Ring structure of splines on triangulations

Transcription

1 Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report

2 RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon n R 2, we consder the space C r ( ) of pecewse polynomal functons that are contnuously dfferentable of order r (C r functons) on. These functons are called splnes, and they have many practcal applcatons ncludng the fnte element method for solvng dfferental equatons. They are also very useful for modelng surfaces of arbtrary topology and are a wdely recognzed tool n sogeometrc analyss, and free-form representaton n Computer Aded Geometrc Desgn. Besdes the nterest that the space of splnes has for applcatons, for every r 0, the set C r ( ) forms a rng under pontwse multplcaton. It was proved that, as a rng, C 0 ( ) s a quotent of the Stanley Resner rng A of [2]. Snce C r+1 ( ) C r ( ), ths result mples that there s a descendng chan of subrngs contaned n A. We use a local characterzaton to study the rng structure of those elements of the Stanley Resner rng whch correspond to splnes of hgher-order smoothness [6], we present some results, and a conjecture for the rng structure of splnes on generc trangulatons. 1. Stanley-Resner rng assocated to a splne space Throughout ths notes wll denote a smplcal complex supported on a smply connected doman R 2, and by Ck r( ) we denote the space of Cr splnes (for some nteger r 0) defned on of degree less than or equal to k. The dual graph G assocated to s the graph wth vertces correspondng to the 2-cells n, and edges correspondng to adjacent pars of 2-cells. Thus, n our settng, G wll be connected and ts number of cycles wll depend on the number of nteror vertces of. For an edge τ, let L τ R[x 1, x 2 ] be the lnear polynomal vanshng on τ, and l τ R := R[x 1, x 2, x 3 ] ts homogenzaton. In [2], t was proved that C 0 ( ) s somorphc to A / n =1 Y 1 as R-algebras, where A := R[Y 1,..., Y n ]/I. The number of varables corresponds to the number of vertces v 1,..., v n (on the boundary and n the nteror) of, and I s the deal of nonfaces of, whch by defnton, s the deal generated by the square-free monomals correspondng to vertex sets whch are not faces of, namely I = Y 1 Y j : {v 1,..., v j } /. The key dea for ths somorphsm s to vew the varables of A as Courant functons centered at the correspondng vertex.e., Y (v j ) = δ j where δ j s the Kronecker delta,, j = 1,..., n. Let us embed n the plane {x 3 = 1} R 3, and form the cone ˆ over wth vertex at the orgn, and consder the splnes C r ( ˆ ) defned on ˆ. The Stanley Resner rng A ˆ has 1

3 2 NELLY VILLAMIZAR one varable more than A, but snce that addtonal varable corresponds to the vertex of the cone t does not appear n any of the generators of I, hence (1) C 0 ( ˆ ) = A. Vewng the varables of the (affne) Stanley Resner rng as Courant functons gves a geometrc pcture of C 0 ( ); the homogenzaton allows to use tools from algebrac geometry for graded rngs for studyng C r ( ) [3 5]. Snce there s a natural embeddng (2) C r+1 ( ˆ ) C r ( ˆ ) for every r 0, the somorphsm (1) mples that there s a descendng chan of subalgebras contaned n A, each correspondng to a subalgebra of splnes of ncreasng orders of smoothness [6]. 2. Local characterzaton and C r -splnes Let σ 1 and σ 2, as n Fg. 1, be two trangles whch meet along an edge τ = σ 1 σ 2. Then, f f, g are polynomals supported on (the homogenzatons) ˆσ 1 and ˆσ 2 respectvely, f and g meet C r smoothly f and only f l r+1 τ (f g) 1 [1]. Ths condton translates nto a condton on the polynomals n A as follows, [6]. For each vertex v n let us denote by (v 1, v 2 ) ts coordnates. For a trangle σ = {v, v j, v k }, let x σ, xσ j, xσ k be the lnear functons that gve the barycentrc coordnates of a pont n R 2 n terms of the vertces of σ. Let X σ, Xσ j, Xσ k be the homogenzaton of xσ, xσ j and x σ k wth respect to x 3, respectvely. Defne A σ := R[X σ, Xσ j, Xσ k ], and let (3) B σ : R[x 1, x 2, x 3 ] A σ be the automorphsm defned by x 1 v 1 X σ + v j1 X σ j + v k1 X σ k x 2 v 2 X σ + v j2 Xj σ + v k2 Xk σ x 3 X σ + Xj σ + Xk σ. If σ 1 and σ 2 are two trangles as n Fg. 1, the change of coordnates from A σ2 to A σ1 s gven by the map (4) B σ2 σ 1 : A σ2 A σ1 defned by l j k v jk v lk + j v lj + k, where v jk denotes the determnant v jk = v 1 v j1 v k1 v 2 v j2 v k Proposton 2.1 ([6]). A polynomal F A corresponds to an element of C r ( ) f and only f ( ) r+1 dvdes F σ1 B σ2 σ 1 (F σ2 ), wth the notaton as n Fg. 1, for each nteror edge τ = σ 1 σ 2 of..

4 RING STRUCTURE OF SPLINES ON TRIANGULATIONS 3 Fgure 1. Trangles σ 1, σ 2, wth σ 1 σ 2 = τ. The prevous condton appled to each nteror edge of a gven trangulaton yelds a characterzaton of the elements n A whch correspond to splnes C r ( ) [6], and leads to the followng result. Proposton 2.2. Let be a 2-dmensonal smplcal complex consstng of least two trangles σ 1 = {v 0, v 1, v 2 } and σ 2 = {v 0, v 2, v 3 }, such that ts graph G s a tree (.e., a connected graph wth no cycles). If has m + 1 vertces then where C r ( ) = R[H 0, H 1, H 2, Y r+1 3,..., Y r+1 m ]/I, H 0 = Y 0 + Y Y m H 1 = v 01 Y 0 + v 11 Y v m1 Y m H 2 = v 02 Y 0 + v 12 Y v m2 Y m. Fgure 2. Example, Proposton 2.2. Sketch of the proof. Let us assume has only two trangles σ 1 = {v 0, v 1, v 2 } and σ 2 = {v 0, v 2, v 3 }. Then, the Stanley Resner rng assocated to s A = R[Y 0, Y 1, Y 2, Y 3 ]/ Y 1 Y 3. Let F A be the lnear polynomal F = a 0 Y 0 + a 1 Y 1 + a 2 Y 2 + a 3 Y 3, wth a R. From Proposton 2.1, f F corresponds to an element n C r ( ), the dfference F σ1 B σ2 σ 1 (F σ2 ) must be dvsble by ( 1 )r+1, wth B σ2 σ 1 as defned n (3). Snce ( ) v F σ1 B σ2σ 1 (F σ2 ) = (a 0 a 0 )X σ1 0 + (a 2 a 2 )X σ1 123 v 013 v a 1 a 0 a 2 a 3 X σ1 1 v 023 v 023 v 023 then (a 0, a 1, a 2, a 3 ) must be n the kernel of the matrx ( ) (5) v123 v 023 v 013 v 012.

5 4 NELLY VILLAMIZAR Ths kernel s spanned by the rows of the matrx v 01 v 11 v 21 v 31 v 02 v 12 v 22 v 32, and these rows defne H 0, H 1, H 2. These polynomals correspond exactly to the trval splnes (f σ = f for all σ ) on, whch generate the rng of polynomals R (contaned n C r ( )). The frst degree we need to consder to get not trval splnes s r + 1. Followng an analogous constructon as before, t s easy to check that Y3 r+1, by Proposton 2.1, corresponds to a splne n C r ( ˆ ) whch s nontrval. Hence, all the polynomals n the rng R[H 0, H 1, H 2, Y3 r+1 ]/I correspond to elements n C r ( ˆ ). The other ncluson follows from the dmenson formula for Ck r ( ) [4], dm C r k ( ) = ( k ) ( k + 1 r + 2 For a smplcal complex wth m 3 trangles, the proposton follows by applyng the prevous procedure recursvely addng one new trangle at the tme. Conjecture 2.3. For a generc central confguraton, where G s a cycle, ). C r ( ˆ ) = R[H 0, H 1, H 2, S 2,..., S m ]/I, where S 1,..., S m are the polynomals n A that correspond to the generators of the module of syzyges of the deal l r+1 1,..., lm r+1 n R generated by the lnear forms l correspondng to the nteror edges of. Followng these deas, a smlar constructon leads to a conjecture for C r ( ) of a generc trangulaton, where G a connected graph wth a fnte number of cycles [7]. There are stll many open problems concernng splne spaces, and knowng about ther algebrac structure mght brng some lght and useful results for computaton. As a consequence of the relaton of C 0 ( ) wth A, the dmensons as vector spaces over R of the subspaces Ck 0 ( ) were derved [2]. Smlarly, the Proposton 2.2 and Conjecture 2.3 may lead to fnd the dmenson of Ck r ( ) for splnes of hgher order of smoothness. References [1] L. J. Bllera, Homology of smooth splnes: generc trangulatons and a conjecture of Strang, Trans. Amer. Math. Soc. 310 (1988), no. 1, [2], The algebra of contnuous pecewse polynomals, Adv. Math. 76 (1989), no. 2, [3] L. J. Bllera and L. L. Rose, A dmenson seres for multvarate splnes, Dscrete Comput. Geom. 6 (1991), no. 2, [4] B. Mourran and N. Vllamzar, Homologcal technques for the analyss of the dmenson of trangular splne spaces, J. Symbolc Comput. 50 (2013), [5], Bounds on the dmenson of trvarate splne spaces: a homologcal approach, to appear n Mathematcs n Computer Scences, specal ssue on computatonal algebrac geometry (2014). [6] H. Schenck, Subalgebras of the Stanley-Resner rng, Dscrete Comput. Geom. 21 (1999), no. 4, [7] N. Vllamzar, Algebrac Geometry for Splnes, Doctoral dssertaton, Unversty of Oslo (2012). RICAM, Austran academy of scences. Altenberger strasse 69, 4040 Lnz, Austra. E-mal address: nelly.vllamzar@oeaw.ac.at