574 中四种学被术大李学极 第36卷 whether T-spline blending functions are always over T-meshes. linearly independentta.The reason for these 1.I T-meshes problems is mainly that the spline over every cell of T-meshes are formed by a set of horicontal the mesh is not a polynomial.hut a piecewise line segments and a set of vertical line segments. polynomial. where T-junctions are allowed.We can also think Forcing functions to be a tensor-proxduxct that the T-mesh is a simply connected domain polynomial on every cell and to satisfy the specified formed by the union of a set of rectangles whose smoothness across common edges,Deng er al sides are horizontal or vertical.Fig.I illustrates introduced the spline spaces over T-meshes4. two examples of T-meshes.while in Fig.2 two They derived the dimension formulae for the spline examples of non-T-meshes are shown. spaces over regular T-meshes via a method based on B-nets.Here "regular"means that the whole domains are rectangles without holes inside. Comparing with Sederberg's T-spline,the surface moxdeling based on the spline spaces over T-meshes allows local refinement to he really local.And the evaluation is quick sinee this type of spline is Fig.1 Two examples of T-meshes polynomial,nat rational. However,all these works about spline spaces over T-meshes only take into account the regular T-meshes,and the spline surfaces defined over such T-meshes will have topological equivalent to the disks.In the practice of surface modeling.the b boundary of the domain of a complex surfare may Fig.2 Two examples of ron-T-meshes be an arbitrary polygon.and the domain may also Fig.3 gives a more complicated T-mesh, have one ar several holes.For example,if we fit a where three parts with gray fill-in are excluded spline surface over T-meshes according to the from the T-mesh.Hence there are three holes in range data of some mask,then we need the domain the T-mesh.Here each small rectangle in the T- T-mesh to have at least three holes inside,one for mesh is called a cell ar a facet.The corners of the mouth,and the other two for two eyes.On the these rectangles are called vertices of the T-mesh. other hand,if we want to model a simple close If a vertex is on the boundary,then it is called a surface or a cylindrical surface with a close section. boundary vertex.Otherwise,it is called an interior then we need periodic splines.Hence in this paper, we consider the spline function spares over general T-meshes.For the regular T-meshes.periodic splines are discussed as well.With the method hs 6 based on B-nets proposed by Deng er al.we ，4 derive the dimension formulae of these spline spaces. Spline spaces over T-meshes In this section,we first review some concepts related to T-meshes and the spline function spaces Fig.3 A T-mesh with notatians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aCF6CG%0,2#/>2C+.O?12 I$J2%0,@M?C2J$,/>2CG%0,2CG?12C$H2.F6I2C>2C ?%%$KC%$1?%.2O0,2I2,//$M2.2?%%U%$1?%&-,J/>2 2H?%+?/0$,0Cb+01[C0,12/>0C/UG2$OCG%0,20C G$%U,$I0?%#,$/.?/0$,?%& :$K2H2.#?%%/>2C2K$.[C?M$+/CG%0,2CG?12C $H2.F6I2C>2C$,%U/?[20,/$?11$+,//>2.2@+%?. F6I2C>2C#?,J/>2CG%0,2C+.O?12CJ2O0,2J$H2. C+1>F6I2C>2CK0%%>?H2/$G$%$@01?%2b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’-$.01#$&2$& 3,/>0CC21/0$,#K2O0.C/.2H02KC$I21$,12G/C .2%?/2J/$F6I2C>2C?,J/>2CG%0,2O+,1/0$,CG?12C $H2.F6I2C>2C& FGF 01#$&2$& F6I2C>2C?.2O$.I2JMU?C2/$O>$.0Q$,/?% %0,2C2@I2,/C?,J?C2/$OH2./01?%%0,2C2@I2,/C# K>2.2F6A+,1/0$,C?.2?%%$K2J&L21?,?%C$/>0,[ />?//>2 F6I2C>0C?C0IG%U1$,,21/2J J$I?0, O$.I2JMU/>2+,0$,$O?C2/$O.21/?,@%2CK>$C2 C0J2C?.2>$.0Q$,/?%$.H2./01?%&\0@YS0%%+C/.?/2C /K$2Z?IG%2C$OF6I2C>2C#K>0%20,\0@Y(/K$ 2Z?IG%2C$O,$,6F6I2C>2C?.2C>$K,& D"/GF FK$2Z?IG%2C$OF6I2C>2C D"/G@ FK$2Z?IG%2C$O,$,6F6I2C>2C D"/GH -F6I2C>K0/>,$/?/0$,C \0@Y! @0H2C ? I$.2 1$IG%01?/2J F6I2C># K>2.2/>.22G?./CK0/>@.?UO0%%60,?.22Z1%+J2J O.$I/>2F6I2C>&:2,12/>2.2?.2/>.22>$%2C0, />2F6I2C>&:2.22?1>CI?%%.21/?,@%20,/>2F6 I2C>0C1?%%2J?12%%$.?O?12/&F>21$.,2.C$O />2C2.21/?,@%2C?.21?%%2JH2./012C$O/>2F6I2C>& 3O?H2./2Z0C$,/>2M$+,J?.U#/>2,0/0C1?%%2J? M$+,J?.UH2./2Z&R/>2.K0C2#0/0C1?%%2J?,0,/2.0$. 57T CDEFGHIFFJ K!"L