Parametrization and smooth approximation of surface triangulations

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1 Parametrization and smooth aroximation of surface triangulations Michael S. Floater * Deartamento de Matemática Alicada, Universidad de Zaragoza, Sain December 1995, revised June 1996 Abstract. A method based on grah theory is investigated for creating global arametrizations for surface triangulations for the urose of smooth surface fitting. The arametrizations, which are lanar triangulations, are the solutions of linear systems based on convex combinations. A articular arametrization, called shae-reserving, is found to lead to visually smooth surface aroximations. 1. Introduction A standard aroach to fitting a smooth arametric curve c(t) through a given sequence of oints x i = (x i,y i,z i ) IR 3, i = 1,...,N is to first make a arametrization, a corresonding increasing sequence of arameter values t i. By finding smooth functions x,y,z : [t 1,t N ] IR for which x(t i ) = x i, y(t i ) = y i, z(t i ) = z i, an interolatory curve c(t) = (x(t), y(t), z(t)) results. Two commonly used arametrizations are the uniform and chord length ones, in which (t i+1 t i )/(t i t i 1 ) = 1, (1) and (t i+1 t i )/(t i t i 1 ) = x i+1 x i / x i x i 1, (2) resectively [2], [6]. In this aer we investigate, by way of numerical examles, a method for making arametrizations for a surface triangulation S (having triangular facets and a boundary), based on a method roosed by Tutte [25] for making straight line drawings of lanar grahs. The nodes x i IR 3 of S are maed to oints u i = (u i,v i ) D, for some convex D IR 2, in such a way that the image of S is a lanar triangulation P. The basic idea is to set each u i to be a convex combination of its neighbours. By considering certain articular convex combinations, three articular arametrizations are introduced: uniform, weighted least squares, and shae-reserving. The first generalizes (1) and the second and third both generalize (2). The name shae-reserving is chosen because this arametrization has a reroduction roerty (Proosition 6). Using a suitable scattered data method [11], smooth functions x,y,z : D IR, can be constructed indeendently to satisfy x(u i ) = x i, y(u i ) = y i, z(u i ) = z i. (3) A arametric surface s : D IR 3 satisfying s(u i ) = x i then results from setting s(u) = (x(u),y(u),z(u)), u D. (4) * The author was artly suorted by the EU roject CHRX-CT

2 Though many surface triangulations cannot be maed into IR 2 without considerable deformation, it has been found that surface interolants s based on the shae-reserving arametrization are visually smooth. Moreover choosing D to be the unit square and using a two hase method of scattered data aroximation [11], the entire surface triangulation can be smoothly aroximated by a single tensor-roduct sline surface s, a oular surface tye in alications. Most existing methods for aroximating surface triangulations fit one or more iecewise olynomial atches to each triangular facet so that the overall geometric continuity is G k, for some k. Such methods are surveyed by Lounsberry, Mann and DeRose [15]. However this will generate a large amount of data when the number of triangles is high. If the triangulation is a single atch it may be referable to aroximate it with a single arametric surface, for examle a tensor-roduct sline. Milroy et al. [18] have made such aroximations directly via a non-linear minimization. Techniques for maing and transforming triangulations have been roosed by Maillot et al. [16] and Li et al. [14]. The method in [16] is to minimize an energy functional based on the theory of elasticity, while that in [14] is a numerical aroximation, using finite elements or finite differences of ellitic equations. 2. Grahs and triangulations We shall draw some standard definitions from grah theory; see [17]. A (simle) grah G = G(V,E) is a set of nodes V = {i : i = 1,...,N} and a set of edges E, a subset of the set of all unordered airs of nodes (i,j), (i j). A node j is a neighbour of node i if (i,j) E. The degree d i of the i-th node is the number of its neighbours. A grah H is a subgrah of a grah G if both its nodes and edges belong to G. A grah G is said to be lanar if it can be embedded in the lane such that (a) each node i is maed to a oint in IR 2, (b) each edge (i,j) E is maed to a curve whose endoints are i and j, (c) the only intersections between curves are at common endoints. Such a lanar embedding of G is referred to by Nishizeki [19] as a lane grah. A lane grah artitions the lane into connected regions called faces. In articular the unbounded face is called the outer face. Different embeddings of a lanar grah may artition the lane in different ways and this justifies the searate term lane grah. The subsequent definitions have been chosen to suit the articular alications we have in mind. Let G be a lane grah and define G as the subgrah consisting of all nodes and edges which are incident on the outer face. If G is a simle lanar curve then we shall say that G is simly connected and that G is the boundary of G. We shall say that a lane grah is triangulated if all its bounded faces are triangular, i.e. have three edges (no demand is laced on the outer face). In this aer G will always be a simly-connected triangulated lane grah, with N 3. An examle is shown in Figure 1. Though the methods to be discussed can certainly be generalized to a wider range of grahs, including many having faces with more than three edges, this might comlicate the exosition unduly and moreover triangulated grahs enjoy some articularly nice roerties (see Proosition 4). We can now describe the triangulations we are interested in. 2

3 G z v P S y x u Fig. 1. A grah, a surface triangulation, and arametrization. Definition 1. A lanar triangulation P is a simly-connected triangulated lane grah whose edges are straight lines. Let F be the set of bounded (triangular) faces of G and we write G = G(V,E,F). Definition 2. A surface triangulation S = S(G,X) is an embedding in IR 3 of a simlyconnected triangulated lane grah G(V,E,F), V = {i : i = 1,...,N}, with node set X = {x i IR 3 : i = 1,...,N}, and with straight lines for edges and triangular facets for faces. Figure 1 shows an examle. We remark that our surface triangulation is less general than the triangulations discussed by Schumaker [23] where the toology can be arbitrary. 3. Parametrizations Since the boundary G of a simly-connected lane grah G is a simle closed lanar curve it has a well-defined orientation, which we will take as the anticlockwise direction. Definition 3. Two simly-connected triangulated lane grahs G 1 and G 2 are isomorhic if there is a 1-1 corresondence between their nodes, edges, and faces in a such a way that: corresonding edges join corresonding oints; corresonding faces join corresonding oints and edges; and the two anticlockwise sequences of nodes in G 1 and G 2 corresond. We are now ready to formulate our notion of a arametrization for a surface triangulation; see Figure 1. Definition 4. A arametrization of a surface triangulation S(G, X) is any lanar triangulation P which is isomorhic to G. In this aer we are concerned with finding a arametrization P for a given surface triangulation S for the urose of aroximating S by a smooth arametric surface. We note from Definition 4 that P need not in general deend on the geometry of S, defined by the oints in X. However our numerical examles will demonstrate that in order to aroximate S by a smooth surface, the local geometry of P ought to mimic the local geometry of S. 3

4 Examle 1. Let u 1,...,u N, N 3, be arbitrary airwise distinct oints in IR 2, not all colinear, and let z 1,...,z N IR be arbitrary. Let P be any triangulation, for examle the Delaunay triangulation [20], [21], of the u i and set x i = (u i,v i,z i ), for i = 1,...,N, where u i = (u i,v i ). Then P is a lanar triangulation and S(P,X) is a surface triangulation with arametrization P. In Examle 1, a natural arametrization P is rovided for aroximating S. If z i = f(u i,v i ) for some unknown C 1 function f : IR 2 IR, it is usual to aroximate f by a sline function g : IR 2 IR. Yet, esecially if f has stee gradients, a much better result might be obtained by aroximating S and hence the surface (u,v) (u,v,f(u,v)) by a arametric surface s : IR 2 IR 3. This allows the ossibility of choosing other arametrizations of S which may yield smoother surface aroximations. Examle 2. Let s : [0,1] [0,1] IR 3 be a smooth surface and let u 1,...,u N, N 3, be arbitrary airwise distinct oints in [0, 1] [0, 1], not all colinear. Let P be any triangulation of the u i. If x i = s(u i ), X = {x i : i = 1,...,N} then S(P,X) is a surface triangulation and P is a arametrization of it. We remark that S in Examle 2 need not be injectively rojectable onto the lane as in Examle 1. Surface triangulations also arise as crude aroximations of various kinds of nonexlicit data sets such as cross-sectional data [22], [9] and densely scattered data samled from three-dimensional objects [13], rovided the toology is suitable. In these cases usually no arametrization is available and it is esecially imortant to be able to construct one. 4. Method for arametrization Finding any arametrization P for a given surface triangulation S can be seen to be equivalent to finding a lanar triangulation (with straight line edges) isomorhic to a given triangulated lane grah G. In the language of grah theory this is a matter of constructing a straight line drawing of a lane grah. It was first shown by Fáry [8], using a roof by induction, that every (simle) lane grah has a straight line drawing. Later, in [24,25], Tutte gave a constructive roof emloying a method known as the barycentric maing for making such a drawing. The boundary nodes of G are maed to the boundary of any convex olygon and every internal node in the drawing is defined to be the centre of mass of its neighbours, their barycentre; see Figure 2. The reader is referred to Chiba et al. [3] and Nishizeki [19] for discussions and further methods for constructing straight line drawings of lanar grahs. Tutte s barycentric maing is the start oint for the remainder of this aer. In the following ages we will roceed by (i) first observing that the barycentric maing can be made much more general by allowing each internal node to be any convex combination of its neighbours, and (ii) rovide an algorithm for choosing the convex combinations so that P locally reserves the shae of S. Let S(G,X), with G = G(V,E,F), be a surface triangulation with node set X = {x i = (x i,y i,z i )}, 1 i N. Assume, by relabelling nodes if necessary, that x 1,...,x n are the 4

5 Fig. 2. The barycentric maing. internal nodes and x n+1,...,x N are the boundary nodes in any anticlockwise sequence with resect to the boundary of G. Let K = N n. Consider the following definitions. (1) Choose u n+1,...,u N, (5) to be the vertices of any K-sided convex olygon D IR 2 in an anticlockwise sequence. (2) For each i {1,...,n}, choose any set of real numbers λ i,j for j = 1,...,N such that λ i,j = 0, (i,j) E, λ i,j > 0, (i,j) E, N λ i,j = 1, (6) j=1 and define u 1,...,u n to be the solutions of the linear system of equations, Finally let u i = N λ i,j u j, i = 1,...,n. (7) j=1 P = P(G,U b,λ), with U b = {u n+1,...,u N }, and Λ = (λ i,j ) i=1,...,n, j=1,...,n, (8) be the embedding of G(V,E,F) in IR 2 with nodes u 1,...,u n, straight lines for edges, and triangles for faces. Note that the equations in (7) demand that every internal oint u i be a strict convex combination of its neighbours. We wish to show that P is a arametrization of S. From Definitions 3 and 4 we see, rovided P is well-defined, that since P is isomorhic to G, P is a arametrization rovided it is a valid lanar triangulation. It is thus only necessary to show that the nodes u i P are well-defined, distinct and that no two edges intersect excet at common end oints. Tutte [25] was interested in a method of making a straight line drawing of a lanar grah. For a large class of lane grahs G, including triangulated ones, he roosed equations (7) in the secial case when λ i,j = 1/d i for all (i,j) E, i = 1,...,n (i.e. u i is the barycentre of its neighbours). He roved in this case that they have a unique 5

6 solution and that P is a straight line drawing, that is to say that u 1,...,u n are unique, distinct and no two edges of P intersect excet at common end oints. In our case G is a simly-connected triangulated lane grah. Thus P is a arametrization of S(G, X) whenever λ i,j = 1/d i for all (i,j) E, i = 1,...,n. We now consider the case of general λ i,j in (6). The imortant thing is to show that (7) has a unique solution. To this end, note that it can be rewritten in the form u i n λ i,j u j = j=1 N j=n+1 λ i,j u j, i = 1,...,n. (9) By considering the two comonents u i and v i of u i searately this is equivalent to the two matrix equations Au = b 1, Av = b 2. (10) Here u and v are the column vectors (u 1...u n ) T and (v 1...v n ) T resectively. The matrix A is n n having elements a i,i = 1, a i,j = λ i,j, j i. The existence and uniqueness of a solution to (7) is thus equivalent to the non-singularity of the matrix A. The roof of the following result is based on a roof in [1] of the invertibility of a matrix occuring from a finite difference aroximation of Poisson s equation u = f in two dimensions. Proosition 1. The matrix A is non-singular. Proof. That A is non-singular is equivalent to the roerty that the only solution of the equation Aw = 0 is w = 0 [1]. From (7) the equation Aw = 0 can be written as w i = N λ i,j w j, i = 1,...,n, (11) j=1 where w n+1 = = w N = 0. Let w max be the maximum of the w 1,...,w n and suose that w max = w k. Consider any neighbouring node j of the node k. Since w k = w max and because of (6), the only way that (11) can be satisfied is if w j = w max. In a similar way every neighbour j of a neighbour of k must satisfy w j = w max and so on. Eventually, since G is connected, a boundary node must be reached with the result that w j = w max for some j {n+1,...,n}. This imlies that w max = 0. A similar argument shows that w min = 0 and therefore that w = 0. It remains to show that P is a triangulation. The roof in [25] that P is a straight line drawing when λ i,j = 1/d i for (i,j) E, deends on only one self-exlanatory roerty of barycentres. This roerty is that if a oint w IR 2 is the barycentre of any oints w 1,...,w d IR 2 and l is any line assing through w, then either (a) all the w i lie on l or (b) there is at least one w i on either side of l. Since this roerty evidently still holds when w is any strict convex combination of its neighbours, the theorem in [25] also alies for the most general λ i,j in (6). We conclude as follows. 6

7 D l w u i S D Fig. 3. Fig. 4. Corollary 2. Let S(G,X) be a surface triangulation and let P = P(G,U b,λ) be an embedding of G as in (8). Then P is a arametrization of S. The roof in [25] involves a considerable amount of grah theory. We rovide here a simle roof that all the internal u i P (i = 1,...,n) belong to D, even though it is a consequence of Corollary 2, as it emhasises the imortance of the convexity of D. Proosition 3. Every internal node u i of P, i {1,...,n}, belongs to D. Proof. By Proosition 1, u 1,...,u n are well-defined. To obtain a contradiction suose that there is at least one internal node of P not belonging to D. Let u i be any such node whose shortest distance to D is maximal and, since D is convex, let w be the unique oint in D nearest to it. The oint w may be either a vertex of D or a oint on one of its edges, as in Figure 3. Passing through u i is an infinite line l, erendicular to the vector u i w. It divides IR 2 into two oen half saces and no oint u j lies in the half sace S not containing w. Now, since u i is a strict convex combination of its neighbours and none of them belong to S, they must all lie on l. By the same reasoning, the neighbours of the neighbours of u i must lie on this line too, and so on. Eventually, since G is connected, this imlies that a boundary node, which is a oint in D, lies on l which is a contradiction. The convexity of D is also a necessary condition for all solutions of (7) satisfying (6) to belong to D. Even though the single internal node in Figure 4 is a convex combination of its neighbours it nevertheless lies outside D (though not outside the convex hull of D). The class of arametrizations of the form (8) is very large as we will now see. Proosition 4. Let P(S) be the class of all arametrizations, according to Definition 4, of a given surface triangulation S. Define T(S) P(S) to be those arametrizations of the form (8) and let C(S) P(S) be those arametrizations whose boundary nodes are the vertices of a convex olygon in an anticlockwise sequence. Then T(S) = C(S). Proof. Due to (5) and Corollary 2, T(S) C(S). Now suose P C(S). Then since P is a lanar triangulation, every internal node lies strictly inside the convex hull of its neighbours. Indeed, otherwise one of the faces incident on the node would not be a convex olygon and so could not be a triangle. Consequently every node can be exressed as a strict convex combination of its neighbours, i.e. in the form (7). Since moreover D is a convex olygon, P T(S). 7

8 Condition (5) demands that (u i+1 u i ) (u i+2 u i+1 ) > 0 for all i = n + 1,...,N, where we define (a 1,a 2 ) (b 1,b 2 ) = a 1 b 2 a 2 b 1 and u N+k = u n+k, for k = 1,2. In the numerical examles we have relaxed this to (u i+1 u i ) (u i+2 u i+1 ) 0. In this way D can be taken to be the unit square, with several boundary nodes lying along each of the four sides. This relaxation caused no roblems in ractice. 5. Secific arametrizations Tutte s barycentric maing (λ i,j = 1/d i, for (i,j) E) could be regarded as a generalization of uniform arametrization for oint sequences. In this section we wish to discuss this and some further secific arametrizations. In order to motivate these, let us review arametrizations for oint sequences. Suose X = {x 1,...,x N } is a sequence of oints in IR 3. Then any increasing sequence of values T = {t 1 < t 2 < < t N } is called a arametrization for X. It is usual to choose t 1 arbitrarily, aroriate ositive values L 1,...,L N 1, a constant µ > 0, and then define t i+1 = t i +µl i recursively, i = 1,...,N 1. For examle, when L i is constant, T is called a uniform arametrization, while if L i = x i+1 x i then it is called chord length [6]. It is well known that choosing arameter values by chord length tends to lead to smoother curve aroximations than using uniform esecially when the data oints are unevenly distributed; see Foley and Nielson [10]. For this reason we search for something analogous for surface triangulations. First observe that there are other ways of determining the t i from the L i. Proosition 5. Suose t 1 < t 2 < t N, t i IR and L 1,L 2,...,L N 1 > 0, L i IR. The following statements are equivalent: (i) t i+1 = t i + µl i, for i = 1,...,N 1, some µ > 0, (ii) if s 1 = t 1, s N = t N, then t 2,...,t N 1 minimize the functional F(s 2,...,s N 1 ) = N 1 i=1 w i(s i+1 s i ) 2, where w i = 1/L i, (iii) t i = λ i,1 t i 1 + λ i,2 t i+1, for i = 2,...,N 1, where λ i,1 = L i /(L i 1 + L i ) and λ i,2 = L i 1 /(L i 1 + L i ). Proof. We first show that (ii) imlies (iii). Since F is convex, it is minimized when F/ s i = 0, for all i. These equations are recisely those in (iii). Conversely, (iii) imlies (ii) since the minimum is unique. To see that (i) and (iii) are equivalent, one can easily show that they are both rearrangements of the equation (t i+1 t i )/(t i t i 1 ) = L i /L i 1. Thus if L i is constant, t i = (t i 1 + t i+1 )/2, i.e. t i is the barycentre of its neighbours. This justifies the term uniform arametrization for P when λ i,j = 1/d i, for (i,j) E. When L i = x i+1 x i, Proosition 5 suggests two ways of generalizing the chord length arametrization to surface triangulations: by minimizing squares of lengths of edges as in (ii); and by choosing the λ i,j directly in a similar way to (iii). We now briefly consider the first of these, treating the other, which is much more effective, more thoroughly in the next section. 8

9 Suose the boundary oints u n+1,...,u N have been chosen and are fixed. Then one could let the internal u i be chosen to minimize the functional F(u 1,u 2,...,u n ) = w i,j u i u j 2, (i,j) E where w i,j = w j,i > 0 for all (i,j) = (j,i) E. Since F : IR 2n IR is convex, it has a global minimum which is attained when F/ u i = F/ v i = 0 for all i. Since u i u j 2 = (u i u j ) 2 + (v i v j ) 2, we find F u i = 2 j:(i,j) E w i,j (u i u j ), F v i = 2 j:(i,j) E w i,j (v i v j ), and consequently, u i = j:(i,j) E w i,ju j / j:(i,j) E w i,j. This is equivalent to solving (7) with λ i,j = w i,j / j:(i,j) E w i,j. In articular when w i,j is constant for all (i,j) E, then λ i,j = 1/d i. Thus Tutte s barycentric maing, or uniform arametrization, minimizes the sum of the squares of lengths of edges in P, with resect to a fixed boundary. Surface aroximations were comuted based on the arametrizations obtained by setting w i,j = 1/ x i x j q, for several values of q including 1. Though the surfaces were generally smoother than when using the uniform arametrization they still suffered from unwanted oscillations. This is robably due to the degrees of freedom being restricted by the condition w i,j = w j,i. 6. Shae-reserving arametrization The chord length arametrization for a sequence of oints x 1,...,x N IR 3 has the roerty that if the x i lie on a straight line then there is some affine maing φ : IR 3 IR 3 such that t i = φ(x i ) (where we identify t i with the oint (t i,0,0)). It is a roerty of this kind we wish to retain in the bivariate case and we now describe a shae-reserving arametrization. Let S(G,X), where G = G(V,E,F), be a surface triangulation and suose that every triangular facet is non-degenerate (has affinely indeendent vertices). For each i, let G i be the subgrah of G whose nodes are i and its neighbours in G, and whose edges are edges in E which connect airs of nodes in G i. Further let x j1,...,x jdi be the neighbours of x i in any anticlockwise sequence, relative to G, and define X i = {x i,x j1,...,x jdi }. Then, referring to Figure 5, our basic idea is to (i) find a local shae-reserving arametrization P i for S i = S i (G i,x i ), maing x i into IR 2 and x j1,...,x jdi into suitable 1,..., di IR 2 and, (ii) choose λ i,j > 0, for j : (i,j) E to satisfy = λ i,jk k, k=1 k=1 λ i,jk = 1. (12) Ste (i) There are different ways of maing S i into the lane. The obvious aroach is to roject it onto the least squares lane through x i and its neighbours or onto the 9

10 x j 6 x j 5 x j 4 x j1 x i x j x j r(l) r(l)+1 l Fig. 5. The subtriangulation S i (G i,x i ) and a local arametrization P i. lane whose normal is an average of the normals of the triangular facets in S i. However both these methods can be unstable when there are large angles between the triangular facets in S i. We have adoted instead a neat method, due to Welch and Witkin [27], for making local arametrizations for surface triangulations which emulates the so-called geodesic olar ma, a local maing known in differential geometry which reserves arc length in each radial direction. The method requires only non-degenerate facets. Let ang(a,b,c) denote the angle between the vectors a b and c b, signed if they are in IR 2. Welch and Witkin [27] choose any IR 2 and 1,..., di IR 2 satisfying, for k = 1,...,d i, k = x jk x i, ang( k,, k+1 ) = 2π ang(x jk,x i,x jk+1 )/θ i, (13) where θ i = d i k=1 ang(x j k,x i,x jk+1 ) and x = x jdi +1 j 1, di +1 = 1. One can see that and 1,..., di are unique u to translations and rotations in IR 2. In the imlementation we have set = 0 and 1 = ( x j1 x i,0) to use u the sare degrees of freedom and then comuted 2,..., di in sequence. Ste (ii) The oints 1,..., di are the vertices of a star-shaed olygon with in its kernel. We now wish to find suitable λ i,jk satisfying (12). If d i = 3 they are the unique barycentric coordinates of with resect to : λ i,j1 = area(, 2, 3 ) area( 1, 2, 3 ), λ i,j 2 = area( 1,, 3 ) area( 1, 2, 3 ), λ i,j 3 = area( 1, 2,) area( 1, 2, 3 ). (14) For d i > 3 there is some choice. The following definition has been imlemented and yields good results in numerical examles. Regarding Figure 5, for each l {1,...,d i }, the straight line through l and intersects the olygon at a unique second oint which is either a vertex r(l) or lies on a line segment with endoints r(l) and r(l)+1. In either case, there is a unique r(l) {1,...,d i } and unique δ 1, δ 2, δ 3 such that δ 1 > 0, δ 2 > 0, δ 3 0, δ 1 + δ 2 + δ 3 = 1 and = δ 1 l + δ 2 r(l) + δ 3 r(l)+1. Define µ k,l, for k = 1,...,d i, by µ l,l = δ 1, µ r(l),l = δ 2, µ r(l)+1,l = δ 3, and µ k,l = 0 otherwise. Then for each l we now find = µ k,l k, k=1 k=1 10 µ k,l = 1, µ k,l 0.

11 Finally define λ i,jk = 1 µ k,l, k = 1,...,d i. (15) d i l=1 Since µ l,l and µ r(l),l are non-zero for every l we note that λ i,jk > 0 for all k. One also finds that = 1 = 1 1 µ k,l k = µ k,l k = λ i,jk k, d i d i and k=1 l=1 λ i,jk = l=1 k=1 1 d i k=1 l=1 Note also that when d i = 3, r(l) = l + 1, and d i k=1 l=1 k=1 µ k,l = 1 µ k,l = 1 1 = 1. d i d i l=1 k=1 λ i,jk = µ k,1 = µ k,2 = µ k,3, and thus (15) conforms with (14). Further, since affine combinations are invariant under translations and rotations, the µ k,l and therefore the λ i,jk are uniquely determined by (13) and hence also by X i. Moreover each λ i,jk deends continuously (but not smoothly) on x i and x j1,...,x jdi. One could also consider taking a weighted average in (15) but this has not been imlemented. The choice of λ i,jk in (15) rovides the following reroduction roerty. In what follows we identify a oint u = (u,v) IR 2 with the oint (u,v,0) IR 3. Proosition 6. Suose that S is lanar and that its boundary nodes x 1,...,x N form a convex olygon in the lane containing S. Let P(G,U b,λ) be a arametrization in which Λ is defined in (15) and such that the boundary oints x n+1,...,x N are maed affinely into u n+1,...,u N U b. Then the arametrization P is an affine maing of S. Proof. Let φ : IR 3 IR 3 be the affine maing for which u i = φ(x i ) for i = n + 1,...,N. It is required to show that also u i = φ(x i ) for i = 1,...,n. Using vector and matrix notation we can exress φ as φ(x) = Mx + b for some non-singular 3 3 matrix M and vector b, such that the third coordinate of φ(x) is zero for all x in the lane containing S. Due to (13), since each S i is lanar, each internal node x i and its neighbours are affinely maed into and its neighbours. It follows from (15) that the λ i,j satisfy Therefore, φ(x i ) x i N λ i,j x j = 0, j=1 N λ i,j φ i (x j ) = Mx i + b j=1 i = 1,...,n. l=1 N λ i,j (Mx j + b) = M x i j=1 N λ i,j x j = 0. Because the solutions to (7) are unique it follows that φ(x i ) = u i for i = 1,...,n as required. 11 j=1

12 The choice of the boundary D should deend on the alication and the kind of aroximation. If one requires a tensor-roduct sline aroximation the natural choice is the unit square. One could use the unit circle if an aroximation in the form of triangular atches or radial basis functions is desirable. When constructing a shae-reserving arametrization, numerical examles have suggested that a good lacement of the boundary oints u n+1,...,u N around D is by chord length. 7. Numerical examles It was found after some exerimentation that the matrix equations (10) can adequately be solved by LU decomosition [12] for n u to 500. Beyond that the structure of the matrix A suggests an iterative method would be better. The matrix A in (10) is diagnonally dominant though not strictly diagonally dominant in any row i if all neighbours of x i are internal nodes. It is also sarse but it will not in general be ossible to arrange the non-zero entries in a diagonal band structure. This kind of matrix occurs frequently in the numerical solution of differential equations and it was found that an iterative method called Bi-CGSTAB [26] was highly effective. This is a variant of the conjugate gradient method for non-symmetric matrices. Bi-CGSTAB was used to solve each equation in (10) for n of the order of 25000, setting every internal u i = (u i,v i ) to be a oint in the centre of D as an initial guess. The algorithm could be stoed after only a few hundred iterations and no instabilities were exerienced. A suitable data structure for the triangulations is that roosed by Cline and Renka [4] for efficient storage of triangulations. Figure 6 shows a Delaunay triangulation of a set of 27 scattered data oints in the lane. The oints were maed uniformly (λ i,j = 1/d i for (i,j) E) into the unit disc in Figure 7, lacing the eight boundary nodes uniformly around the unit circle. In Figure 8, the same oints were maed into the unit square [0,1] [0,1]. This time four chosen corner boundary oints were maed to the corners of the square and the remaining boundary oints maed uniformly along each side. The oints were maed using the shae-reserving arametrization (15) in Figure 9, the sides being maed by chord length. Figure 10 shows a surface triangulation S of 1000 oints samled from a salt dome, a geological object with an overhang. Three arametrizations P in the unit square are dislayed in Figures 11, 13, and 15, using resectively (i) uniform, (ii) weighted least squares of edge lengths with w i,j = 1/ x i x j, and (iii) shae-reserving. As in the revious examle, four corner oints were maed to the corners of the square, and the sides were maed uniformly in the first and by chord length in the second two. In each case three C 1 iecewise-cubic interolants x, y, z, satisfying (3), were then made for each comonent using the Clough Tocher slit on P [7], [11]. The resulting interolatory surface s(u,v), of the form (4) was then samled on a square grid and interolated by a C 2 cubic tensor-roduct sline s (u,v). These are shown in Figures 12, 14, and 16, resectively. It is very clear that the third surface, based on the shae-reserving arametrization, is visually smoother than the first two. Finally by retriangulating the u i P, with a new triangulation ˆP, one obtains a new surface triangulation Ŝ, arametrized by ˆP, having the same nodes x i as S. If one chooses 12

13 ˆP to satisfy some desirable roerty, this roerty can be transmitted to Ŝ, roviding an interesting way of otimizing surface triangulations. Such otimizations are usually based on recursively swaing edges in an existing one according to some goodness criterion as described by Schumaker [21], and Dyn, Levin, and Ria [5]. Moreover relacing P by ˆP may have a beneficial effect on the surface aroximation when it is based on iecewise olynomials on ˆP. In Figure 17 the Delaunay triangulation ˆP, which maximizes the minimum angle of its triangles, was comuted for the u i aearing in Figure 15. The tensor-roduct surface aroximation ŝ of Ŝ is dislayed in Figure 18. The new surface aears to be somewhat smoother than that in Figure 16, benefitting from more well-roortioned triangles. The corresonding surface retriangulation Ŝ is dislayed in Figure 19 and it has as might be exected, less long thin triangles than the original in Figure 10. However note that any retriangulation of the arametrization deends on the choice of the boundary. 8. Final remarks A method for making shae-reserving arametrizations of surface triangulations has been resented and used to generate well-behaved smooth surface interolations and aroximations. In this aer G was assumed to be a triangulated grah. Tutte s barycentric maing alies to more general grahs, having faces with more than three edges, for examle rectangular networks or the grah in Figure 2. With care, the shae-reserving arametrization could similarly be extended to various networks of oints in IR 3. Acknowledgement. I wish to thank Hermann Kellermann at SINTEF for hel with the imlementation of the numerical examles. I am also grateful to Jesús Miguel Carnicer for suggesting imrovements to the layout of the aer. 9. References 1. Atkinson K. E., An introduction to numerical analysis, Wiley, new York, de Boor C., A Practical Guide to Slines, Sringer-Verlag, New York, Chiba N., K. Onoguchi, T. Nishizeki, Drawing lane grahs nicely, Acta Informatica 22 (1985), Cline A. K., R. L. Renka, A storage-efficient method for construction of a Thiessen triangulation, Rocky Mount. J. Math. 14 (1984), Dyn N., D. Levin, S. Ria, Data deendent triangulations for iecewise linear interolation, IMA J. Numer. Anal. 10 (1990), Farin G., Curves and surfaces for comuter aided geometric design, Academic Press, San Diego, Farin G., Triangular Bernstein-Bézier atches, Com. Aided Geom. Design 3 (1986), Fáry I., On straight line reresentatation of lanar grahs, Acta Sci. Math. Szeged 11 (1948), Floater M., G. Westgaard, Smooth surface reconstruction from cross-sections using imlicit methods, rerint, SINTEF, Oslo. 13

14 10. Foley T., G. Nielson, Knot selection for arametric sline interolation, in Mathematical Methods in Comuter Aided Geometric Design II, T. Lyche and L. Schumaker (eds.), Academic Press, New York, 1992, Franke R., L. Schumaker, A bibliograhy of multivariate aroximation, in Toics in Multivariate Aroximation, C. Chui, L. Schumaker, and F. Uteras (eds.), Academic Press, New York, 1986, Golub G. H., C. F. Van Loan, Matrix Comutations, John Hokins Univ. Press, Hoe H., T. DeRose, T. DuCham, J. McDonald, W. Stuetzle, Surface reconstruction from unorganized oints, Comuter Grahics 26 (1992), Li Z., C. Suen, T. Bui, Q. Gu, Harmonic models of shae transformations in digital images and atterns, CVGIP: grah. models and imag. roc., 54 (1992), Lounsberry M., S. Mann, T. DeRose, Parametric surface interolation, IEEE Com. Grah. and A., (1992), Maillot J., H. Yahia, A. Verroust, Interactive texture maing, SIGGRAPH Com. Grah. Proc. (1993), Marshall C. W., Alied Grah Theory, Wiley, New York, Milroy M., C. Bradley, G. Vickers, D. Weir, G 1 continuity of B-sline surface atches in reverse engineering, Com. Aided Design 27 (1995), Nishizeki T., Planar grah roblems, Comuting Sulementum 7 (1990), Prearata F. P., M. I. Shamos, Comutational geometry, Sringer-Verlag, New York, Schumaker L. L., Triangulation methods, in Toics in Multivariate Aroximation, C. K. Chui, L. L. Schumaker, F. Utreras (eds.), Academic Press, New York (1987), Schumaker L. L., Reconstructing 3D objects from cross-sections, in Comutation of Curves and Surfaces, W. Dahmen, M. Gasca, and C. Micchelli eds., Kluwer, Dordrecht (1990), Schumaker L. L., Triangulations in CAGD, IEEE Comuter Grahics and Alications (1993), Tutte W. T., Convex reresentations of grahs, Proc. London Math. Soc. 10 (1960), Tutte W. T., How to draw a grah, Proc. London Math. Soc. 13 (1963), van der Vorst H. A., Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAMsci, 13 (1992), Welch W., A. Witkin, Free-form shae design using triangulated surfaces, Comuter Grahics, SIGGRAPH 94 (1994),

15 Fig. 6. Delaunay triangulation. Fig. 7. Uniform arametrization. Fig. 8. Uniform arametrization. Fig. 9. Shae-reserving arametrization. Fig. 10. Triangulated salt dome. 15

16 Fig. 11. Uniform arametrization. Fig. 12. Surface aroximation. Fig. 13. Weighted least squares arametrization. Fig. 14. Surface aroximation. Fig. 15. Shae-reserving arametrization. Fig. 16. Surface aroximation. 16

17 Fig. 17. Delaunay retriangulation. Fig. 18. Surface aroximation. Fig. 19. Surface retriangulation. 17