第6潮 Dimensions of spline spaces over generl T-mishes 577 3 Dimension formula of spline spaces E-E-E.=V. (3) (N) over general T-meshes +=+.+座=+德(4) In this sertion,we will derive a dimensian Proof formula for the spline space s(m.)over a I)Eg.(1)holds obviously according to the general T-mesh whenm≥2a+1andn≥2p+l. definition of A:and a.. 3.1 Some notations for a T-mesh (I)Since every cell has two horizontal Before we derive the dimension formula.we boundary lines,each of which is part of some introduce some notations for a T-mesh as shown in horizontal cedge,and every horizontal nom Tab.1. boundary c-edge has A+1 adjacent cells,by the Tabs.1 Notations for a T-mesh contribution of boundary eedges,it follows that motations description E H mher of horiatal houndary edges 2F=A+ (+1). D mber of verticel boundary edges By Eq.(1).we have mhr f horiannia interinr le 2F-座-E=E. E er of wrtiral inierint age Similarly, A her of horiza boundary dges 2F-的-E.=E. B mamber of werticel boundary c-edues )Observe that A mdr of hotiamll worleudhry cules E-E-E,=(E-E)+(E-E,). B madr of werinl wrloundry®s Since E.-Ex and E.-E.represent the numbers of nalr of intetior ocs an the ith botioontal non- all interior T-vertices on all horizontal and vertical unly,j■L,a。Da cedges,respectively,it follows that maber of interor odges on the ith verticul not- 西 E-E一E=Vh. cundiry.i-1,…s (N)If there are no singular b-vertires in the E maeber of interior edges I-mesh.according to the simple connectivity,it 件 mber of interior coss vertes follows that I ms ol interinr T-wertices “=成+ mf interint wrtins Note that we can give a little perturbation to each mber of free b-vertices singular h-vertex to isolate the two cells W her of sirolar b-vertices intersected at it.Then we have m of boundary vertics V+V作=+ maber of ells in the mesh It is easy to know that the number of inner b Similar to Lemma 4.1 for a regular T-mesh in vertices is equal to the number of boundary edges Ref.[2].for a general T-mesh,we have the but not boundary cedges.Hence we have following topological cquations. 四+？=+ Lemma 3.1 Given a T-mesh with the notations in Tah.1.then 3.2 The dimension formula (I) Now we are ready to prove the dimension E formula for the spline space s(m.n,.)when 4=E (1) m之2m+1dn之23+1. (Π) Theorem 3.2 Given a general T-mesh and a 2F-E-E=E,2F-E-E=E.(2) corresponding spline space s(m.nB), (Ⅲ) suppose m2+1 andn28+1.then H !"#$%&"’%(’.#<*+’(&)*"%$&)+,$& ’-$./$%$.+*01#$&2$& 3,/>0CC21/0$,!K2 K0%%J2.0H2?J0I2,C0$, O$.I+%?O$./>2CG%0,2CG?12#"’!(!"!#!!#$H2.? @2,2.?%F6I2C>!K>2,’ (("DS?,J(((#dS< HGF >’#$%’5+5"’%&(’.+01#$&2 N2O$.2K2J2.0H2/>2J0I2,C0$,O$.I+%?!K2 0,/.$J+12C$I2,$/?/0$,CO$.?F6I2C>?CC>$K,0, F?MYS& 0+4GF ’$/?/0$,CO$.?F6I2C> ,$/?/0$,C J2C1.0G/0$, E:, ,+IM2.$O>$.0Q$,/?%M$+,J?.U2J@2C E:1 ,+IM2.$OH2./01?%M$+,J?.U2J@2C E, ,+IM2.$O>$.0Q$,/?%0,/2.0$.2J@2C E1 ,+IM2.$OH2./01?%0,/2.0$.2J@2C E):, ,+IM2.$O>$.0Q$,/?%M$+,J?.U162J@2C E):1 ,+IM2.$OH2./01?%M$+,J?.U162J@2C E), ,+IM2.$O>$.0Q$,/?%,$,6M$+,J?.U162J@2C E)1 ,+IM2.$OH2./01?%,$,6M$+,J?.U162J@2C ’@ ,+IM2.$O0,/2.0$.2J@2C$,/>2@/>>$.0Q$,/?%,$,6 M$+,J?.U162J@2!@;S!$!E), (@ ,+IM2.$O0,/2.0$.2J@2C$,/>2@/> H2./01?%,$,6 M$+,J?.U162J@2!@;S!$!E)1 E ,+IM2.$O0,/2.0$.2J@2C FD ,+IM2.$O0,/2.0$.1.$CC0,@H2./012C F* ,+IM2.$O0,/2.0$.F6H2./012C F ,+IM2.$O0,/2.0$.H2./012C F:* ,+IM2.$OO.22M6H2./012C F:/ ,+IM2.$OC0,@+%?.M6H2./012C F: ,+IM2.$OM$+,J?.UH2./012C G ,+IM2.$O12%%C0,/>2I2C> W0I0%?./$D2II?T&SO$.?.2@+%?.F6I2C>0, X2OY%(&!O$.?@2,2.?% F6I2C>!K2>?H2/>2 O$%%$K0,@/$G$%$@01?%2b+?/0$,C& ;$##+ HJF <0H2, ? F6I2C> K0/> />2 ,$/?/0$,C0,F?MYS!/>2, ""# % E), @;S ’@ ;E,!% E)1 @;S (@ ;E1< "S# "## (GCE): , CE), ;E,!(GCE): 1 CE)1 ;E1<"(# "$# ECE), CE)1 ;F* < "!# "%# F: DF: / ;E: , DE: 1!F: * DF: / ;E): , DE): 1<"T# L.’’( ""#BbY"S#>$%JC$MH0$+C%U?11$.J0,@/$/>2 J2O0,0/0$,$O’-?,J(-< "##W0,12 2H2.U 12%% >?C /K$ >$.0Q$,/?% M$+,J?.U%0,2C!2?1>$O K>01>0CG?./$OC$I2 >$.0Q$,/?% 162J@2! ?,J 2H2.U >$.0Q$,/?% ,$,6 M$+,J?.U162J@2>?C’- DS?JA?12,/12%%C!MU/>2 1$,/.0M+/0$,$OM$+,J?.U162J@2C!0/O$%%$KC/>?/ (G ;E): , D% E), @;S "’@DS#< NUBbY"S#!K2>?H2 (GCE): , CE), ;E,< W0I0%?.%U! (GCE): 1 CE)1 ;E1< "$#RMC2.H2/>?/ ECE), CE)1 ; "E, CE),#D"E1 CE)1#< W0,12E,CE),?,JE1CE)1.2G.2C2,//>2,+IM2.C$O ?%%0,/2.0$.F6H2./012C$,?%%>$.0Q$,/?%?,JH2./01?% 162J@2C!.2CG21/0H2%U!0/O$%%$KC/>?/ ECE), CE)1 ;F* < "%#3O/>2.2?.2,$C0,@+%?.M6H2./012C0,/>2 F6I2C>!?11$.J0,@/$/>2C0IG%21$,,21/0H0/U!0/ O$%%$KC/>?/ F: ;E: , DE: 1< ’$/2/>?/K21?,@0H2?%0//%2G2./+.M?/0$,/$2?1> C0,@+%?. M6H2./2Z /$ 0C$%?/2 />2 /K$ 12%%C 0,/2.C21/2J?/0/&F>2,K2>?H2 F: DF: / ;E: , DE: 1< 3/0C2?CU/$[,$K/>?//>2,+IM2.$O0,,2.M6 H2./012C0C2b+?%/$/>2,+IM2.$OM$+,J?.U2J@2C M+/,$/M$+,J?.U162J@2C&:2,12K2>?H2 F: * DF: / ;E): , DE): 1< ’ HG@ 02$9"#$%&"’%(’.#<*+ ’$K K2?.2.2?JU/$G.$H2/>2J0I2,C0$, O$.I+%?O$./>2CG%0,2CG?12#"’!(!"!#!!#K>2, ’ (("DS?,J(((#DS< 02$’.$#HG@ <0H2,?@2,2.?%F6I2C>?,J? 1$..2CG$,J0,@ CG%0,2 CG?12 #"’!(!"!#!!#! C+GG$C2’ (("DS?,J(((#DS!/>2, K"2 40I2,C0$,C$OCG%0,2CG?12C$H2.@2,2.?%F6I2C>2C 577