Katholieke Universiteit Leuven Department of Computer Science

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1 Interpolation with quintic Powell-Sabin splines Hendrik Speleers Report TW 583, January 2011 Katholieke Universiteit Leuven Department of Computer Science Celestijnenlaan 200A B-3001 Heverlee (Belgium)

2 Interpolation with quintic Powell-Sabin splines Hendrik Speleers Report TW 583, January 2011 Department of Computer Science, K.U.Leuven Abstract We discuss local interpolation schemes for C 2 -continuous quintic Powell-Sabin splines represented with a normalized B-spline basis. As part of the construction of the interpolation rules, we make use of tensor algebra and blossoming to describe and manipulate polynomials in Bernstein-Bézier representation. Keywords : interpolation, quintic Powell-Sabin splines, normalized B-spline representation MSC : Primary : 65D05, Secondary : 65D07, 41A05

3 Interpolation with quintic Powell-Sabin splines Hendrik Speleers Department of Computer Science, Katholieke Universiteit Leuven Celestijnenlaan 200A, B-3001 Leuven, Belgium Abstract We discuss local interpolation schemes for C 2 -continuous quintic Powell-Sabin splines represented with a normalized B-spline basis. As part of the construction of the interpolation rules, we make use of tensor algebra and blossoming to describe and manipulate polynomials in Bernstein-Bézier representation. Keywords: interpolation, quintic Powell-Sabin splines, normalized B-spline representation AMS classification: 65D05, 65D07, 41A05 1 Introduction The construction of interpolating functions through scattered data is a common problem in many application areas. This problem can be addressed by working with smooth spline functions defined on a triangulation where its vertices are located at the data points. A number of authors considered C 2 -continuous quintic spline functions over triangulations with Powell-Sabin refinement (see [1, 8, 9, 10, 13]). Each triangle in such a triangulation is split into six subtriangles. A normalized basis for particular quintic Powell-Sabin splines was developed in [18]. The locally supported basis functions form a convex partition of unity, i.e., they are nonnegative and they sum up to one. This B-spline representation allows a natural definition of control polynomials and control points, which are useful for geometric modelling. A spline in such a representation can also be evaluated in a stable way, using only a sequence of simple convex combinations. A similar basis was developed for quadratic Powell-Sabin splines in [4] and for reduced cubic Clough-Tocher splines in [19]. In this paper we construct interpolation rules for quintic Powell-Sabin splines in this normalized B-spline representation. The interpolation requires data at the vertices of the triangulation and at the Powell-Sabin split points. For the computation of the rules, we make use of tensor algebra to describe and manipulate blossoms of polynomials in Bernstein-Bézier form. Higher-order tensors are a natural generalization of matrices. We also consider interpolation based on several reduced spline spaces. These reduced spline interpolants only need data at the vertices of the triangulation. Interpolation and approximation schemes based on quadratic Powell-Sabin splines can be found in [5, 11, 14, 15] and shape-preserving schemes in [12, 16, 20, 21]. The paper is organized as follows. In Section 2 we recall some relevant concepts of tensor algebra. Section 3 is devoted to Bernstein-Bézier polynomials on triangles, and we point out the relation to tensors. In Section 4 we define the quintic Powell-Sabin splines and we describe the construction of the normalized basis. Section 5 describes the interpolation rules for the quintic Powell-Sabin splines, and in Section 6 we discuss some reduced spline spaces. Finally, a numerical example is presented in Section 7. 1

4 2 TENSOR ALGEBRA 2 2 Tensor algebra In this section we introduce some basic definitions of multilinear algebra. A tensor, also known as a multiway array or a multidimensional matrix, is a higher-order generalization of a vector (first order tensor) and a matrix (second order tensor). Tensors are multilinear mappings over a set of vector spaces. The order of tensor A R I1 I2... IN is N. An element of A is denoted by a i1i 2...i N where 1 i n I n. The n-mode vectors of A are the I n -dimensional vectors obtained from A by varying the index i n while keeping the other indices fixed. In this terminology, column vectors of a matrix are referred to as 1-mode vectors and row vectors as 2-mode vectors. A tensor can be represented in matrix form by unfolding the tensor along a single dimension. The matrix unfolding A (n) R In (In+1...IN I1...In 1) of a tensor A R I1... IN is a matrix composed of all the n-mode vectors of the tensor. It contains the element a i1i 2...i N at the position with row number i n and column number equal to 1 + n 1 (i j 1) j=1 n 1 k=j+1 I k + N j=n+1 (i j 1) N k=j+1 n 1 I k k=1 I k. These unfoldings are illustrated in Figure 1 for a tensor of order three. A generalization of the product of two matrices is the product of a tensor and a matrix. The n-mode product of a tensor A R I1... IN and a matrix U R Jn In is defined as the tensor B R I1... In 1 Jn In+1... IN of which the elements are given by b i1...i n 1j ni n+1...i N = I n i n=1 a i1...i n 1i ni n+1...i N u jni n. (2.1) The product is denoted by and, using matrix unfolding it is equal to B = A n U, (2.2) More details on tensor algebra can be found in [3]. B (n) = U A (n). (2.3) 3 Bivariate polynomials on triangles 3.1 Barycentric coordinates and the Bernstein-Bézier representation of polynomials Let T (V 1, V 2, V 3 ) be a non-degenerate triangle. Any point P in the plane of the triangle can be uniquely expressed in terms of the barycentric coordinates τ = (τ 1, τ 2, τ 3 ) with respect to T, where τ 1 + τ 2 + τ 3 = 1. Given two points P 1 and P 2 in the plane of the triangle, the barycentric direction δ = (δ 1, δ 2, δ 3 ) of the vector P 2 P 1 with respect to T is defined as the difference of the barycentric coordinates of both points. If the Euclidean distance P 2 P 1 2 = 1, then δ is called a unit barycentric direction. A barycentric direction δ always satisfies δ 1 + δ 2 + δ 3 = 0. Let (x i, y i ) be the Cartesian coordinates of vertex V i, i = 1, 2, 3, then the unit barycentric directions along the x and y direction are found as δ x = (y 2 y 3, y 3 y 1, y 1 y 2 )/E, δ y = (x 3 x 2, x 1 x 3, x 2 x 1 )/E, (3.1a) (3.1b)

5 3 BIVARIATE POLYNOMIALS ON TRIANGLES 3 I 3 I 3 I 2 I 1 I 1 A I 2 A (1) (a) I 3 I 1 I 2 I 2 I 1 A I 3 A (2) (b) I 3 I 2 I 2 I 3 I 1 A I 1 A (3) (c) Figure 1: Unfolding of the (I 1 I 2 I 3 )-tensor A to (a) the (I 1 I 2 I 3 )-matrix A (1), (b) the (I 2 I 3 I 1 )-matrix A (2) and (c) the (I 3 I 1 I 2 )-matrix A (3) with I 1 = I 2 = I 3 = 3. with E = x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 Let Π d denote the linear space of bivariate polynomials of total degree less than or equal to d. Any polynomial p d Π d on triangle T has a unique Bernstein-Bézier representation p d (τ) = b ijk Bijk(τ). d (3.2) Here, i+j+k=d Bijk d d! (τ) = i!j!k! τ 1 i τ j k 2 τ 3 (3.3) are the Bernstein polynomials of degree d. The coefficients b ijk are called Bézier ordinates. The Bézier domain points ξ ijk are defined as the points with barycentric coordinates ( i d, j d, ) k d. By associating the Bézier ordinates b ijk with the Bézier domain points ξ ijk, one can display the Bernstein-Bézier representation schematically as in Figure 2 for the case d = 3. We refer to [6] for more details..

6 3 BIVARIATE POLYNOMIALS ON TRIANGLES 4 b 003 V 3 b 102 b 012 b 201 b 111 b 021 V 1 V 2 b 300 b 210 b 120 b 030 Figure 2: Schematic representation of the Bézier ordinates of a cubic bivariate polynomial. 3.2 The blossom of polynomials The blossom (or polar form) of p d (τ) is denoted by P d (τ 1, τ 2,..., τ d ), and it is completely characterized by the following three properties (see [17]): 1. The blossom is symmetric with respect to any permutation π of its d arguments, P d (τ 1, τ 2,..., τ d ) = P d (π(τ 1, τ 2,..., τ d )). 2. The blossom is multi-affine, i.e. affine in each of its d arguments, P d (a τ + b σ, τ 2,..., τ d ) = a P d (τ, τ 2,..., τ d ) + b P d (σ, τ 2,..., τ d ), a + b = The restriction of the blossom to the diagonal gives the polynomial p d, p d (τ) = P d (τ, τ,..., τ). We consider the case d = 3 in more detail, as we are particularly interested in it further on. Let τ l = (τ l 1, τ l 2, τ l 3), l = 1, 2, 3, then the blossom of p 3 on domain triangle T (V 1, V 2, V 3 ) is given by P 3 (τ 1, τ 2, τ 3 ) = b 300 (τ 1 1 τ 2 1 τ 3 1 ) + b 030 (τ 1 2 τ 2 2 τ 3 2 ) + b 003 (τ 1 3 τ 2 3 τ 3 3 ) + b 210 (τ 1 2 τ 2 1 τ τ 1 1 τ 2 2 τ τ 1 1 τ 2 1 τ 3 2 ) + b 201 (τ 1 3 τ 2 1 τ τ 1 1 τ 2 3 τ τ 1 1 τ 2 1 τ 3 3 ) + b 021 (τ 1 3 τ 2 2 τ τ 1 2 τ 2 3 τ τ 1 2 τ 2 2 τ 3 3 ) + b 120 (τ 1 1 τ 2 2 τ τ 1 2 τ 2 1 τ τ 1 2 τ 2 2 τ 3 1 ) + b 102 (τ 1 1 τ 2 3 τ τ 1 3 τ 2 1 τ τ 1 3 τ 2 3 τ 3 1 ) + b 012 (τ 1 2 τ 2 3 τ τ 1 3 τ 2 2 τ τ 1 3 τ 2 3 τ 3 2 ) + b 111 (τ 1 1 τ 2 2 τ τ 1 1 τ 2 3 τ τ 1 2 τ 2 3 τ τ 1 2 τ 2 1 τ τ 1 3 τ 2 1 τ τ 1 3 τ 2 2 τ 3 1 ), (3.4) with b ijk, i + j + k = 3, the Bézier ordinates of p 3 on T. The multi-affine de Casteljau algorithm can be used to evaluate the blossom. Subdivision of a polynomial (3.2) can be compactly described through blossoming. One can use the blossom to find the Bernstein-Bézier representation of polynomial p 3 with respect to another (finer) triangle. Consider a second triangle, where its vertices W l, l = 1, 2, 3, have σ l = (σ l 1, σ l 2, σ l 3)

7 3 BIVARIATE POLYNOMIALS ON TRIANGLES 5 as barycentric coordinates with respect to the former triangle T (V 1, V 2, V 3 ). The Bézier ordinates d ijk of p 3 on the new triangle are obtained using (3.4) as d 300 = P 3 (σ 1, σ 1, σ 1 ), d 210 = P 3 (σ 1, σ 1, σ 2 ), d 201 = P 3 (σ 1, σ 1, σ 3 ), d 030 = P 3 (σ 2, σ 2, σ 2 ), d 021 = P 3 (σ 2, σ 2, σ 3 ), d 120 = P 3 (σ 1, σ 2, σ 2 ), d 003 = P 3 (σ 3, σ 3, σ 3 ), d 102 = P 3 (σ 1, σ 3, σ 3 ), d 012 = P 3 (σ 2, σ 3, σ 3 ), d 111 = P 3 (σ 1, σ 2, σ 3 ). (3.5) Blossoming can also be used to describe directional derivatives of a polynomial (3.2). The r-th order directional derivative of polynomial p d with respect to the unit barycentric directions δ i, i = 1,..., r, on triangle T (V 1, V 2, V 3 ) can be compactly expressed as D r δ 1,...,δ rp d(τ) = d! (d r)! P d(τ,..., τ, δ 1,..., δ r ). (3.6) Here, the r barycentric directions δ i and (d r) times the barycentric coordinates τ are taken as the arguments of the blossom. 3.3 A representation based on tensors The blossom of a polynomial in Bernstein-Bézier representation can be described using tensor algebra. In this section we focus on the case d = 3, but the approach can be generalized to any degree d. We define the symmetric (3 3 3)-tensor B by its matrix unfolding B (1) as B (1) = b 300 b 210 b 201 b 210 b 120 b 111 b 201 b 111 b 102 b 210 b 120 b 111 b 120 b 030 b 021 b 111 b 021 b 012 b 201 b 111 b 102 b 111 b 021 b 012 b 102 b 012 b 003. (3.7) This tensor consists of the Bézier ordinates b ijk, i + j + k = 3, of the polynomial p 3 (τ) in (3.2) defined on domain triangle T (V 1, V 2, V 3 ). Note that the three matrix unfoldings of B are equal, i.e., B (1) = B (2) = B (3). The barycentric coordinates τ l = (τ1, l τ2, l τ3), l l = 1, 2, 3, with respect to T can be arranged into the vectors T l = [ τ1 l τ 2 l τ 3 l ], l = 1, 2, 3. The blossom in (3.4) can then be written as the product of a tensor and three vectors: P 3 (τ 1, τ 2, τ 3 ) = B 1 T 1 2 T 2 3 T 3. (3.8) It follows that a polynomial p d in Bernstein-Bézier form (3.2) can be written as p d (τ 1 ) = B 1 T 1 2 T 1 3 T 1. (3.9) Let the (3 3 3)-tensor D be defined in a similar way as B in (3.7) using the Bézier ordinates d ijk, i + j + k = 3, and let the matrix S be defined as S = σ1 1 σ2 1 σ3 1 σ1 2 σ2 2 σ3 2. (3.10) σ1 3 σ2 3 σ3 3 The subdivision rules in (3.5) can then be compactly written as D = B 1 S 2 S 3 S. (3.11)

8 3 BIVARIATE POLYNOMIALS ON TRIANGLES 6 We now consider the following Hermite interpolation problem: find the Bézier ordinates b ijk, i + j + k = 3, of the polynomial p 3 (x, y) that satisfies a+b x a y b p 3(P ) = h x a yb, a 0, b 0, a + b 3, (3.12) for a given point P and a given set of h x a y b-values. We could determine the ten unknowns b ijk by formulating and solving a linear system of ten equations. Alternatively, we use the following tensor approach. Let (3 a b)! ĥ x a yb = h 6 x a yb, (3.13) we define the (3 3 3)-tensor H by its matrix unfolding H (1) as H (1) = ĥ ĥ x ĥ y ĥ x ĥ x 2 ĥ xy ĥ y ĥ xy ĥ y 2 ĥ x ĥ x 2 ĥ xy ĥ x 2 ĥ x 3 ĥ x 2 y ĥ xy ĥ x 2 y ĥ y ĥ xy ĥ y 2 ĥ xy ĥ x 2 y ĥ xy 2 ĥ y 2 ĥ xy 2 ĥ y 3 ĥ xy 2. (3.14) Assume the point P has barycentric coordinates τ = (τ 1, τ 2, τ 3 ) with respect to the domain triangle. We combine these barycentric coordinates together with the unit barycentric directions along the x and y direction, i.e. (3.1), into the matrix T as T = τ 1 τ 2 τ 3 δ1 x δ2 x δ3 x. (3.15) δ y 1 δ y 2 δ y 3 From (3.6) we then obtain that interpolation problem (3.12) can be reformulated as The solution is found as H = B 1 T 2 T 3 T. (3.16) B = H 1 T 1 2 T 1 3 T 1. (3.17) Instead of solving a linear system of ten equations, by using the tensor approach we only need to compute the inverse of a (3 3)-matrix. We now compute the matrix T 1. Let (x P, y P ) be the Cartesian coordinates of the point P. From the definition of barycentric coordinates and the expressions (3.1), we derive the relation τ 1 τ 2 τ 3 δ x 1 δ x 2 δ x 3 δ y 1 δ y 2 δ y 3 x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 = x P y P We then obtain that T 1 = x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 x P y P = 1 x 1 x P y 1 y P 1 x 2 x P y 2 y P 1 x 3 x P y 3 y P. (3.18) By using tensors of order d, one can apply the same technique to solve the Hermite interpolation problem similar to (3.12) for polynomials with arbitrary degree d.

9 4 QUINTIC POWELL-SABIN SPLINES 7 4 Quintic Powell-Sabin splines 4.1 The PS5-spline space Given a simply connected subset Ω R 2 with polygonal boundary Ω, let be a conforming triangulation of Ω. Let n v, n t and n e be the number of vertices, triangles and edges in, respectively. A Powell-Sabin (PS-) refinement of partitions each triangle T j into six smaller triangles in the following way: 1. Choose an interior point Z j in each triangle T j, so that if two triangles T i and T j have a common edge, then the line joining Z i and Z j intersects the common edge at a point R ij between its vertices. 2. Join each point Z j to the vertices of T j. 3. For each edge of the triangle T j (a) which is common to a triangle T i : join Z j to the intersection point R ij of that edge and the line Z i Z j. (b) which belongs to the boundary Ω: join Z j to an arbitrary point R ij on that edge. The obtained subtriangles are denoted as T. The space of piecewise quintic polynomials on with global C 2 -continuity is denoted as S 2 5( ) = { s C 2 (Ω) : s T Π 5, T }. (4.1) We consider a particular subspace of S5 2( ) with additional smoothness around some vertices and edges. Let V = {V i } nv i=1 be the set of vertices in, let Z = {Z i } nt i=1 be the set of split points in, and let E be the set of all edges in that connect a split point Z i to a point R ij. The quintic Powell-Sabin (PS5-) spline space is defined as Ŝ 2 5 ( ) = { s S 2 5 ( ) : s C 3 (W ), W (V Z ); s C 3 (e), e E }. (4.2) Here, C µ (W ) means that the polynomials on triangles in sharing the vertex W have common derivatives up to order µ at that vertex. Analogously, C µ (e) means that the polynomials on triangles in sharing the edge e have common derivatives up to order µ along that edge. In [10] it is shown that the dimension of the space Ŝ2 5 ( ) equals 10n v + n t. The following interpolation problem can then be considered: there exists a unique spline s(x, y) Ŝ2 5 ( ) such that and a+b x a y b s(v l) = f xa y b,l, l = 1,..., n v, a 0, b 0, a + b 3, (4.3a) for any given set of f x a y b,l-values and g m -values. s(z m ) = g m, m = 1,..., n t, (4.3b) 4.2 A normalized B-spline representation A procedure for the construction of a normalized basis for Ŝ2 5( ) was developed in [18]. Every PS5-spline can be represented as n v 10 n t s(x, y) = c v i,j Bi,j(x, v y) + c t k Bk(x, t y). (4.4) i=1 j=1 k=1

10 4 QUINTIC POWELL-SABIN SPLINES 8 We call Bi,j v (x, y) and Bt k (x, y) a B-spline with respect to vertex V i and triangle T k, respectively. The basis functions Bi,j v (x, y) and Bt k (x, y) have a local support and they form a convex partition of unity, i.e., B v i,j(x, y) 0, B t k(x, y) 0, and n v 10 i=1 j=1 n t Bi,j(x, v y) + Bk(x, t y) = 1. (4.5) k=1 Let M i be the molecule (also called 1-ring) of vertex V i, defined as the union of all triangles in that contain V i. In order to construct the B-splines, we first define for each vertex V i ten linearly independent decuplets { α ab i,j, 0 a + b 3}, j = 1,..., 10, as follows: 1. For each vertex V i, identify the corresponding PS5-points. They are defined as S il = 2 5 V i V l, (4.6) for all vertices V l that are situated at the boundary of the molecule M i of V i. The vertex V i itself is also a PS5-point. 2. For each vertex V i, find a triangle t i (Q v i,1, Qv i,2, Qv i,3 ) that contains all the PS5-points of V i. Denote its vertices by Q v i,j = (Xv i,j, Y i,j v ). The triangles t i, i = 1,..., n v, are called PS5- triangles. We remark that the PS5-triangles are not uniquely defined. In [18] an optimization strategy was proposed to select triangles of minimal area. 3. Consider the ten Bernstein polynomials (3.3) of degree three defined on PS5-triangle t i. Each decuplet { α ab i,j, 0 a + b 3}, j = 1,..., 10, is related to the function and derivative values up to order three of one of these cubic Bernstein polynomials evaluated at the point V i : for all 0 a + b 3. ( ) a+b α ab i,1 = 20 3 a+b (5 a b) (4 a b) 5 x a y b B3 300 (V i), ( ) a+b α ab 20 3 a+b i,2 = (5 a b) (4 a b) 5 x a y b B3 030(V i ), ( ) a+b α ab 20 3 a+b i,3 = (5 a b) (4 a b) 5 x a y b B3 003(V i ), ( ) a+b α ab i,4 = 20 3 a+b (5 a b) (4 a b) 5 x a y b B3 210 (V i),..., ( ) a+b α ab i,10 = 20 3 a+b (5 a b) (4 a b) 5 x a y b B3 111 (V i), (4.7) A triangle T (V 1, V 2, V 3 ) with its PS-refinement is shown in Figure 3. Figure 4 depicts a PS5-triangle t 1 containing the PS5-points V 1, S 12 and S 13 on the triangle T. The basis functions Bi,j v (x, y) and Bt k (x, y) are then constructed as follows: 1. The B-spline Bi,j v (x, y) with respect to vertex V i is defined as the unique solution of interpolation problem (4.3) with all f xa y b,l = 0, except for l = i, where f xa y b,i = α ab i,j, and with all g m = 0, except for any m such that T m M i, where g m = βi,j m 0. The B-spline is zero outside the molecule of vertex V i. The values α ab i,j are determined by the PS5-triangle

11 4 QUINTIC POWELL-SABIN SPLINES 9 V 3 R R Z k V 1 R 12 V 2 Figure 3: A Powell-Sabin split of a triangle T k (V 1, V 2, V 3 ) drawn with dashed lines. t i, see (4.7). The values βi,j m are specified with the aid of the Bernstein-Bézier representation of the B-spline. We consider the macro-triangle T k (V 1, V 2, V 3 ), as shown in Figure 3. On each of the six subtriangles, the spline is a quintic polynomial that can be represented in its Bernstein-Bézier formulation, i.e. with d = 5 in equations (3.2) and (3.3). The corresponding Bézier ordinates are schematically represented in Figure 5(a). Non-zero Bézier ordinates are denoted by filled bullets, and zero ordinates are denoted by open bullets î. Because of the C 3 -smoothness at vertex V 1, the Bézier ordinates in the 3-disk around vertex V 1 are completely determined by the values α ab i,j. The Bézier ordinates in the 3-disk around the split point Z k are found by subdivision of a single cubic polynomial p 3 defined on the triangle with vertices P 1 = 3 5 V Z k, P 2 = 3 5 V Z k, P 3 = 3 5 V Z k. (4.8) The Bernstein-Bézier representation of this polynomial p 3 has Bézier ordinates that are all zero, except for the three ordinates in the neighbourhood of P 1. Their values are also determined by α ab i,j. The values of the remaining non-zero Bézier ordinates in Figure 5(a) can be computed from the C 3 -smoothness conditions across the interior edges Z k R 12 and Z k R 31. The value of βi,j k is obtained by evaluating the cubic polynomial p 3 at split point Z k. 2. The B-spline B t k (x, y) with respect to triangle T k is defined as the unique solution of interpolation problem (4.3) with all f x a y b,l = 0 and with all g m = 0, except for m = k, where g k = β k 0. It is easy to prove that such a B-spline vanishes outside T k. The value β k is determined with the aid of the Bernstein-Bézier representation of the B-spline. We consider again the triangle T k (V 1, V 2, V 3 ) shown in Figure 3. The corresponding Bézier ordinates are schematically represented in Figure 5(b). Most of these ordinates are zero. The Bézier ordinates in the 3-disk around the split point Z k are found by subdivision of a cubic polynomial p 3 defined on the triangle with vertices (4.8). The Bézier ordinates in the Bernstein-Bézier representation of this polynomial p 3 are all zero, except for the central ordinate b 111 = 1. The value of β k is obtained by evaluating p 3 at split point Z k. For more details we refer to [18]. Since each B-spline Bi,j v (x, y), j = 1,..., 10, with respect to vertex V i is related to a Bernstein polynomial of degree three, see (4.7), the corresponding coefficients c v i,j of the PS5-spline in (4.4) can be represented schematically as in Figure 6 with respect to PS5-triangle t i. The cubic polynomial defined on t i with the coefficients c v i,j as Bézier ordinates is called the control polynomial with respect to vertex V i, and it is denoted by T i (x, y). This control polynomial is tangent to the PS5-spline s(x, y) at vertex V i.

12 5 PS5-SPLINE INTERPOLATION 10 Q v 1,3 V 3 e 1,3 S 13 e 1,8 e 1,9 e 1,5 e 1,10 e 1,6 S 12 V 1 V 2 e 1,1 e 1,4 e 1,7 e 1,2 Q v 1,2 Q v 1,1 Figure 4: A PS5-triangle t 1 (Q v 1,1, Qv 1,2, Qv 1,3 ) of vertex V 1 containing the PS5-points V 1, S 12 and S 13 on triangle T (V 1, V 2, V 3 ), together with the schematic representation of the Bézier ordinates e 1,j, j = 1,..., 10, of the subdivided control polynomial T 1 (x, y) onto the triangle with as vertices the points V 1, S 12 and S PS5-spline interpolation 5.1 Interpolation at the vertices In this section we will show how to satisfy the interpolation conditions (4.3a) at the vertices. We first arrange the spline coefficients c v i,j, j = 1,..., 10, associated with vertex V i = (x i, y i ) into a (3 3 3)-tensor C. Its matrix unfolding C (1) is given by C (1) = c v i,1 c v i,4 c v i,5 c v i,4 c v i,7 c v i,10 c v i,5 c v i,10 c v i,8 c v i,4 c v i,7 c v i,10 c v i,7 c v i,2 c v i,6 c v i,10 c v i,6 c v i,9 c v i,5 c v i,10 c v i,8 c v i,10 c v i,6 c v i,9 c v i,8 c v i,9 c v i,3. (5.1) The barycentric coordinates of V i with respect to its PS5-triangle t i are denoted by (γ i,1, γ i,2, γ i,3 ). The unit barycentric directions along the x and y direction with respect to t i are denoted by (γi,1 x, γx i,2, γx i,3 ) and (γy i,1, γy i,2, γy i,3 ), respectively. They can be arranged into the matrix G as G = By the definition of t i and by (3.18) we have G 1 = γ i,1 γ i,2 γ i,3 γ x i,1 γ x i,2 γ x i,3 γ y i,1 γ y i,2 γ y i,3 1 X v i,1 x i Y v i,1 y i 1 X v i,2 x i Y v i,2 y i 1 X v i,3 x i Y v i,3 y i. (5.2). (5.3)

13 5 PS5-SPLINE INTERPOLATION 11 (a) (b) Figure 5: Schematic representation of the Bézier ordinates of (a) a B-spline with respect to a vertex and (b) a B-spline with respect to a triangle. Non-zero Bézier ordinates are denoted by filled bullets, and zero ordinates are denoted by open bullets î. Let ˆf x a y b ( ) a+b (5 a b)! 5 = f xayb,i, (5.4) we define the (3 3 3)-tensor F by its matrix unfolding F (1) as ˆf ˆfx ˆfy ˆfx ˆfx 2 ˆfxy ˆfy ˆfxy ˆfy 2 F (1) = ˆf x ˆfx 2 ˆfxy ˆfx 2 ˆfx 3 ˆfx2 y ˆf xy ˆfx2 y ˆf y ˆfxy ˆfy 2 ˆfxy ˆfx 2 y ˆf xy 2 ˆfy 2 ˆfxy 2 ˆfy 3 ˆf xy 2. (5.5) Combining (3.12) (3.16) with (4.7), it follows that interpolation problem (4.3a) at vertex V i can be reformulated as F = C 1 G 2 G 3 G. (5.6)

14 5 PS5-SPLINE INTERPOLATION 12 c v i,3 Q v i,3 c v i,8 c v i,9 c v i,5 c v i,10 c v i,6 Q v i,1 c v i,1 c v i,4 c v i,7 Q v i,2 c v i,2 Figure 6: Schematic representation of the B-spline coefficients c v i,j, j = 1,..., 10, with respect to PS5-triangle t i (Q v i,1, Qv i,2, Qv i,3 ). Thus, C = F 1 G 1 2 G 1 3 G 1. (5.7) The values of the coefficients c v i,j can then be computed using (2.1), (5.1), (5.3), (5.5) and (5.7). For a compact notation of these expressions, we make use of a function R i (P 1, P 2, P 3 ) with three points P l = (x Pl, y Pl ), l = 1, 2, 3, as arguments. This function is defined as R i (P 1, P 2, P 3 ) = f i + 1 ( (xp1 x i ) + (x P2 x i ) + (x P3 x i ) ) f x,i ( (yp1 y i ) + (y P2 y i ) + (y P3 y i ) ) f y,i ( (xp1 x i )(x P2 x i ) + (x P2 x i )(x P3 x i ) + (x P3 x i )(x P1 x i ) ) f x 36 2,i + 5 ( (xp1 x i )(y P2 y i ) + (x P2 x i )(y P3 y i ) + (x P3 x i )(y P1 y i ) 36 + (y P1 y i )(x P2 x i ) + (y P2 y i )(x P3 x i ) + (y P3 y i )(x P1 x i ) ) f xy,i + 5 ( (yp1 y i )(y P2 y i ) + (y P2 y i )(y P3 y i ) + (y P3 y i )(y P1 y i ) ) f y 36 2,i (x P 1 x i )(x P2 x i )(x P3 x i ) f x 3,i + 25 ( (xp1 x i )(x P2 x i )(y P3 y i ) + (y P1 y i )(x P2 x i )(x P3 x i ) (x P1 x i )(y P2 y i )(x P3 x i ) ) f x 2 y,i + 25 ( (yp1 y i )(y P2 y i )(x P3 x i ) + (x P1 x i )(y P2 y i )(y P3 y i ) (y P1 y i )(x P2 x i )(y P3 y i ) ) f xy 2,i (y P 1 y i )(y P2 y i )(y P3 y i ) f y 3,i. (5.8) We then arrive at the following theorem.

15 5 PS5-SPLINE INTERPOLATION 13 d 33 d 34 d 36 d 35 d41 d 37 d 38 d 40 d 39 d 48 d 42 d d d 44 d 46 d d60 45 d d d 70 d66 d 73 d 82 d 81 d d d d d d d 74 d 80 d d 83 d d d 61 d 69 d 64 d 72 d d 79 d d 84 d 91 d 87 d 27 d d 16 d 28 d d d 85 d 29 d d d 76 d 77 d d d 23 d 9 d 13 d 18 8 d 19 d 24 d 4 d d 12 d 55 d 56 d 57 d 30 7 d 3 d 6 d 11 d 52 d 53 d 54 d 31 d 1 d 2 d 5 d 10 d 49 d 50 d 51 d 32 d 25 d 20 d 17 Figure 7: Schematic representation of the Bézier ordinates of a PS5-spline. Theorem 5.1. The PS5-spline s(x, y) in (4.4), with the coefficients c v i,j, j = 1,..., 10, corresponding to vertex V i chosen as c v i,1 = R i (Q v i,1, Q v i,1, Q v i,1), c v i,2 = R i (Q v i,2, Q v i,2, Q v i,2), c v i,3 = R i (Q v i,3, Q v i,3, Q v i,3), c v i,4 = R i(q v i,1, Qv i,1, Qv i,2 ), cv i,5 = R i(q v i,1, Qv i,1, Qv i,3 ), cv i,6 = R i(q v i,2, Qv i,2, Qv i,3 ), c v i,7 = R i(q v i,2, Qv i,2, Qv i,1 ), cv i,8 = R i(q v i,3, Qv i,3, Qv i,1 ), cv i,9 = R i(q v i,3, Qv i,3, Qv i,2 ), c v i,10 = R i (Q v i,1, Q v i,2, Q v i,3), (5.9) satisfies interpolation problem (4.3a) at vertex V i. 5.2 Interpolation at the split points To compute the value of a PS5-spline at a split point Z k we make use of the Bernstein-Bézier representation. We consider the macro-triangle T k (V 1, V 2, V 3 ), as shown in Figure 3, and we assume that the points indicated in the figure have the following barycentric coordinates: V 1 = (1, 0, 0), V 2 = (0, 1, 0), V 3 = (0, 0, 1), Z k = (z 1, z 2, z 3 ), R 12 = (λ 12, λ 21, 0), R 23 = (0, λ 23, λ 32 ), R 31 = (λ 13, 0, λ 31 ). (5.10) The Bézier ordinates are schematically represented in Figure 7. In [18] it is shown how these ordinates can be computed from the spline coefficients c v i,j and ct k in a stable way, using a sequence of convex combinations. We will briefly review the computation steps without going into the details. We will use the same notation as in [18] for the representation of all (intermediate) ordinates.

16 5 PS5-SPLINE INTERPOLATION 14 The Bézier ordinates d 1,..., d 16 in the neighbourhood of V 1 are completely determined by control polynomial T 1 (x, y). These ordinates can be found through subdivision of T 1 (x, y). A two-stage subdivision approach was proposed in [18]. In the first stage the Bernstein-Bézier form of the control polynomial is computed onto the triangle with the PS5-points V 1, S 12 and S 13 as its three vertices. The Bézier ordinates of the control polynomial on this finer triangle are denoted by e 1,j, j = 1,..., 10, as indicated in Figure 4. In the second stage the values of d 1,..., d 16 are computed from the ordinates e 1,j. Let e 2,j, j = 1,..., 10, be the Bézier ordinates of control polynomial T 2 (x, y) restricted on the triangle with vertices (V 2, S 23, S 21 ), and let e 3,j, j = 1,..., 10, be the Bézier ordinates of control polynomial T 3 (x, y) restricted on the triangle with vertices (V 3, S 31, S 32 ). The values of Bézier ordinates d 17,..., d 48 in the neighbourhood of vertices V 2 and V 3 are then computed from e 2,j and e 3,j. The values of the Bézier ordinates d 49,..., d 75 are obtained from the C 3 -smoothness conditions of the PS5-spline across the interior edges Z k R 12, Z k R 23 and Z k R 31. The remaining Bézier ordinates d 76,..., d 91 are found by subdivision of a single cubic polynomial p 3 defined on the triangle with vertices (4.8). This polynomial p 3 is defined by the following ten Bézier ordinates: b 300 = d 7, b 210 = d 12, b 201 = d 14, b 030 = d 23, b 021 = d 28, b 120 = d 30, b 003 = d 39, b 102 = d 44, b 012 = d 46, b 111 = c t k. (5.11) The values of d 7, d 12 and d 14 are given by d 7 = z 1 (z 1 e 1,1 + z 2 e 1,4 + z 3 e 1,5 ) + z 2 (z 1 e 1,4 + z 2 e 1,7 + z 3 e 1,10 ) + z 3 (z 1 e 1,5 + z 2 e 1,10 + z 3 e 1,8 ), (5.12a) d 12 = z 1 (z 1 e 1,4 + z 2 e 1,7 + z 3 e 1,10 ) + z 2 (z 1 e 1,7 + z 2 e 1,2 + z 3 e 1,6 ) + z 3 (z 1 e 1,10 + z 2 e 1,6 + z 3 e 1,9 ), (5.12b) d 14 = z 1 (z 1 e 1,5 + z 2 e 1,10 + z 3 e 1,8 ) + z 2 (z 1 e 1,10 + z 2 e 1,6 + z 3 e 1,9 ) + z 3 (z 1 e 1,8 + z 2 e 1,9 + z 3 e 1,3 ). (5.12c) The values of d 23, d28 and d 30 are computed from e 2,j, j = 1,..., 10, and their expressions are similar to (5.12). Likewise, the expressions for d 39, d44 and d 46 are based on e 3,j, j = 1,..., 10. For more details we refer to [18]. The value of the PS5-spline s(x, y) at split point Z k can be computed by evaluating the polynomial p 3 with Bézier ordinates (5.11) at Z k, i.e., s(z k ) = d 91 = z 1 3 d z 1 2 z 2 d z 1 2 z 3 d14 + z 2 3 d z 2 2 z 3 d z 2 2 z 1 d30 + z 3 3 d z 3 2 z 1 d z 3 2 z 2 d z 1 z 2 z 3 c t k. (5.13) The following theorem follows directly from (5.13). Theorem 5.2. The PS5-spline s(x, y) in (4.4), with the coefficient c t k corresponding to triangle T k (V 1, V 2, V 3 ) chosen as c t k = 1 6 z 1 z 2 z 3 ( g k z 1 3 d 7 3 z 1 2 z 2 d12 3 z 1 2 z 3 d14 z 2 3 d 23 3 z 2 2 z 3 d28 3 z 2 2 z 1 d30 z 3 3 d 39 3 z 3 2 z 1 d44 3 z 3 2 z 2 d46 ), (5.14) satisfies interpolation problem (4.3b) at the split point Z k in triangle T k. Combining Theorems 5.1 and 5.2 gives us the full interpolation scheme for PS5-splines in representation (4.4).

17 6 REDUCED PS5-SPLINE INTERPOLANTS 15 6 Reduced PS5-spline interpolants 6.1 C 3 -smoothness across an interior edge In this section we derive the restriction on the spline coefficient c t k such that the PS5-spline will be C 3 -continuous across the interior edge Z k V 1 inside macro-triangle T k (V 1, V 2, V 3 ) (see Figure 3). This condition is satisfied when the values of d 3, d 6, d 7, d 8, d 11, d 12, d 13, d 14, d 15, d 52, d 55, d 72, d 75, d 76, d 84 and d 85 (see Figure 7) can be regarded as Bézier ordinates of a single cubic polynomial after subdivision. We consider the cubic polynomial with Bézier ordinates f abc, a + b + c = 3, defined on the triangle with vertices P 1 = 4 5 V Z k, P 2 = 1 5 V V Z k, P 3 = 1 5 V V Z k. (6.1) In case of the required additional C 3 -smoothness, it follows that the values of d 3, d 6, d 7, d 8, d 11, d 12, d 13, d 14, d 15, d 52, d 55, d 72, d 75, d 76, d 84 and d 85 can be computed from f abc using the formulae (3.4) (3.5). This gives us new expressions for these d i in terms of f abc. Comparing the new expressions with the expressions given in [18], we get the restriction on the spline coefficient c t k. We recall from [18] the following relations: d 3 = z 1 e 1,1 + z 2 e 1,4 + z 3 e 1,5, (6.2a) d 6 = d 6 λ 12 d 3 λ 21 = z 1 e 1,4 + z 2 e 1,7 + z 3 e 1,10, (6.2b) d 8 = d 8 λ 13 d 3 = z 1 e 1,5 + z 2 e 1,10 + z 3 e 1,8, λ 31 (6.2c) d 11 = d 11 λ 12 d 6 λ 12 λ 21 d6 2 = z 1 e 1,7 + z 2 e 1,2 + z 3 e 1,6, λ 21 (6.2d) d 15 = d 15 λ 13 d 8 λ 13 λ 31 d8 2 = z 1 e 1,8 + z 2 e 1,9 + z 3 e 1,3, λ 31 (6.2e) d 13 = d 12 λ 12 d 7 λ 21 z 1 d6 λ 21 z 2 d11 λ 21 z 3 = z 1 e 1,10 + z 2 e 1,6 + z 3 e 1,9. (6.2f) In case of C 3 -smoothness across the interior edge Z k V 1, it is easy to verify that we must have d 3 = f 300, d6 = f 210, d8 = f 201, We also obtain from [18] that d 11 = f 120, d 15 = f 102, d13 = f 111. (6.3) d 53 = d 52 λ 12 d 11 = λ 12 (λ 12 d6 + λ 21 d11 ) + λ 21 (λ 12 d31 + λ 21 d24 ), λ 21 (6.4a) d 56 = d 55 λ 12 d 12 = λ 12 (z 1 d6 + z 2 d11 + z 3 d13 ) + λ 21 (z 1 d31 + z 2 d24 + z 3 d29 ), λ 21 (6.4b) d 71 = d 72 λ 13 d 15 = λ 13 (λ 13 d8 + λ 31 d15 ) + λ 31 (λ 13 d43 + λ 31 d38 ), λ 31 (6.4c) d 74 = d 75 λ 13 d 14 λ 31 = λ 13 (z 1 d8 + z 2 d13 + z 3 d15 ) + λ 31 (z 1 d43 + z 2 d45 + z 3 d38 ), (6.4d)

18 6 REDUCED PS5-SPLINE INTERPOLANTS 16 where and d 24 = d 24 λ 21 d 19 = z 1 e 2,8 + z 2 e 2,5 + z 3 e 2,10, λ 12 (6.5a) d 31 = d 31 λ 21 d 24 λ 21 λ 12 d24 2 = z 1 e 2,3 + z 2 e 2,8 + z 3 e 2,9, λ 12 (6.5b) d 29 = d 30 λ 21 d 23 λ 12 z 1 d31 λ 12 z 2 d24 λ 12 z 3 = z 1 e 2,9 + z 2 e 2,10 + z 3 e 2,6. (6.5c) d 38 = d 38 λ 31 d 35 = z 1 e 3,7 + z 2 e 3,10 + z 3 e 3,4, λ 13 (6.6a) d 43 = d 43 λ 31 d 38 λ 31 λ 13 d38 2 = z 1 e 3,2 + z 2 e 3,6 + z 3 e 3,7, λ 13 (6.6b) d 45 = d 44 λ 31 d 39 λ 13 z 1 d43 λ 13 z 3 d38 λ 13 z 2 = z 1 e 3,6 + z 2 e 3,9 + z 3 e 3,10. (6.6c) In case of C 3 -smoothness across Z k V 1, we find that d 53 = λ 12 (λ 12 f λ 21 f 120 ) + λ 21 (λ 12 f λ 21 f 030 ), d 56 = λ 12 (z 1 f z 2 f z 3 f 111 ) + λ 21 (z 1 f z 2 f z 3 f 021 ), d 71 = λ 13 (λ 13 f λ 31 f 102 ) + λ 31 (λ 13 f λ 31 f 003 ), d 74 = λ 13 (z 1 f z 2 f z 3 f 102 ) + λ 31 (z 1 f z 2 f z 3 f 003 ). (6.7a) (6.7b) (6.7c) (6.7d) Combining (6.3), (6.4) and (6.7), we find the following necessary conditions for the additional C 3 -smoothness λ 12 d31 + λ 21 d24 = λ 12 f λ 21 f 030, z 1 d31 + z 2 d24 + z 3 d29 = z 1 f z 2 f z 3 f 021, λ 13 d43 + λ 31 d38 = λ 13 f λ 31 f 003, z 1 d43 + z 2 d45 + z 3 d38 = z 1 f z 2 f z 3 f 003. (6.8a) (6.8b) (6.8c) (6.8d) We can reformulate (6.8) as Finally, we deduce from [18] that f 030 = d 24 + λ 12 ( ) d31 d11, (6.9a) λ ( 21 f 021 = d z z ) 2 λ 12 ( ) d31 d11, (6.9b) z 3 z 3 λ 21 f 003 = d 38 + λ 13 ( ) d43 d15, (6.9c) λ ( 31 f 012 = d z z ) 3 λ 13 ( ) d43 d15. (6.9d) z 2 z 2 λ 31 d 77 = d 76 λ 12 d 13 λ 21 = z 1 (z 1 d6 + z 2 d11 + z 3 d13 ) + z 2 (z 1 d31 + z 2 d24 + z 3 d29 ) + z 3 c t k. (6.10)

19 6 REDUCED PS5-SPLINE INTERPOLANTS 17 The additional C 3 -smoothness across Z k V 1 implies that d 77 = z 1 (z 1 f z 2 f z 3 f 111 ) + z 2 (z 1 f z 2 f z 3 f 021 ) + z 3 (z 1 f z 2 f z 3 f 012 ), (6.11) resulting in the following condition on the spline coefficient c t k : c t k = z 1 f z 2 f z 3 f 012. (6.12) Filling (6.2), (6.3), (6.5) (6.6) and (6.9) into (6.12), we get an expression for c t k in terms of e i,j, i = 1, 2, 3 and j = 1,..., 10. Since the ordinates e i,j can be computed through subdivision from the three control polynomials T i (x, y) corresponding to the three vertices V i, i = 1, 2, 3, we have an expression for c t k in terms of the vertex coefficients cv i,j, i = 1, 2, 3 and j = 1,..., 10. A similar procedure can be used to derive the restrictions on c t k for additional C3 -smoothness across the other interior edges Z k V 2 or Z k V Reduced PS5-splines In this section we consider some reduced PS5-spline spaces that preserve full approximation order. The B-spline coefficients c t k, k = 1,..., n t, in representation (4.4) are determined in a specific way such that the obtained PS5-splines are still able to reproduce quintic polynomials. A first approach is requiring additional C 3 -smoothness across an interior edge for all macrotriangles in the triangulation. Such a reduced PS5-spline space was described in [1]. For each macro-triangle T k (V 1, V 2, V 3 ) in, there are three possible choices to impose the additional smoothness, i.e., across the edge Z k V 1, Z k V 2 or Z k V 3. In case of Z k V 1, we get the following theorem. Theorem 6.1. The PS5-spline s(x, y) in (4.4), with the coefficient c t k corresponding to triangle T k (V 1, V 2, V 3 ) chosen as c t k = z 1 d13 + z 2 d29 + z 3 d45 + z ( ) 2 λ 12 ( ) z 1 z 2 d31 d11 z 3 λ 21 + z ( ) 3 λ 13 ( ) z 1 z 3 d43 d15, (6.13) z 2 λ 31 reproduces quintic polynomials. Proof. From (6.3), (6.9) and (6.12), it follows that the spline s(x, y) is C 3 -continuous across interior edge Z k V 1. From [1] we know that such a spline reproduces quintic polynomials. The next theorem gives a symmetric condition on the coefficient c t k. This reduced spline space was developed in [18]. Theorem 6.2. The PS5-spline s(x, y) in (4.4), with the coefficient c t k T k (V 1, V 2, V 3 ) chosen as corresponding to triangle reproduces quintic polynomials. c t k = z 1 (z 1 e 1,10 + z 2 e 1,6 + z 3 e 1,9 ) + z 2 (z 1 e 2,9 + z 2 e 2,10 + z 3 e 2,6 ) + z 3 (z 1 e 3,6 + z 2 e 3,9 + z 3 e 3,10 ), (6.14)

20 7 A NUMERICAL EXAMPLE 18 We now consider reduced PS5-splines defined on a uniform PS-triangulation, where the barycentric coordinates in (5.10) satisfy on all triangles. z 1 = z 2 = z 3 = 1/3 and λ 12 = λ 21 = λ 23 = λ 32 = λ 31 = λ 13 = 1/2 (6.15) Theorem 6.3. Let be a uniform PS-refinement of a uniform triangulation, satisfying (6.15), then the reduced PS5-splines described in Theorems 6.1 and 6.2 are equivalent on. Moreover, these splines are locally C 3 -continuous on each macro-triangle in. Proof. The first three terms in (6.13) are equivalent to (6.14). The remaining terms in (6.13) are zero because of the uniformity of the PS-triangulation. The local C 3 -continuity on each macrotriangle follows from the symmetry of formula (6.14). The reduced spline space in Theorem 6.3 was also considered in [8] for the uniform three-direction mesh. Theorem 5.1 can be used to construct an interpolation scheme based on the reduced PS5-splines described in Theorems 6.1 and 6.2. These reduced spline interpolants have the advantage that only data are needed located at the vertices of the triangulation. 7 A numerical example For a practical example we consider the function f(x, y) = 3 4 exp ( (9x 2)2 + (9y 2) ( 2 exp (9x 7)2 + (9y 3) 2 4 ) + 34 ( exp (9x + 1)2 49 9y + 1 ) 10 ) 1 5 exp( (9x 4) 2 (9y 7) 2), (7.1) on the domain Ω = [0, 1] [0, 1]. This function is known as Franke s test function (see, e.g., [2, 7]). Let {V i } nv i=1 be the set of 25 data points given in Figure 8. The figure shows the corresponding Delaunay triangulation with a particular PS-refinement. The dimension of the general PS5-spline space on this mesh is 282, and the dimension of the reduced spline space is 250. The data f x a y b,l and g m in (4.3) are sampled from the function f(x, y). To measure the accuracy of a spline fit s, we have computed the maximum error and the mean error on a uniform grid G on Ω, i.e., E max = max f(x, y) s(x, y), E mean = (x,y) G (x,y) G f(x, y) s(x, y) The PS5-spline interpolant, defined by Theorems 5.1 and 5.2 on the mesh shown in Figure 8, has a maximum error equal to , and its mean error is The reduced PS5-spline interpolant, defined by Theorems 5.1 and 6.2, has a maximum error of and a mean error of This reduced PS5-spline interpolant is depicted in Figure 9.

21 8 CONCLUDING REMARKS Figure 8: A triangulation with PS-refinement Figure 9: A reduced PS5-spline (with the triangular mesh lines) and its contour plot. The spline is defined on the mesh in Figure 8, and it interpolates the function (7.1) at the vertices. 8 Concluding remarks In this paper we provided local interpolation rules for quintic Powell-Sabin splines represented with a normalized B-spline basis. The general interpolation scheme uses data at the vertices of the triangulation and at the Powell-Sabin split points. We also discussed interpolation based on several reduced spline spaces. These reduced spline interpolants only need data at the vertices of the triangulation. The presented interpolation schemes make use of derivative information (up to order three) at the vertices. When no derivative information is available, these interpolation schemes can be incorporated into a two-stage method. In the first stage the needed derivative information is then locally estimated from the data, e.g., by doing local least squares with polynomials or radial basis functions.

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