, of as a transition between the simple tensor- some of them, such as, evaluation and local product meshes and the complicated triangular refinement, are not simple. In the T-spline meshes. Over the T-meshes, we can define many theory, the basis functions are rational which lead types of spline spaces. Currently, there are basically two types of spline spaces defined over T- to evaluations and are not simple; on the other meshes: one is from Sederberg et al, and the hand, the local refinement is dependent on the other is from Deng et al2]. structure of the mesh, and its complexity is Ref. [1] invented T-spline. It is a point-based uncertain. Furthermore, it is an ongoing problem Received: 2005-11-10; Revised: 2006-05-17 Foundation item: Supported by the Outstanding Youth Grant of NSF of China (60225002), National Key Basic Research Project of China (2004CB318000), NSF of China (60473132), the TRAPOYT in Higher Education Institute of MOE of China, and SRF for ROCS, SEM. Biography: HUANG Zhang-jin, male, born in 1980, PhD. Research field:CAGD and CG. E-mail: zjh@mail. ustc. edu. cn Corresponding author: DENG Jian-song, Prof. E-mail: dengjs@ustc. edu. cn! "!"#""$ #$%&!"!’$&" ())"%"& !"#$%&’"(#%)*+$,)-."(,/)+%/+&%0-+/1%"’"2."(/1)%& *+,&())" -./01%234")(5!6(778#())"$)"6)57!6)9 !"#$%&"’%&’(&)*"%$&)+,$&’-$./$%$.+*01#$&2$&! :;-’<=>?,@6A0,!4B’<*0?,6C$,@!D3E0, #!"#$%&’"(&)*+$&,"’$&-./!0(-1"%/-&2)*3.-"(."$(45".,()6)72)*8,-($!9"*"-(!))("!8,-($$ 34&5.+,5"F>2CG%0,2CG?12C$H2.@2,2.?%F6I2C>2C?,J/>2G2.0$J01CG%0,2CG?12C$H2..2@+%?.F6 I2C>2CK2.2C/+J02J&L0/>/>2I2/>$JM?C2J$,N6,2/C/>2J0I2,C0$,O$.I+%?2K2.2G.$G$C2J?C K2%%& 6$78’.9&"F6I2C>%CG%0,2%NPQ02.$.J0,?/2%J0I2,C0$, :;:%<#4$."RS7T&T%R(TS&5 !’,<#$%5,’9$"- 3=>><4?$,5:*+&&"(",+5"’%#@AAA$""54)7 !"0#$%&’()*+, !"#!$%&!’ ( #’()*+,-*.*/!0123(!))("$ -.!!"#$%F&’()*+,-./0F&’()12*+,-!3456N&)789:#;< )=>?@& /01!F&’"*+"NPQ02.AB"=> A B%5.’9<,5"’% F6I2C>2C?.2O$.I2JMU?C2/$O>$.0Q$,/?% %0,2C2@I2,/C?,J?C2/$OH2./01?%%0,2C2@I2,/C! K>2.2F6A+,1/0$,C?.2?%%$K2J&3/1$+%JM2/>$+@>/ $O?C ?/.?,C0/0$, M2/K22, />2 C0IG%2/2,C$.6 G.$J+1/ I2C>2C ?,J />2 1$IG%01?/2J /.0?,@+%?. I2C>2C&RH2./>2F6I2C>2C!K21?,J2O0,2I?,U /UG2C $O CG%0,2 CG?12C& V+..2,/%U!/>2.2 ?.2 M?C01?%%U/K$/UG2C$OCG%0,2CG?12CJ2O0,2J$H2.F6 I2C>2C"$,20CO.$I W2J2.M2.@"&$6&S’!?,J/>2 $/>2.0CO.$I42,@"&$6&(’ & X2OY&S’0,H2,/2JF6CG%0,2&3/0C?G$0,/6M?C2J CG%0,2$H2.F6I2C>2C!0&2&!O$.2H2.UH2./2Z!? M%2,J0,@O+,1/0$,0CJ2O0,2J&B?1>$O/>2M%2,J0,@ O+,1/0$,C1$I2CO.$IC$I2/2,C$.6G.$J+1/CG%0,2 CG?12&F>$+@>/>0C/UG2$OCG%0,2C+GG$./CI?,U H?%+?M%2$G2.?/0$,CK0/>0,?1$,C0C/2,/O.?I2K$.[! C$I2 $O/>2I!C+1> ?C!2H?%+?/0$, ?,J %$1?% .2O0,2I2,/!?.2 ,$/ C0IG%2&3, />2 F6CG%0,2 />2$.U!/>2M?C0CO+,1/0$,C?.2.?/0$,?%K>01>%2?J /$2H?%+?/0$,C?,J?.2,$/C0IG%2%$,/>2$/>2. >?,J!/>2%$1?%.2O0,2I2,/0CJ2G2,J2,/$,/>2 C/.+1/+.2 $O/>2 I2C>!?,J 0/C 1$IG%2Z0/U 0C +,12./?0,&\+./>2.I$.2!0/0C?,$,@$0,@G.$M%2I ! C$,$"-$9"())56SS6S)%C$-"&$9"())"6)56S7 D’<%9+5"’%"5$#"W+GG$./2JMU/>2R+/C/?,J0,@]$+/><.?,/$O’W\$OV>0,?#")((5))($!’?/0$,?%^2UN?C01X2C2?.1>_.$A21/$OV>0,? #())TVN!S8)))$!’W\$OV>0,?#")T7!S!($!/>2FX-_R]F0,:0@>2.BJ+1?/0$,3,C/0/+/2$O‘RB$OV>0,?!?,J WX\O$.XRVW!WB‘& E"’/.+)27":;-’<=>?,@6A0,!I?%2!M$.,0,S98)!_>4&X2C2?.1>O02%J"V-<4?,JV<&B6I?0%"QA>!I?0%&+C/1&2J+&1, :’..$&)’%9"%/+<52’."4B’<*0?,6C$,@!_.$O&B6I?0%"J2,@AC!+C/1&2J+&1