Wegelerstraße Bonn. Germany phone fax Maharavo Randrianarivony

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1 Wegelerstraße Bonn. Germany phone fax Maharavo Randranarvony COONS GENERALIZATION FOR POLYTOPES INS Preprnt

2 1. Introducton Over the last decades, several efforts have been done to obtan effcent representatons of geometrc data wth the help of scarce nformaton. That can be llustrated by the representaton of complete Bézer and splne [6] curves or surfaces where one stores only a fnte number of control ponts. Smlarly, the orgnal transfnte nterpolaton ntated by Coons can be used to descrbe a surface by usng only the curves whch bound t. We want to address n ths document the problem of constructng a transfnte nterpolaton n a convex doman. Let us frst consder a few earler works related to polyhedra or transfnte nterpolatons. Polyhedra have already been nvestgated n the ancent era. The smplest examples mght be the Platonc solds whch have been examned by Plato. The study of polyhedra and polytopes are usually closely connected to the Euler-Poncaré characterstc [?, 13] as well studed by Schläfl [19]. Zegler has also done a lot of nvestgaton about polytopes and ther propertes [?]. The frst who ntated the dea of such transfnte nterpolatons was Coons [3] n the year sxtes. Later, Forrest has ntensvely used transfnte nterpolaton n order to generate curved coordnate systems [7]. Addtonally, Farn has ntroduced an mprovement [6] of Coons patches related to varatonal prncples. The tradtonal approach n generatng transfnte nterpolatons [3] takes the sum of the combnatons of data on opposte faces whch s then subtracted by some mxed quanttes. Gordon has expressed [?] that n terms of operators and boolean sums such as P Q = P +Q PQ. The tetrahedral transfnte nterpolaton of Mansfeld [?] stll used that constructon by combnng a node wth ts opposte face. Addtonally, the work of Mansfeld does not contan any topologc nformaton whatsoever. The major drawback of Mansfeld s approach s that t looses ts valdty when appled to more general frameworks. In fact, for more complcated polyhedra lke the ones llustrated n Fg. 1, the noton of opposte faces does not have any meanng except for a few specal cases. Perronnet has consdered [12] treatments of transfnte nterpolaton on a few specal polyhedra whch are not necessarly of tensor-product type such as tetrahedron and pentahedron. He has treated each polyhedron one by one and he dd not provde any general way of representng hs results. In addton, hs formulas are so lengthy that t s dffcult to use them for analytcal purpose. As for CAD preparaton, Brunnett and Randranarvony have nvested a lot to develop and mplement a method whch s approprate for surfaces n ntegral equatons [15]. That conssts manly n decomposng trmmed surfaces [1] nto four-sded patches [18] and n usng Coons patches to generate mappngs. Ther methods have already been successfully mplemented to CAD and molecular surfaces [16, 17]. Harbrecht and Randranarvony

3 3 [11] have used those surface CAD models for applcatons n Wavelet Boundary Element Method (BEM) solvers. In the framework of multfaceted surfaces, Loop et al. [?] have devsed a method of generalzng Bézer on multfaceted polygons. Besdes, the usual Bézer curve usng Bernsten polynomals [14] has already been generalzed [20] by Sedel to multdmensonal smplex. In the opnon of the author, the works of Loop and Sedel ft well wth the presented method n ths document because ther approach enables the defnton of the curved boundary faces accurately. The formula whch s presented here does not use combnatons of data from opposte faces. Instead, t consders the topologc enttes whch can stll be used even for domans whch are not n tensor product structures. In order to present that result, we organze ths paper as follows. We start n Secton 2 by recallng some nterestng facts from the usual 2D Coons map on the unt square. We show there another formulaton of the Coons map from topologc perspectve. That motvates the constructon of the transfnte nterpolaton n the subsequent sectons. That wll be followed by a precse demonstraton of the problem settng for more general convex domans n Secton 3 where we fnd also some ntroducton of mportant notatons. To prepare the formulaton of the transfnte nterpolaton, we ntroduce n Secton 4 varous auxlary results n form of lemmas. Among others, we wll meet there a projecton onto the faces of a convex doman and pertanng propertes. The formulaton and the proof of the man results are found n Secton 5 n whch we show also the affne stablty of the nterpolant. Toward the end of ths document, we show eventually some llustratve practcal results. 2. Coons map revsted In ths secton, we consder the usual [3, 10] Coons patch defned on [0, 1] 2 from topologc perspectve. That wll serve as a motvaton of the subsequent sectons. Let us consder four parametrc curves (2.1) α, β, γ, δ : [0, 1] R 2 whch are supposed to fulfll the next compatblty condtons that are llustrated n Fg. 2(a): (2.2) α(0) = δ(0), α(1) = β(0), γ(1) = β(1), γ(0) = δ(1). The Coons transfnte nterpolaton conssts n generatng a parametrc surface x(u, v) defned on the unt square [0, 1] 2 such that the boundary of the mage of x concdes wth the gven four curves: (2.3) x(u, 0) = α(u), x(u, 1) = γ(u) u [0, 1], x(0, v) = δ(v), x(1, v) = β(v) v [0, 1].

4 4 (a) (b) (c) (d) (e) (f) Fgure 1. A few examples of polytopes. The tradtonal method of creatng a transfnte nterpolaton by blendng data from opposte faces cannot be drectly appled to the general case.

5 5 N 4 N 3 γ(0) = δ(1) γ δ α α(0) = δ(0) γ(1) = β(1) β α(1) = β(0) v 1 v λ 2 λ 1 u = (u, v) λ 3 λ 4 N 2 N 1 u 1 u (a) (b) (c) Fgure 2. (a)boundary of a Coons patch, (b)examples of a Coons patch, (c)the four shaded areas determne the barycentrc coordnates λ 1, λ 2, λ 3, λ 4 of u = (u, v) wth respect to [N 1, N 2, N 3, N 4 ] As an llustraton, we can see n Fg. 2(b) the mage of a unform grd on the unt square by a Coons map x. Let f 0 and f 1 denote two arbtrary smooth functons satsfyng (2.4) f (j) = δ j, j = 0, 1 and f 0 (t) + f 1 (t) = 1 t [0, 1]. The functons f 0, f 1 whch are better known as blendng functons can be chosen n several ways [7]. Among others, three methods are usual: lnear, cubc and trgonometrc blendng functons: (2.5) f 0 (t) := 1 t and f 1 (t) := t, f 0 (t) := B0(t) 3 + B1(t) 3 and f 1 (t) := B2(t) 3 + B3(t), 3 f 0 (t) := cos 2 (0.5πt) and f 1 (t) := sn 2 (0.5πt). A constructon of a soluton to (2.3) whch was due to Coons frst blends the opposte curves β and δ. Then, t s added by a blend between the opposte curves α and γ. Fnally, those two blends are subtracted by mxed terms to ensure good nterpolaton at the corners. That can be expressed n matrx form as: (2.6) x(u, v) := [ [ ] [ ] δ(v) [ f 0 (u) f 1 (u) + α(u) γ(u) β(v) ] [ ] [ ] α(0) γ(0) f 0 (v) f 0 (u) f 1 (u). α(1) γ(1) f 1 (v) ] [ f 0 (v) f 1 (v) ] The above constructon s dffcult to generalze to more general polytopes except to some specal cases such as a smplex. That s because the noton of opposte faces s not obvous to fgure out. Now, we want to consder the above transfnte nterpolaton on the unt square but from topologc pont of vew. To that end, consder a functon

6 6 B defned on the boundary of D := [0, 1] 2 as follows (2.7) B(u, 0) := α(u), B(u, 1) := γ(u), B(0, v) := δ(v), B(1, v) := β(v). For a pont u = (u, v) nsde D, the expressons λ 1 := (1 u)(1 v), λ 2 := u(1 v), λ 3 := uv, λ 4 := (1 u)v determne barycentrc coordnates wth respect to N 1 := (0, 0), N 2 := (1, 0), N 3 := (1, 1), N 4 := (0, 1) as shown n Fg.2(c). Let us now consder the four functons (2.8) b 1 (u, v) := f 0 (u)f 0 (v), b 3 (u, v) := f 1 (u)f 1 (v), b 2 (u, v) := f 1 (u)f 0 (v), b 4 (u, v) := f 0 (u)f 1 (v). We deduce from (2.4) that they sum to unty and that b (u, v) = 0 when the barycentrc coordnate λ = 0. By usng the propertes of the blendng functons, the above expresson of x can be reformulated n terms of b as follows x(u, v) = b 1 (u, v) [ α(u) + δ(v) α(0) ] + b 2 (u, v) [ α(u) + β(v) α(1) ] (2.9) + b 3 (u, v) [ β(v) + γ(u) γ(1) ] or equvalently + b 4 (u, v) [ γ(u) + δ(v) γ(0) ], x(u, v) = ( 1) 3 { b 1 (u, v) [ ( 1) 1 B(u, 0) + ( 1) 1 B(0, v) + ( 1) 0 B(0, 0) ] + b 2 (u, v) [ ( 1) 1 B(u, 0) + ( 1) 1 B(1, v) + ( 1) 0 B(1, 0) ] + b 3 (u, v) [ ( 1) 1 B(1, v) + ( 1) 1 B(u, 1) + ( 1) 0 B(1, 1) ] + b 4 (u, v) [ ( 1) 1 B(u, 1) + ( 1) 1 B(0, v) + ( 1) 0 B(0, 1) ]}. The role of ths document s to generalze the above formula to general convex domans n R d havng M nodes N 1,...,N M. In fact, we wll develop a transfnte nterpolaton of the form M (2.10) ( 1) d+1 b (λ) ( 1) dm(π) B P Π,N (λ) =1 Π G() where G() s the set of topologc enttes (nodes and edges n 2D) such that the node N s ncdent upon them and P s a certan projecton whch wll be specfed subsequently. 3. Nomenclatures and problem settng There are several ways of defnng a polytope but the two most appled settngs are the one usng convex hulls and the one usng hyperplanes. They correspond to the

7 7 H 4 H 1 H2 H 3 H H 5 H H H 4 (a) (b) Fgure 3. (a)the large ponts, the bold lne segments and the shaded doman represent some faces 4 =1 Hσ where σ {0, +, }, (b)some chambers dentfed by ther sgn sequence σ 1 σ 2 σ 3 σ 4 splt R d \ { 4 =1H }. so-called V-polytope and H-polytope. Thus, a polytope D can be expressed n the followng forms (3.11) (3.12) (V polytope) : D = HULL { N R d : = 1,..., M }, (H polytope) : D = L H +. In both cases, we assume that polytopes are closed and bounded. The famous theorem [?,?,?] of Weyl-Mnkowsk states the equvalence of those two formulatons. Consequently, we wll use both settngs and choose at each crcumstance the one whch facltates the presentaton. In order that ths paper s readable by nonspecalts n polytopes, let us specfy those two settngs more accurately by ntroducng mportant notatons and propertes Hyperplane arrangements and polytopes. Let us consder an arrangement A whch [?] s defned to be a fnte set of (affne) hyperplanes H n R d where L. Each hyperplane H splts R d nto two half-spaces H + and H whch are both supposed to be topologcally open. Thus, the hyperplane H belongs to nether H + nor H. In order to manage the sgn dstrbutons, let us denote H 0 := H. More precsely, there are some a 1,, a d, a d+1 R such that (3.13) H 0 := {u = (u 1,..., u d ) R d : a 1 u a d u d + a d+1 = 0}, (3.14) H + := {u = (u 1,..., u d ) R d : a 1 u a d u d + a d+1 > 0}, (3.15) H := {u = (u 1,..., u d ) R d : a 1u a du d + a d+1 < 0}.

8 8 The hyperplanes H, J decompose the whole space R d nto nonempty subsets F of the form (3.16) F = L H σ, σ {0, +, } whch are called faces as llustrated n Fg. 3(a). A chamber (see Fg. 3(b)) s a face for whch σ 0 for all L. Note that chambers are open and convex and they form a partton of R d \ ( L H ) as llustrated n Fg. 3(b). A polytope D s defned to be the closure of a bounded chamber: (3.17) D = L H σ σ {+, }. The faces of D havng dmenson d 1 wll be termed facets whle those havng zero dmenson smply nodes or vertces. For three ndex subsets I 1, I 2, I 3 L where I I j =, one defnes the dmenson of a subset of the form (3.18) B = [ ] [ ] [ ] I0 H 0 I1 H + I2 H R d to be the dmenson of I0 H 0 as llustrated n Fg. 4(a). If I 0 =, then we suppose I0 H 0 := Rd so that dm(b) = d. In partcular, as seen from (3.17) the dmenson of a polytope D n R d s d. Let us now consder the ncdence n the set F of faces of an arrangement A. An element Π 1 F s called a face of Π 2 F f for each L, one has (see Fg. 4(b)) (3.19) σ (Π 1 ) = 0 or σ (Π 1 ) = σ (Π 2 ). In such a case, we denote Π 1 Π 2 and we say Π 1 s ncdent upon Π 2. The relaton between a face and the faces ncdent upon t s [?] that (3.20) C = B. B C Snce we are only nterested n one sngle polytope D, we may suppose wthout loss of generalty that all the hyperplanes H n the arrangement A are ncdent upon D. A polytope s sad to be n a general poston f the correspondng hyperplanes admt the next propertes: (3.21) (3.22) If k d, then dm(h 1 H k ) = d k, If k > d, then H 1 H k =. The Euler-Poncaré characterstc operator χ s defned to be an nteger valued functon defned for each V whch s a fnte unon of faces as n (3.16) such that χ fulflls the followng condtons (E1) χ( ) = 0, (E2) χ(f) = ( 1) dm(f) for F of the form (3.16),

9 Π 1 Π 2 Π 3 9 H 3 H 1 B = H 1 H + 2 H 3 H 2 (a) (b) Fgure 4. (a) The subset B = H 1 H 2 + H 3 R 2 has dmenson dm(b) = dm(h 1 ) = 1. (b) We have Π 1 Π 2 and Π 3 Π 2. (E3) χ(v W) = χ(v ) + χ(w) χ(v W). From now on, we wll generally adopt the notaton of Coxeter and Schläff by usng Π for desgnng faces of a polytope except the nodes whch we denote by N p Barycentrc coordnates and problem settng. Consder a polytope D n R d. Accordng to the Weyl-Mnkowsk property [?,?], the closure of the polytope D s the convex hull of the nodes whch are ncdent upon D,.e. (3.23) D = HULL { N D : dm(n) = 0 }. We wll denote the nodes of the polytope D by N 1,..., N M. A barycentrc coordnate λ s a sequence of functons (λ 1,...,λ M ) (3.24) λ : D R + {0} such that for each u D we have M (3.25) λ (u) = 1, and u = =1 M λ (u)n. We also assume the unqueness property: for gven µ 1,..., µ M R + {0} such that µ µ M = 1, there s a unque u D such that (3.26) λ (u) = µ. As opposed to the smplex case, there mght be several methods of defnng the functons λ(u) whch fulfll the two crtera (3.25) and (3.26) n a general convex doman. The purpose of ths document s not a revew of barycentrc coordnates. Joe Warren has done a good survey [?] for barycentrc coordnates on convex domans. Throughout ths document, we fx one way of defnng barycentrc coordnate functons. Because of the unqueness property (3.26), we wll wrte nterchangeably u D or λ = (λ ) = (λ (u)) D for notatonal convenence. =1

10 10 Let us now formulate the problem settng accurately. Suppose that we have a boundary functon (3.27) B : D \ D R p. In ths document, our ntent s to fnd a transfnte nterpolant of B defned on the whole D (3.28) T [B] : D R p such that T [B](λ) = B(λ) for all λ D \ D and that T [B] s stable under affne transform such that (3.29) T [ A(B) ] = A ( T [B] ) for any affne transform A. The affne stablty property s very mportant n applcatons because the boundary faces are usually gven n terms of dscrete grds such as control nets of splnes or Bézer patches [?, 20]. Suppose that the transfnte nterpolaton T of the ntal faces has been computed. In order to fnd the transfnte nterpolaton wth respect to the transformed control ponts, we need only to apply the affne transform to T. 4. Topologc relatons and projectons Before beng able to solve the problem n (3.28), we need frst to concentrate n ths secton on the ncdence propertes of topologc enttes. In partcular, we wll nvestgate a set G(q) whch s composed of the faces havng a node N q as an ncdent node. Among others, we wll analyze the parttonng of G(q). Afterwards, a projecton wll be ntroduced and we examne some related propertes Incdence nformaton. Consder an arrangement A of hyperplanes H R d where L. Let D be a polytope of dmenson d whch s the closure of the chamber: (4.30) D = L H σ σ {+, }. For each fxed node N q of the polytope D, we defne (4.31) G(q) := { Π D s.t. N q Π and Π D }. Snce the node N q s a face of D, there must exst J = J (q) L such that (4.32) N q = [ J (q) H ] [ L\J (q) H σ ],

11 11 N q = Π Π spec N q Π Π spec Π Π (a) (b) Fgure 5. The node N q s ncdent upon both Π and Π whch are faces of D. We have Π Π spec whle Π Π spec as ntroduced n (3.16) where dm( J H ) = 0. Hence, from defnton (3.19) we obtan that every face Π of D such that N q s ncdent upon Π must be of the form Π = Π K = [ ] [ ] K H J \K H σ (4.33) Wq G(q), where (4.34) W q n whch K J. := L\J H σ Consder any fxed facet Π spec of the polytope D. Snce Π spec s of dmenson d 1, there must exst some r L such that [ ] (4.35) Π spec = H r L\{r} H σ where σ 0 from (4.30) for all L \ {r}. Suppose that a node N q s ncdent upon Π spec. Let us ntroduce two sets G 1 = G 1 (q, Π spec ) and G 2 = G 2 (q, Π spec ) such as { [ ] [ ] } G 1 := H r P\{r} H J \P H σ (4.36) Wq : r P J { [ ] [ ] } G 2 := Hr σr P\{r} H J \P H σ (4.37) Wq : r P J where the sgns σ are the same as that n the polytope D from (4.30). Fg. 5 depcts examples of faces Π and Π whch belong respectvely to G 1 and G 2. Before beng able to prove the man result of ths document, we need to treat several prelmnary notons. Let us start from the followng easy but mportant observatons. Lemma 4.1. Consder a polytope D n general poston. For each facet Π spec of the polytope D and a node N q ncdent upon Π spec, the subsets G 1 (q, Π spec ) and G 2 (q, Π spec ) from relatons (4.36) and (4.37) possess the next propertes: (C0) Π spec G 1 (q, Π spec ) and D G 2 (q, Π spec ).

12 12 (C1) The sets G 1 \ {Π spec } and G 2 \ {D} form a partton of G(q) \ {Π spec }. (C2) For each Π G 1 (q, Π spec ) \ {Π spec }, there exsts a unque Π G 2 (q, Π spec ) \ {D} such that (4.38) ( 1) dm(π) ( 1) dm( Π) = 1. PROOF. The membershps n (C0) are readly obtaned from (4.30), (4.35) and the defntons of G 1 and G 2. As for (C1), we observe that every element Π of G 1 can be expressed as Π K n (4.33) by defnng K := P. That holds also for G 2 n whch we defne K := P \ {r}. Thus, we obtan the nclusons (4.39) G 1 \ {Π spec } G(q) \ {Π spec }, and G 2 \ {D} G(q) \ {Π spec }. We want now to show that G(q) (G 1 \ {Π spec }) (G 2 \ {D}). Let Π K G(q) be gven as n (4.33). We dstngush two cases wth respect to K. Case 1: If K contans r, then (4.40) Π K = H r [ K\{r} H ] [ J \K H σ Hence, Π K belongs to G 1 where P := K. Case 2: If r K, then (4.41) Π K = H σr r Hence, Π K belongs to G 2 where P := K {r}. ] Wq. [ ] [ ] K H J \(K {r}) H σ Wq. Snce G 1 and G 2 are obvously dsjont, we obtan (C1). Let Π G 1 \ {Π spec } be gven. Thus, there s P J such that r P and (4.42) Π = Π P := H r [ P\{r} H ] [ J \P H σ ] W q G 1. Let us consder (4.43) Π = ΠP := H σr r [ P\{r} H ] [ J \P H σ ] W q G 2. Snce Π and Π dffer only wth respect to one sngle hyperplane H r, we deduce from the hypothess of general poston (3.21) (4.44) dm( Π) = dm(π) + 1. Hence, we obtan (C2). Q.E.D.

13 Projectons onto faces. For a polytope D, let us consder an arbtrary face Π and let C desgnate the ndces of the nodes ncdent upon Π. Consder now any node N q where q C. Let us ntroduce a projecton dependng on both Π and N q : (4.45) P := P Π,Nq : D Π such that for λ = (λ 1,..., λ M ) D, the barycentrc coordnates of the mage µ := P(λ) = (µ 1,..., µ M ) are defned as (4.46) (4.47) (4.48) µ j := 0 f N j Π µ j := λ j f N j Π and j q (. e. j C \ {q}) µ q := 1 λ. C\{q} We can mmedately observe that d j=0 µ j = 1. Lemma 4.2. Consder a face Π spec havng dmenson (d 1) of a polytope D R d. Suppose that the nodes ncdent upon Π spec are N p(1),...,n p(z). For every vertex N q Π spec, we have (4.49) P Π spec,n q (λ) = λ, for each λ = (λ 1,..., λ M ) such that λ = 0 for N {N p(1),..., N p(z) }. PROOF. Consder λ = (λ 1,..., λ M ) such that λ = 0 for N {N p(1),..., N p(z) } and let us denote µ := P Π spec,n q (λ). We want to show that µ j = λ j n whch we examne several stuatons about j = 1,..., M. Frst, consder j {p(1),..., p(z)}. Thus, we have λ j = 0. Snce N j Π spec, we deduce from (4.46) that µ j = 0. Hence, µ j = λ j. Consder now an ndex j {p(1),..., p(z)} such that N j N q. By the defnton of P n (4.47), we obtan µ j = λ j. On the other hand, due to the partton of unty, we obtan: (4.50) µ q = 1 µ = 1 (4.51) = 1 {p(1),...,p(z)}\{q} N λ = λ q. =1 q {p(1),...,p(z)}\{q} That means, for any j = 1,..., M we always have µ j = λ j. As a consequence, we obtan P Π spec,n q (λ) = λ for such λ. λ Q.E.D.

14 14 Lemma 4.3. Consder a face Π spec havng dmenson (d 1) of a polytope D R d whch s n general poston. Let a node N q be ncdent upon Π spec. Suppose that Π spec s supported by the hyperplane H r. Consder two faces (4.52) (4.53) Π = Π P := H r [ P\{r} H ] [ J \P H σ ] W q, Π = Π P := Hr σr [ P\{r} H ] [ J \P H σ ] W q such that dm(π) and dm( Π) have dfferent party. We have the followng dentty (4.54) ( 1) dm(π) B P Π,Nq (λ) + ( 1) dm( Π) B P Π,N q (λ) = 0 for all λ = (λ 1,..., λ M ) such that λ = 0 for N Π spec. PROOF. Observe frst that Π Π spec whereas Π Π spec. Consder µ := P Π,Nq (λ) and µ := P Π,N q (λ). We ntend to demonstrate that µ j = µ j for each j = 1,..., M. We wll consder several cases wth respect to j = 1,..., M. Case 1: If N j Π spec then by hypothess we have λ j = 0. Snce Π Π spec, N j s not ncdent upon Π ether. Hence, accordng to (4.46) about the defnton of P we obtan µ j = 0. As for the determnaton of µ j, we consder two stuatons. If N j Π, we obtan from (4.47) that µ j = λ j = 0. In the stuaton N j Π, we deduce from (4.46) that µ j = 0. Thus, n both stuatons, we have µ j = µ j. Case 2: Suppose now that N j Π spec and that N j N q. We consder agan two stuatons. In the stuaton where N j Π, we have µ j = 0. Snce Π Π, we have also N j Π. Thus, µ j = 0. Hence, µ j = µ j. Consder now the stuaton where N j Π. That s µ j = λ j. Snce N j s a face of D, we have from (3.16) (4.55) N j = L H z where z {0, σ }. Snce N j Π spec whch s supported by H r, z r = 0 so that relaton (4.55) becomes (4.56) N j = H r [ L\{r} H z ]. We apply the same process to N j Π n order to obtan (4.57) N j = H r [ P\{r} H ] [ L\P H z ] where z {0, σ } f L \ P. That s to say N j Π. As a consequence, µ j = λ j or equvalently µ j = µ j.

15 15 Case 3: Suppose now that N j = N q. From the frst two cases, we mmedately obtan (4.58) µ q = 1 M µ = 1 =1 q M µ = µ q. =1 q Therefore, we obtan P Π,Nq (λ) = P Π,N q (λ). As a result, we deduce (4.59) ( 1) dm(π) B P Π,Nq (λ) = ( 1) dm(π) B P Π,Nq (λ). Q.E.D. 5. Constructon of the transfnte nterpolaton In the prevous sectons, we saw dfferent defntons and lemmas for the generaton of the projectons on faces. Now, we want to use them to construct a transfnte nterpolaton on the whole polytope D Blendng functons and topologc formula. To generalze the case of 2D Coons map, let us ntroduce the noton of barycentrc blendng functons defned for all λ = (λ 1,..., λ M ) n the reference doman D. In fact, we defne them to be a set of functons b, = 1,..., M admttng the followng three propertes: (5.60) (5.61) (5.62) (P1) If λ p = 0 n λ = (λ 1,..., λ M ) then b p (λ) = 0, (P2) For each p = 1,..., M, we have b p (λ) = 1 f λ j = δ j,p, M (P3) b (λ) = 1 λ = (λ 1,..., λ M ) D. =1 As n the case of 2D transfnte nterpolatons, the smplest way of choosng the blendng functons s to use the lnear ones whch are n the current case: (5.63) b (λ) := λ, = 1,..., M. Theorem 5.1. Consder a polytope D R d n general poston havng M nodes N 1,..., N M. The functon M (5.64) T [B](λ) = ( 1) d+1 b (λ) ( 1) dm(π) B P Π,N (λ) =1 Π G() determnes a transfnte nterpolaton. Thus, for a facet Π spec of D such that the nodes of Π spec are N p(1),...,n p(z), we have (5.65) T [B](λ) = B(λ) for every λ = (λ 1,..., λ M ) such that λ q = 0 for all q {p(1),..., p(z)}.

16 16 PROOF. Let us consder λ such that λ q = 0 for all q V := {p(1),..., p(z)}. We obtan from property (P1) of the barycentrc blendng functons b n (5.60) that (5.66) (5.67) M T [B](λ) = ( 1) d+1 b (λ) ( 1) dm(π) B P Π,N (λ) =1 = ( 1) d+1 q V Π G() b q (λ) Π G(q) ( 1) dm(π) B P Π,Nq (λ). Let us denote by F(q, Π spec ) the set G(q) \ {Π spec } so that we obtan T [B](λ) = ( 1) [ d+1 b q (λ) ( 1) d 1 B P Π spec,n q(λ) + q V ] ( 1) dm(π) B P Π,Nq (λ). Π F(q,Π spec ) We use now Lemma 4.1 to partton F(q, Π spec ) nto F 1 (q, Π spec ) := G 1 (q, Π spec )\{Π spec } and F 2 (q, Π spec ) := G 2 (q, Π spec )\{D}. Consequently, we break up the above summaton as T [B](λ) = ( 1) [ d+1 b q (λ) ( 1) d 1 B P Π spec,n q(λ) + q V ] ( 1) dm(π) B P Π,Nq (λ) + ( 1) dm(π) B P Π,Nq (λ). Π F 1 (q,π spec ) As a result, we obtan T (λ) = ( 1) d+1 q V Π F 1 (q,π spec ) Π F 2 (q,π spec ) [ b q (λ) ( 1) d 1 B P Π spec,n q (λ) + ( )] ( 1) dm(π) B P Π,Nq (λ) + ( 1) dm( Π) B (λ). P Π,Nq where Π F 2 (q, Π spec ) s the unque face correspondng to Π F 1 (q, Π spec ) as specfed n (4.38). Due to a combnaton of property (C2) n Lemma 4.1 and Lemma 4.3, the last two terms cancel out smultaneously, so that we obtan (5.68) T (λ) = ( 1) d+1 q V ( 1) d 1 b q (λ) B P Π spec,n q (λ). Accordng to relaton (4.49) of Lemma 4.2, P Π spec,n q (λ) = λ rrespectve of q V. Hence, (5.69) T (λ) = B(λ) q V b q (λ) = B(λ). Q.E.D.

17 Affne stablty. In ths secton, we would lke to concentrate on the proof of the affne stablty of the proposed transfnte nterpolaton. Theorem 5.2. For the d-dmensonal polytope D n general poston havng nodes N 1,..., N M, the nterpolant M (5.70) T [B](λ) := ( 1) d+1 b q (λ) ( 1) dm(π) B P Π,Nq (λ) s affnely stable. Thus, we have q=1 Π G(q) (5.71) T [ A(B) ] = A ( T [B] ) for any affne transform A. PROOF. Let ( us consder an affne transform A : R p R p. Thus, we have the property that ) A j I µ jv j = j I µ ja(v j ) for any coeffcents (µ j ) j I such that j I µ j = 1. For each node N q ncdent upon the polytope D, let us agan consder J (q) L such that N q = [ J (q) H ] [ L\J (q) H σ ] as ntroduced n relaton (4.32). Hence, every face Π of D upon whch N q s ncdent must be of the form (5.72) Π = Π K := [ K H ] [ J \K H σ ] Wq where K J. On the one hand, by applyng (3.20) to the subarrangement (5.73) B := {H where J } A, we obtan the topologc closure (5.74) = J H σ K J ( [ ] [ ] ) K H J \K H σ. (Observe that (5.74) s not the unon over K J of Π K because we have L \ K n (5.72) an ntersecton wth W q ). As a consequence, snce we took the unon of dsjont sets n (5.74) we obtan ( ) χ H σ = [ ] ] χ( ) K H [ J \K H σ K J J = K J = K J = K J [ ] ] exp 1 dm( ) K H [ J \K H σ [ ] ] ) exp 1 dm( K H [ L\K H σ Wq ( 1) dm ( Π K ).

18 18 On the other hand, ( χ J ) H σ = χ(r d ) χ ( R d \ ) H σ J = ( 1) d χ ( J H ρ ) where ρ := σ. From the ncluson-excluson property of the Euler-Poncaré operator, we have χ( H ρ ) = χ(h ρ ) χ(h ρ H ρ j j ) + J J <j,j J + <j<k,j,k J χ(h ρ H ρ j j H ρ k k ) + ( 1)card(J ) χ( H ρ ). J For any subset R J, we have R Hρ because each H contans N q for J. Therefore, we have dm( R Hρ ) = d because ρ 0. Hence, χ( R Hρ ) = ( 1)d. As a consequence, we obtan (5.75) (5.76) χ( card(j ) H ρ ) = ( ) card(j ) ( 1) j 1 ( 1) d j J j=1 card(j ) ( ) card(j ) = ( 1) d+1 ( 1) j j (5.77) = ( 1) d+1 ( 1) = ( 1) d. Hence, χ( ( J Hρ ) ) = ( 1)d. That mples χ J Hσ ( ) K J ( 1)dm Π K = 0. That s, (5.78) ( 1) dm(d) + ( 1) dm(π) = 0 whch means (5.79) ( 1) d+1 Π G(q) Π G(q) j=1 ( 1) dm(π) = 1. As a consequence, we obtan that the coeffcents sum to unty such as M (5.80) ( 1) d+1 b q (λ) ( 1) dm(π) = q=1 Π G(q) whch mples the affne stablty (5.71). M b q (λ) = 1. q=1 = 0. Hence, we obtan Q.E.D.

19 19 (a) (b) (c) (d) Fgure 6. Transfnte nterpolaton defned on some convex polytopes. In (d) we recognze the usual Coons patch where the polygon of defnton s the unt square Concluson and llustratve results. In order to solve the problem of transfnte nterpolaton, we have proposed a topologc formula of the form M (5.81) ( 1) d+1 b (λ) ( 1) dm(π) B P Π,N (λ) =1 Π G() where G() s the set of topologc enttes upon whch the node N s ncdent. In contrast to some other methods, the presented technque enables the treatment of multfaceted domans whch are not necessarly of tensor product structure. Dong further analytcal works wth such short formula should be more practcal than usng long ones. Therefore, we beleve that the above method has good applcatons n theoretcal analyss such

20 20 as search for upper bounds or error estmatons. Unfortunately, we could only prove the result under the condton of general poston of the polytopes. We thnk that such a condton s only redundant and the formula remans vald wthout any postonal restrcton but the proof for that seems to be very dffcult. That wll be an open queston whch we mght treat n the future. On the other hand, the above method can also be used n appled stuatons. For nstance, let us brefly show some practcal results from the formerly proposed approach. Note that the results shown here are not meant to be a substtuton of the demonstratons. They are only for llustratve purpose. We have consdered some transfnte nterpolatons where the boundary curves or surfaces are gven. We appled the above transfnte nterpolaton to fnd the mages nsde the convex domans. In Fg.6, we have gathered the results whch consst of the mage by the transfnte nterpolaton of some dscretzaton wthn some convex domans. Acknowledgment Durng the PhD work of the author, Prof. Brunnett (Unversty of Chemntz) has helped hm n the topc of computatonal geometry. Prof. Schneder (Unversty of Berln) and Prof. Harbrecht (Unversty of Bonn) have supported the author n hs Post-doctoral postons. The author s grateful to the HCM (Hausdorff Center for Mathematcs) and ts Excellence Cluster program for supportng hs vst at the Unversty of Bonn. References [1] G. Brunnett: Geometrc desgn wth trmmed surfaces. Comput. Suppl. 10, (1995). [2] CFD General Notaton System (Standard Interface Data Structures), [3] S. Coons: Surfaces for computer aded desgn of space forms. Project MAC, Department of Mechancal Engneerng n MIT, Revsed to MAC-TR-41, (1967). [4] H. Coxeter: Regular polytopes, The Macmllan Company, New York, [5] D. Eppsten: Nneteen Proofs of Euler s formula: V E + F = 2. The geometry junkyard, eppsten/junkyard/euler, [6] G. Farn, D. Hansford: Dscrete Coons patches, Comput. Aded Geom. Des. 16, No. 7, (1999). [7] A. Forrest: On Coons and other methods for the representaton of curved surfaces. Comput. Graph. Img. Process. 1, (1972). [8] W. Gordon, C. Hall: Constructon of curvlnear co-ordnate systems and applcatons to mesh generaton. Int. J. Numer. Methods Eng. 7, (1973). [9] W. Gordon, C. Hall: Transfnte element methods: blendng-functon nterpolaton over arbtrary curved element domans. Numer. Math. 21, (1973).

21 21 [10] J. Gregory: Interpolaton to boundary data on the smplex. Comput. Aded Geom. Des. 2, (1985). [11] H. Harbrecht, M. Randranarvony: From Computer Aded Desgn to wavelet BEM. To appear n Journal Comput. Vs. Sc., (2008). [12] A. Perronnet: Interpolaton transfne sur le trangle, le tetraèdre et le pentaèdre. Applcaton à la créaton de mallage et à la condton de Drchlet. C. R. Acad. Sc., Pars, Sér. I, Math. 326, No (1998). [13] H. Poncaré: Sur la généralsaton d un Théorème d Euler relatf aux polyèdre. Comptes rendus hebdomadares des séances de l Academe des Scences. Pars, 117, (1893). [14] H. Prautzsch, W. Boehm, M. Paluszny: Bézer and B-Splne technques, Sprnger, Berln, [15] M. Randranarvony, G. Brunnett: Preparaton of CAD and molecular surfaces for meshfree solvers. In: Proc. Meshfree Methods for Partal Dfferental Equatons, Bonn, LNCS Sprnger, (2007). [16] M. Randranarvony: Geometrc processng of CAD data and meshes as nput of ntegral equaton solvers. PhD thess, Technsche Unverstät Chemntz, [17] M. Randranarvony, G. Brunnett: Molecular surface decomposton usng geometrc technques. In: Proc. Conf. Bldverarbetung für de Medzne, Berln, (2008). [18] M. Randranarvony: Quadrlateral Decomposton by two-ear property resultng n CAD segmentaton. In: Proc. Ffth Internatonal Conference on Computatonal Geometry, Vence, (2008). [19] L. Schläfl: Theore der velfachen Kontnutät. Denkschrften der Schwezerschen naturforschenden Gesellschaft. 38, (1901). [20] H. Sedel: Polar forms for geometrcally contnuous splne curves of arbtrary degree. ACM Trans. Graph. 12, No. 1, 1 34 (1993). Maharavo Randranarvony, Insttut für Numersche Smulaton, Unverstät Bonn, Wegelerstr. 6, Bonn, Germany. E-mal address: randran@ns.un-bonn.de