A B-SPLINE-LIKE BASIS FOR THE POWELL-SABIN 12-SPLIT BASED ON SIMPLEX SPLINES

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1 MATHEMATICS OF COMPUTATION Volume, Number, Pages S 5-578(XX)- A B-SPLINE-LIKE BASIS FOR THE POWELL-SABIN -SPLIT BASED ON SIMPLEX SPLINES ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Abstract. We introduce a simplex spline basis for a space of C - quadratics on the well known Powell-Sabin -split triangular region. Among its many important desirable properties, we show that it has an associated recurrence relations for evaluation and differentiation. Also developed are a Marsden-like identity, quasi-interpolants, approximation methods exhibiting unconditional stability, a subdivision scheme, and smoothness conditions across macroelement edges. Contents. Introduction.. Background.. Notation 4.3. PS-Split 5. Simplex Splines over PS-Split 6.. Recurrence relations: basis form 3. A Marsden-like Identity Proof of the Marsden-like Identity Linear independence Differentiation 9 4. Evaluation Algorithms 5. Subdivision Computing the knot insertion matrix Coefficient averaging algorithm 7 6. Stability of the S-Basis and the Distance to the Control Points Linear and quadratic quasi-interpolant Stability of the quadratic S-spline basis Distance between a surface and its control points 3 7. Smooth Surfaces Joins: Conditions on PS Refinements of Triangulations 3 8. Approximation Methods Lagrange interpolation Hermite interpolation Quasi-interpolants 36 Received by the editor October 5, and in revised form, November 3,. Mathematics Subject Classification. 4A5, 65D7,65D7,65D5. The authors would like to thank the Centre of Mathematics for Applications (CMA) at the University of Oslo for support and encouragement of this work. c XXXX American Mathematical Society

2 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD 9. Concluding Remarks 37 Appendix A. Proof of Recurrence Relation Theorem.3 37 Appendix B. Acknowledgements 4 References 4. Introduction Surfaces defined over triangulations have widespread application in many areas ranging from finite element analysis and physics and engineering applications to the entertainment industry. For many of these applications piecewise linear surfaces do not offer sufficient smoothness. To obtain C smoothness, one must either use quintic polynomials with degrees of freedom over each triangle or use lower degree macroelements that subdivide each triangle into a number of subtriangles. Thus far, the second approach has largely been based on using the Bernstein-Bézier basis on each subtriangle, manually enforcing the smoothness internal to each triangle and solving the resulting constrained system. In this paper we introduce the S-spline basis, a B-spline-like basis, over a single macroelement known as the Powell-Sabin -split (PS-split). Internal to the macroelement, each of the basis elements is C and the basis is unconditionally stable independent of the shape of the macroelement. Analogous results for analytical and shape properties, so inextricably intertwined in the B-spline/Bézier formulation of surfaces, are shown for the S-spline basis... Background. Consider a triangulation T of a domain Ω R and integers r, d with r < d. Let S r d(t ) := {f C r (Ω) : f T is a polynomial of degree d, T T } be the space of piecewise C r polynomials of degree d over T. To evaluate f S r d (T ) the standard approach is to represent f piecewise using a Bernstein Bézier representation on each triangle in T and use the de Casteljau Recursive Algorithm [3] for evaluation. It was shown in [5] that the L q normalized Bernstein-Bézier basis on a triangle is stable in any L q norm. Now suppose T is obtained from a coarser triangulation T by splitting each triangle in T into subtriangles. Suppose for each triangle T there exists a basis B of the space S r d (T ) restricted to that has: stable recurrence relations differentiation formula minimal support local linear independence nonnegative partition of unity explicit dual functionals L q stable basis, q simple conditions for C joins to neighboring triangles well conditioned collocation matrices for Lagrange and Hermite interpolation using certain sites, and subdivision algorithms of Lane-Riesenfeld type. These characteristics yield a B-spline-like basis within each, but they do not give a B-spline basis for the whole triangulation T. So B behaves like the triangular

3 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT 3 Figure. Left: PS6-split, and right: PS-split configurations. Bernstein Bézier basis across the edges of T, but like a B-spline internal to each triangle of T. The characteristics of this basis make it unnecessary to convert it to a Bernstein Bézier representation on each of the subtriangles of. Instead there is a single control mesh to facilitate control and early visualization of the surface over each triangle in T. We develop a quadratic C spline basis with the above desired behavior on each element of T that has been split according to the scheme of the Powell-Sabin - Split (PS) []. Each triangle in T is split into subtriangles delineated by the complete graph connecting all vertices and edge midpoints. (See Fig. ). The union of the bases over T can be used to represent the space of C quadratic splines on T. An interpolatory subdivision scheme for the PS split introduced in [9] can be used to evaluate a quadratic PS specified surface on an arbitrary triangulation. There are other notable approaches to constructing spline spaces over triangulations, in particular: The nodal basis, dual to the degrees of freedom for the Hermite constraints, is commonly employed [3] in finite element calculations. Elaborate constructions exist based on perturbing the knots of a Bernstein- Bézier triangle representation in [6, 7,, ]. However, the space thusly obtained, represented by a simplex spline basis over the perturbed knot set, depends on the particular perturbations applied. Also, evaluation based on simplex spline recursion is slow [, ]. An approach employing Delaunay configurations based on points in general position is introduced and studied by Neamtu in [8, 9]. This approach uses a sum of simplex splines as basis functions and generalizes many B- spline properties; however, there is no associated recurrence relation relating these functions to basis functions of lower degree. Box splines [] can be applied to uniform triangulations. Schemes using a different split rule, called the Powell-Sabin 6-Split (PS6)[], create T from T by dividing each triangle of T into 6 subtriangles by connecting the incenter of a triangle to its vertices and to the incenters of its adjacent triangles. Dierckx [8] introduced a B-spline like basis for C quadratics on a T split of T according to the PS6 rules. Since there is no

4 4 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD recurrence relation for this basis, evaluation is done by transforming them to the Bernstein-Bézier representation on each of the 6 subtriangles. Being nonnegative and forming a partition of unity, the basis functions can be manipulated with their corresponding control points. They form a stable basis in the L q norm, q, where the stability constant depends on the smallest angle in the underlying triangulation [6, 3]. The Dierckx basis was recently extended in [5] to a space of C quintics on triangulations amenable to the PS6-split. In this paper we introduce the S-spline basis, a quadratic simplex spline basis S, for the PS macroelement. In addition to having simplex spline properties, we prove that the S-spline basis has many desirable B-spline properties. In particular, they form a partition of unity, provide a recurrence relation down to hat functions, satisfy a Marsden-like identity, and exhibit L q stability for a scaled version. Furthermore, the restriction of each basis element to the boundary edges of the macroelement reduces to a standard univariate B-spline. A control mesh can be formed that mimics the shape of the surface and exhibits distance O(h ) to any one of its control points from its surface, where h is the length of the longest edge. We obtain a pyramidal evaluation algorithm in terms of the control points that is strikingly reminiscent of the analogous one for triangular Bézier surfaces. An unusual hybrid, the control mesh presents both triangular and quadrilateral connectivity. The S-spline basis can be used to represent surfaces over arbitrary triangulations. We derive conditions for C smoothness across macrotriangle edges in terms of control points similar to the triangular Bézier case. We obtain two algorithms for the subdivision approach considered. One subdivides each macroelement into 4 submacroelements so that the S-spline basis elements are subdivided giving formulas analogous to Oslo Algorithm [4]. The other is an analog of the the Lane-Riesenfeld Algorithm for Bézier surfaces [4]. Repeated subdivision converges quadratically due to the aforementioned distance result and can be used for evaluation, rendering and other computations commonly associated with freeform geometric modeling. Now we preface the ensuing mathematical development with a description of the notation consistently used throughout... Notation. We use small boldface letters to denote vectors and capital boldface letters for matrices, while calligraphic fonts like R, S indicate sets. With function spaces we associate symbols like S. The symbol R m,n denotes the class of m n real matrices A, R,.... We denote the unit vectors in R m by e,..., e m, the identity matrix with I, and e := [,,..., ] T R m. We write #A for the number of columns of A, and #S for the cardinality of set S. If A R m,n and i = [i,..., i r ] T, j = [j,..., j s ] T with i < < i r m, j < < j s n, then A(i, j) R r,s denotes the matrix whose k, l element is a ik,j l. In particular, c(i) denotes the vector whose jth element is c ij. For a vector of functions f the symbol f(i) denotes the subvector of functions whose jth element is f ij. The support of a function f, denoted by supp(f), is the closure of the set of values in the domain of f at which f is nonzero. We denote by Π d (R ) the space of bivariate polynomials with real coefficients of total degree d, i. e., the span of all monomials of the form x i xi, where the nonnegative integers satisfy i + i d. The dimension of Π d (R ) is ν d :=

5 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT Figure. Reference labels for vertices and induced subtriangles for the PS-split. (d + )(d + )/. S r d ( ) denotes the space of piecewise polynomials of degree d and smoothness C r on a triangulation of R. To denote the closed or open convex hull of a set S we use ch(s) or ch(s) o, respectively. When S, consisting of the columns of A, spans R m, we say simply A spans R m..3. PS-Split. Prominent among the known macroelements is the Powell-Sabin -split (PS-split). Given three noncollinear points p, p, p 3 in R, the triangle (.) := ch({p, p, p 3 }) with vertices p, p, p 3 will serve as our macrotriangle. The PS-split divides into subtriangles delineated by the complete graph formed by the 3 original triangle vertices and its 3 edge midpoints (see Fig. ). We number the vertices p,..., p and the triangles,..., of the PS- split as depicted in Figure. Note that p 4 := (p + p ), p 5 := (p + p 3 ), p 6 := (p + p 3 ), (.) p 7 := (p 4 + p 6 ), p 8 := (p 4 + p 5 ), p 9 := (p 5 + p 6 ), p := 3 (p + p + p 3 ). P S is the PS triangulation of. We follow the convention in [], to decide in which subtriangle each edge and vertex belongs. The author defines a point p to be in the half-open convex hull of k if and only if there is a vector η with positive slope, and a scalar ε > such that {p + se + tη : < s, t < s + t < ε} is completely contained in the interior of k. The resulting configuration is shown in

6 6 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Figure 3. Scheme that uniquely assigns each x to a unique (possibly half-open) subtriangle k. Figure 3 for a typical triangle. Note that in what subtriangle each edge and vertex is contained depends on the orientation of the triangle. This convention extends to associate every edge and vertex in the PS-refinement of an arbitrary triangulation with a unique subtriangle. Thus this uniquely and completely defines a piecewise polynomial at all points of the PS-refinement of an arbitrary triangulation. The following algorithm determines a unique subtriangle for a point in R with respect to a single PS macroelement. The algorithm is consistent with the convention described above for the triangle in Figure 3. Algorithm.. Let x R and be partitioned by {,..., } as in Figure 3: () Find the barycentric coordinate triple (β, β, β 3 ) of x with respect to. () Compute sw := [ 5, 4, 3,,, ]t, where t := [(β > ), (β ), (β 3 ), (β > β ), (β > β 3 ), (β β 3 )] T. (3) Select k from the table below. sw k (4) x k. x / for other values of sw. Note that t is a Boolean vector consisting of zeros and ones. For example the second to last component is one if x is closer to p than to p 3, constraining it to lie in only one of the triangles k, k =,, 3, 7, 8, 9.. Simplex Splines over PS-Split On the triangulation P S of the triangle, we consider the spline spaces: (.) S d := S d d ( P S ), d =,,.

7 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT Figure 4. Shaded areas indicated support for knot configurations of the linear simplex splines S j, on PS-split. Since there are triangles and vertices in P S (see Fig. ), n := dim(s ) = and n := dim(s ) =, respectively. Similarly, n := dim(s ) =, and there is a unique f S that interpolates function values and gradients at the vertices p, p, p 3 and cross derivatives at the midpoints p 4, p 5, p 6 []. Definition.. For d =,,, we define S-splines for S d by (.) S j,d = v j,d ν d M( K j,d ), j =,..., n d, where M( K j,d ) is a degree d simplex spline normalized to have unit integral, and v j,d := v(ω j,d ) is the -dimensional volume (area) of Ω j,d := supp(s j,d ) = ch(k j,d ). The set (.3) S d := {S,d,..., S nd,d}, d =,,, is the S-spline basis of degree d for the PS-split. The knot sets K j,d are defined as follows: K j, consists of the three vertices of j, while K j, and K j, are defined as follows: (.4) K, = [p (), p 4, p 6 ] K, = [p (), p 5, p 4 ] K 3, = [p () 3, p 6, p 5 ] K 4, = [p, p 4, p, p ] K 5, = [p, p 5, p 3, p ] K 6, = [p 3, p 6, p, p ] K 7, = [p, p 4, p, p 6 ] K 8, = [p, p 5, p, p 4 ] K 9, = [p 3, p 6, p, p 5 ] K, = [p 4, p 5, p 6, p ],

8 8 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Figure 5. Shaded areas indicated support for knot configurations of the quadratic simplex splines S j, on PS-split. (.5) K, = [p (3), p 4, p 6 ] K, = [p (), p 4, p, p 6 ] K 3, = [p, p 4, p, p 5, p 6 ] K 4, = [p (), p 5, p, p 4 ] K 5, = [p (3), p 5, p 4 ] K 6, = [p (), p 5, p 3, p 4 ] K 7, = [p, p 5, p 3, p 6, p 4 ] K 8, = [p () 3, p 6, p, p 5 ] K 9, = [p (3) 3, p 6, p 5 ] K, = [p () 3 p 6, p, p 5 ] K, = [p 3, p 6, p, p 4, p 5 ] K, = [p (), p 4, p 3, p 6 ]. Note that superscript indicates multiplicity so that [p (), p 4, p 6 ] = [p, p, p 4, p 6 ]. Comprised of points in R, the d + 3 columns of K j,d are called the knots of S j,d. A spline of degree d on P S is any linear combination n d j= c js j,d. The supports of the linear S-splines are shown in Figure 4. There are 4 distinctly structured types grouped as follows: the 3 corners S,, S,, S 3,, the 3 midpoint edges S 4,, S 5,, S 6,, the 3 Egyptian pyramids S 7,, S 8,, S 9,, and the center pyramid S,. For quadratic S-splines shown in Figure 5 there are three distinct types, namely, the 3 corners S,, S 5,, S 9,, the 6 edges S,, S 4,, S 6,, S 8,, S,, S, and the 3 with trapezoidal support S 3,, S 7,, S,. Observe that all support sets are shaped either like a triangle or a quadrilateral. The S-splines have the following properties. Theorem.. For d =,,, j =,..., n d, and x : () Smooth piecewise polynomial: S j,d S d. () Nonnegativity and positivity: S j,d (x), S j,d (x) > for x ch(k j,d ) o.

9 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT 9 (3) Partition of unity: n d j= S j,d(x) =. (4) Convex hull: If the S-spline coefficients c j are points in R or R 3 then f(x) := n d j= c js j,d (x) is contained in ch({c j } n d ). (5) Local linear independence: On each of the subtriangles k the nonzero S j,d constitute a basis for Π d (R ). (6) Basis: The S-basis S d := {S,d,..., S nd,d} is a basis for S d. (7) Degree : S j, = χ j, the characteristic function of j. (8) Linear hat: S j, is a piecewise linear hat function with, (.6) S j, (p i ) = δ i,j, i, j =,...,, (9) Edge restriction: For d =, a spline f = n d j= c js j,d reduces to a univariate spline with one interior (midpoint) knot along an edge of. In particular, for d = along the edge ch({p, p }), (.7) f(( t)p + tp ) = c B, (t) + c 4 B, (t) + c B 3, (t), t [, ], where {B j, } 3 j= is the set of 3 consecutive univariate linear B-splines on the knot vector [,,,, ], normalized to form a partition of unity on [, ]. Similarly, for d = (.8) f(( t)p +tp ) = c B, (t)+c B, (t)+c 4 B 3, (t)+c 5 B 4, (t), t [, ], where {B j, } 4 j= is the set of 4 consecutive univariate quadratic B-splines on the knot vector [,,, /,,, ]. Again, they are normalized to form a partition of unity. Proof. As bona fide simplex splines, the elements of the S-basis exhibit the following desirable properties [5, 7]: () S j,d has d + 3 knots and is a piecewise polynomial of degree d. () The support of S j,d is ch(k j,d ). (3) S j,d is positive in the interior of its support. (4) The knot lines of S j,d form the complete graph of the knots. (5) S j,d has d + m continuous derivatives across a knot line containing m collinear knots. Since all knot lines of S j,d are segments of edges in P S it follows from Property that S j,d is a piecewise polynomial of degree d on P S. Moreover, since there are exactly two knots on those knot lines of S j,d containing a point in the interior of, S j,d C d ( P S ) by Property 5. It follows that S j,d S d. There are precisely ν d nonzero S j,d on each triangle k (cf. Figures 4 and 5). In particular, it follows from the support, continuity and partition of unity that for d =, S j, is the characteristic function of j, and that for d = equation (.6) holds. We show a Marsden-type identity in Section 3 that implies that functions in S d span Π d (R ) on every k and this identity also implies partition of unity, local linear independence, and basis properties. The convex hull property follows from nonnegativity and partition of unity. The edge restriction property will be proved using the recurrence relations in Theorem.3. Figure 6 illustrates the edge restriction property along the front edge ch({p, p }), where S j,, for j =,, 4, 5, exhibit the proper shapes of a univariate quadratic B- spline. Along the boundary of the triangle, the S-spline S 3,, whose support is trapezoidal, has function value, but its cross-boundary derivative is nonzero.

10 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Figure 6. The S-splines S j, for j =,, 3, 4, 5, on the unit triangle p = (, ), p = (, ), and p 3 = (, )... Recurrence relations: basis form. Consider the recurrence relation for univariate B-splines B j,d of degree d and with d + knots t j,..., t j+d+, B j,d = µ j,d B j,d + λ j+,d B j+,d, µ j,d (x) := x t j t j+d t j, λ j,d (x) := t j+d x t j+d t j, and where λ j,d = µ j,d := if t j+d = t j. On a knot vector t := (t j ) m j= with t t t m, the recurrence relation for the (r := m d ) B-splines of degree d that are defined on t can be written in matrix form as, (.9) [B,d,..., B r,d ] = [B,d,..., B r+,d ]R d,t, t x < t m, where µ,d λ,d µ,d (.) R d,t =. λ.. 3,d R r+,r.... µr,d λ r+,d If t k x < t k+ for some integer k, then only B k d,d (x),..., B k,d (x) can be nonzero and (.9) reduces to the polynomial version (.) [B k k d,d,..., B k k,d] = [B k k d+,d,..., B k k,d ]R k d,t,

11 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT where Bj,d k is the polynomial of degree d representing B j,d on [t k, t k+ ) and (.) R k d,t := λ k d+,d µ k d+,d λ k,d µ k,d Rd,d+. We now give analogs of equations (.9) and (.) for the S-spline basis. Consider first (.9). Theorem.3. If (.3) s T d := [S,d,..., S nd,d], d =,,, then (.4) s T d = s T d R d, d =,, where R R, and R R, are given by (.5) γ β 3, 4β γ β,3 4β 3 γ β,3 4β 3 γ β 3, 4β γ 3 β, 4β R (x) := γ 3 β, 4β β 3, 4β,3 3γ β,3 4β, 3γ β,3 4β, 3γ β 3, 4β,3 3γ β, 4β 3, 3γ 3 β, 4β 3, 3γ 3 and (.6) R (x) = γ β β 3 β γ β 3 β γ 3 β β,3 3β 3 β,3 β, 3β β 3, β 3, 3β β, β,3 3β 3β β,3 β, 3β 3 3β β 3, β 3, 3β γ 3 γ γ 3β 3 β,. Here β = [β, β, β 3 ] T is the vector of barycentric coordinates of x with respect to the triangle so that β i (p j ) = δ i,j and (.7) γ j := β j, β i,j = β i β j, for i, j =,, 3. Moreover, R d (i, j)s i,d (x) for all i, j and x.

12 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Table. Enumeration of the sets H k d and Gk d. k H k = G k H k G k,6,7,,,,3,,,,4,7 4,5,6,,3,4,, 3,4,8 8,9,,3,4,5,6,7 4,5,8,3,4 3,4,5,6,7,8 5 3,5,9 6,7,8 6,7,8,9,, 6 3,6,9,, 7,8,9,,, 7 6,7,,3,,,3,7,,, 8 4,7, 3,4,6,7,3,4,7,, 9 4,8, 7,8,,,3,4,6,7, 5,8, 3,7, 3,4,6,7,8, 5,9, 3,6,7,8,, 6,9, 3,7,8,,, To prove this result we use the recurrence and knot insertion relations found in [7]. The details of the proof can be found in Appendix A. Using (.4) repeatedly we obtain, Corollary.4. Suppose x k for some k and d. Then (.8) s d (x) T := [S,d (x),..., S nd,d(x)] = e T k Moreover, for a spline with coefficients c = [c,..., c nd ] T, n d (.9) f d (x) := c j S j,d (x) = e T k j= d R i (x). i= d R i (x)c. We need to establish some notation before developing an analog of (.) for the S-spline basis. Definition.5. We define (.) i= G k d := {j : k supp(s j,d )}, d =,,, H k d := {j : R d (k, j) for some x}, k =,..., n d, d =,. We use the symbols g k d and hk d for the vectors consisting of the elements in Gd k and Hd k, respectively, arranged in increasing order. The elements in Gd k single out the indices of the S-splines of degree d that are nonzero over triangle k. It is easily verified that each Gd k contains ν d elements, so that g k d = [i,..., i νd ] T with i i νd (cf. Figures 4 and 5), and we recall that ν d is the dimension of the polynomial space Π d (R ), d =,,. We show the elements in the sets Hd k and Gk d in Table. We also note that (.) G k = H k and G k = H i Hi Hi3, [i, i, i 3 ] = g k, k =,...,.

13 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT 3 Definition.6. For d =,, define S k j,d to be the polynomial representing S j,d on k. Also define, s k d = [S k,d,..., S k n d,d] T, s k d = s k d(g k d) R ν d, which represents the ordered vector whose elements are in the set S k d := {Sk j,d : j G k d }. Definition.7. For k define submatrices R k := R (k, g k ) R,3, R k := R (g k, g k ) R 3,6, where g k d is defined in Definition.5. For example, from Table it follows that g = [, 6, 7] and g = [,, 3,,, ]. Thus, R (x)r (x) = [ ] γ β β 3 γ γ 3, 4β γ 3 3β γ. γ 3 3β We are now ready to state the polynomial version of Corollary.4 (cf. equation (.)). Corollary.8. For k =,...,, d =,, and c T = [c,..., c nd ], d d (.) s k d T = R k i, fd k = R k i c(g k d), d =,,. i= i= Here c(g k d ) is the vector [c i,..., c iνd ] T, where g k d = [i,..., i νd ] T. Moreover, fd k is the function f d restricted to triangle k. Proof. Clearly s k (x) = f k (x) = showing the result for d =. By (.8) S k j, = R (k, j), j =,...,, and (.) follows for d =. For j =,..., Sj, k = e T k R R (:, j) = R (k, i)r (i, j) i= 3β 3 = R (k, i)r (i, j) = R (k, g k )R (g k, j). γ i G k But then (.) follows for d =. The edge property of Theorem. can now be proved. Suppose f d = n d j= c js j,d is a spline of degree d on P S and x = ( t)p + tp. The corresponding barycentric coordinates are β = t, β = t, and β 3 =. Let {B j, } 3 j= be the consecutive univariate linear B-splines on the knot vector [,,,, ], normalized to form a partition of unity on [, ]. If t < / then x, and (.) takes the form f (x) = R (x) [ c c 4 c 7 ] T = B, (t)c + B, (t)c 4, R (x) = [ t t ], while if t, then x 3 and f 3 (x) = R 3 (x) [ c c 4 c 8 ] T = B, (t)c +B 3, (t)c 4, R 3 (x) = [ t t ].

14 4 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD For d = and x let {B j, } 4 j= be the consecutive univariate quadratic B- splines on the knot vector [,,,,,, ], normalized to form a partition of unity on [, ]. Then t t f (x) = R (x) t t [ ] T c c c 3 c 4 c c t 3t t = ( t) c + t( 3t)c + t c 4 = B, (t)c + B, (t)c + B 3, (t)c 4, while for x 3, t t f 3 (x) = R 3 (x) t t [ ] T c c 3 c 4 c 5 c 6 c 7 3 ( t) t t = ( t) c + ( t)(3t )c 4 + ( t) c 5 = B, (t)c + B 3, (t)c 4 + B 4, (t)c 5. This proves the edge property. 3. A Marsden-like Identity In the univariate B-spline case, the Marsden identity is usually written, (z x) d = j B j,d (x) j+d k=j+ (z t k ), where B j,d is a B-spline of degree d with knots t j, t j+,..., t j+d+, and {B j,d } j is normalized to be a partition of unity. Dividing both sides by z d and setting y := z we obtain a form more amenable to multivariate generalization (3.) ( xy) d = j B j,d (x)ψ j,d (y), ψ j,d (y) = j+d k=j+ ( t k y). The ψ j,d are polynomials of degree d called dual polynomials. Consider now the S-spline basis. We define dual polynomials ψ j,d of degree d for j =,..., n d. For d =, ψ j, = and for d >, (3.) ψ j,d (y) := where (3.3) d ( p T j,d,ry), d =,, y R, r= p j,, := p j, j =,...,, [p,,,..., p,,] := [p, p, p 4, p 4, p, p, p 5, p 5, p 3, p 3, p 6, p 6 ], [p,,,..., p,,] := [p, p 4, p, p, p, p 5, p, p 3, p 3, p 6, p, p ]. are called dual points. There are no dual points for d =. Theorem 3. (Marsden-like Identity). For d =,, n d (3.4) ( x T y) d = S j,d (x)ψ j,d (y) = s d (x) T ψ d (y), x, y R, where j= s d (x) T := [S,d (x),..., S nd,d(x)], ψ d (y) := [ψ,d (y),..., ψ nd,d(y)] T.

15 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT 5 Also, there are polynomial versions, one for each subtriangle k, k =,...,. If g k d := [i,..., i νd ] T is given as in Definition.5, then (3.5) ( x T y) d = ν d j= S k i j,d(x)ψ ij,d(y) = s k d(x) T ψ k d(y), k =,...,, x, y R. Here S k j,d is the polynomial representing S j,d on the subtriangle k and (3.6) s k d(x) T := [S k i,d(x),..., S k i νd,d(x)], ψ k d(y) := [ψ i,d(y),..., ψ iνd,d(y)] T. Note the similarity between equation (3.) and equation (3.4). In the linear case the term t j+ y is replaced by p T j y, and in the quadratic case, we move around the boundary of inserting p for the S j, with trapezoidal support. In contrast to the univariate case, the dual points are not always knots of the S-splines. In particular, p is not a knot of any quadratic S-spline. The Marsden-like identity can be used to give explicit representations for monomials. Lemma 3.. For x = (x, x ), (3.7) where (3.8) n d = S j,d (x), d =,,, j= n d x = m j,d S j,d (x), d =,, j= [ ] x x x x x x = H(ψ j, )S j, (x), j= m j, := p j,, = p j, j =,,...,, m j, := (p j,, + p j,,)/, j =,,...,, and p j,,r are given by (3.3). Moreover, (3.9) H(ψ j, ) := Proof. We set y = in equation (3.4) to get, [ D e (ψ j, ) D e D e (ψ j, ) D e D e (ψ j, ) D e (ψ j, ) n d = s d (x) T ψ d () = s d (x) T e = S j,d (x), which proves Property 3 in Theorem., the partition of unity normalization. To show the expression for x in (3.7), we differentiate equation (3.4) with respect to y, and set y = so, dx T = s d (x) T y ψ d (), where y ψ d (y) is the n d matrix whose rth column is ψ d (y)/ y r, r =,. Differentiating and setting y = lead to, x T = s (x) T [p,,,..., p,,] T, j= x T = s (x) T [p,, + p,,,..., p,, + p,,] T, ].

16 6 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Figure 7. Quadratic domain mesh M ( ) and the corresponding control mesh for some surface f. from whence the identity for x follows. Applying second order partial derivatives with respect to y on both sides of equation (3.4) and setting y =, we obtain the last equation. Definition 3.3. The points m j,d given by (3.8) for j =,..., n d are called domain points. The domain mesh M d ( ) is defined by vertices and edges. Two domain points m j, = (p j,, + p j,, )/ and m k, = (p k,, + p k,, )/ define an edge if {p j,,, p j,, } {p k,,, p k,, }. Alternatively, (3.) M d ( ) := n d i= ch({m j,d : j H i d}), where Hd i is given in Definition.5 and ch is the boundary of the convex hull of {m j,d : j Hd i }. A spline f d := n d j= c js j,d has control points (3.) γ j,d := (m j,d, c j ), j =,..., n d, where the m j,d are the domain points given by (3.8). The control mesh Γ d (, c) is defined by lifting the domain mesh by a piecewise bilinear mapping A : M d ( ) Γ d (, c) defined so that A(m j,d ) = γ j,d for all j. The edges of M ( ) are simply the mesh lines in P S as shown in Figure. The domain mesh and the control mesh for d = are shown in Figure 7. The domain mesh defined in Definition 3.3 is also related to the pyramidal Algorithm 4. (d =, r = ) as shown in Figure 8. It is unusual in that it is a hybrid mesh comprised of both triangular and quadrilateral faces. In the univariate B-spline case, the numbers used to represent x in terms of B- splines are known as nodes, knot averages or Greville points. For P S they could analogously be called dual point averages or Greville points. Following the terminology used in [3], we shall call them domain points.

17 ( R (x)ψ (y) ) 7 = β 3l 6 + 4β 3 l 7 3γ l SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT Proof of the Marsden-like Identity. The same matrices R d introduced in subsection. for recurrence relations for S-splines also appear in recurrence relations for the dual polynomials. Theorem 3.4. For x, y R and d =,, (3.) (3.3) R d (x)ψ d (y) = ( x T y)ψ d (y), R k d(x)ψ k d(y) = ( x T y)ψ k d (y), k =,...,, where R and R are given by (.5) and (.6), ψ, ψ, ψ by (3.), R k and R k in Definition.7, and ψ k d = ψ d (g k d ). Moreover, gk d is given in Definition.5. Proof. Let l j = l j (y) = p T j y for y R. Fix x, y R and let β be the vector of barycentric coordinates of x with respect to. Then (3.4) x T y = (β + β + β 3 ) (β p + β p + β 3 p 3 ) T y = β l + β l + β 3 l 3. Consider first d =. Because of the symmetry of the equations for the boundary triangles and analogous similarity for the interior triangles, we consider only a single outer triangle corresponding to ψ, and a single interior triangle 7 corresponding to ψ 7,. Combining (.5),(.), (3.4) and (3.) we obtain, ( R (x)ψ (y) ) = γ l + β 3 l 6 + 4β l 7 and = (β )l + (β 3 β )(l + l 3 ) + β (l + l + l 3 ) = β l + β l + β 3 l 3 = ( x T y)ψ,, = (β 3 β )(l + l 3 ) + (β β 3 )(l + l + l 3 ) (β )(l + l + l 3 ) = β l + β l + β 3 l 3 = ( x T y)ψ 7,. For d = we consider rows, 4, 7,. Again, from (.6), (.), (3.4), and (3.), row becomes ( R (x)ψ (y) ) = γ ψ, + β ψ, + β 3 ψ, = γ l + β l l 4 + β 3 l 6 l = ( (β )l + β (l + l ) + β 3 (l 3 + l ) ) l = ( β l + β l + β 3 l 3 ) l = ( x T y)ψ,, and, row 4 becomes ( R (x)ψ (y) ) 4 = β,3ψ, + 3β 3 ψ 3, + β,3 ψ 4, = β,3 l l 4 + 3β 3 l 4 l + β,3 l 4 l = ( (β β 3 )l + β 3 (l + l + l 3 ) + (β β 3 )l ) ) l 4 = (β l + β l + β 3 l 3 )l 4 = ( x T y)ψ 4,.

18 8 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Inserting l 7 = 4 (l + l + l 3 ) for row 7, we get ( R (x)ψ (y) ) 7 = ( β 3 ψ, + 3β ψ 3, + 3β 3 ψ, + β ψ, ) / = ( β 3 l l 4 + 3β l 4 l + 3β 3 l 6 l + β l 6 l ) / = ( β (l 4 + l 6 )l + β ( (l + l + l 3 )l 4 l l 6 ) + β 3 ( (l + l + l 3 )l 6 l l 4 )) / = ( β l + β l + β 3 l 3 ) l7 = ( x T y)ψ 7, and for row, we produce, ( R (x)ψ (y) ) = γ 3ψ 3, γ ψ 7, γ ψ, = γ 3 l 4 l γ l 5 l γ l 6 l = ( ( β 3 )(l + l ) + ( β )(l + l 3 ) + ( β )(l + l 3 ) ) l / = ( β l + β l + β 3 l 3 ) l = ( x T y)ψ,. Next consider (3.3). Since R (k, j) = for j / G k, it easily follows that R k (x)ψ k (y) = e T k R (x)ψ (y) = e T k ( x T y)ψ (y) = ( x T y)ψ k (y). Finally, consider d =. Let G k = {i, i, i 3 }. Since R (i r, j) = for j / H ir Gk for r =,, 3, e T i e T i R k (x)ψ k (y) = e T i R (x)ψ (y) = e T e T i ( x T y)ψ (y) = ( x T y)ψ k (y). i 3 e T i 3 Invoking the recurrence relations for the dual polynomials we arrive at Theorem 3.5. For d =,, x j R, j =,..., d and y R, (3.5) ( x T y) ( x T d y)ψ (y) = R (x ) R d (x d )ψ d (y). Moreover, (3.6) ( x T y) ( x T d y) = R k (x ) R k d(x d )ψ k d(y), k =,...,, where R k d is given in Definition.7 and ψ k d(y) by (3.6). Proof. We simply apply (3.) and (3.3) repeatedly. Proof of Theorem 3.. We use (3.5) and (3.6) with x = = x d = x. If x, then x k for some k. Using the kth component of (3.5) we find, using Corollary.4, that d ( x T y) d = e T k R i (x)ψ d (y) = s d (x) T ψ d (y). i= This proves (3.4), and then, by (3.6) and (.), ( x T y) d = d R k i (x)ψ k d(y) = s k d(x) T ψ k d(y). i=

19 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT Linear independence. In this section we show linear independence of both the S-splines and subsets of the dual polynomials. Recall in our notation that S k j,d is the polynomial that represents S j,d on k and that S k j,d = if j / Gk d. Theorem 3.6 (Local linear independence). Suppose d =,, and k. Then pairs of sets Sd k := {Sk j,d : j Gk d } and Ψk d := {ψ j,d : j Gd k } are both bases for Π d (R ). Proof. The fact that Sd k forms a basis for Π d(r ) follows directly from conjoining the two requisite conditions, namely, Lemma 3. affords the spanning property for the set Sd k, and their linear independence follows from the observation that the set Gd k contains exactly ν d elements. We now show that Ψ k d is also a basis. Inasmuch as Ψ k d has the same cardinality as Π d (R ), it remains only to show that Ψ k d spans Π d (R ). Setting x = in equation (3.5) yields = s k d ()T ψ k d(y), so the constant can be expressed as a linear combination of ψ s. For d >, differentiating (3.5) with respect to x and then setting x = to get dy = x s k d ()T ψ k d(y), establishes that y is in the span of the ψ s. Taking the Hessian with respect to x on either side of equation (3.5) and setting x =, we obtain the quadratic case. Theorem 3.7. For d =,,, the set of S-splines {S j,d } n d j= forms a basis for S d. Proof. Let n d = dim(s d ). By Theorem. the n d functions S j,d belong to S d. The result then follows as a consequence of local linear independence. Marsden s identity and linear independence also imply the following symmetry property. Lemma 3.8. For x, z R, (3.7) (3.8) R k (x)r k (z) = R k (z)r k (x), k =,...,, R (x)r (z) = R (z)r (x). Proof. For any x, y, z R and k, R k (x)r k (z)ψ k (y) = ( z T y)r k (x)ψ k (y) = ( z T y)( x T y)ψ k (y) = ( x T y)( z T y)ψ k (y) = ( x T y)r k (z)ψ k (y) = R k (z)r k (x)ψ k (y). Since ψ k is a basis for Π (R ), (3.7) follows. Furthermore, for x, z k, (3.8) follows from (3.7). Thus, (3.8) holds for all x, z R Differentiation. In this section we derive a formula for the directional derivative of the S-splines S j,d. Let D u := u = u x + u x with u = [u, u ] T be a directional derivative. The unique solution α := [α, α, α 3 ] T of (3.9) α + α + α 3 =, α p + α p + α 3 p 3 = u is called the directional coordinates of u. If u = q q, with q i R for i =, then α j := β,j β,j, j =,, 3, where β i := [β i,, β i,, β i,3 ] T is the vector of barycentric coordinates of q i, i =,. We have the following differentiation formula:

20 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Theorem 3.9. If u R has directional coordinates α, then for d =, (3.) (3.) D u s T d = ds T d U d,u, D u s k T d = ds k T d U k d,u, k =,...,, where with α i,j := α i α j, (3.) α α 3, 4α α α,3 4α 3 α α,3 4α 3 α α 3, 4α α 3 α, 4α U,u := α 3 α, 4α α 3, 4α,3 6α, α,3 4α, 6α α,3 4α, 6α α 3, 4α,3 6α α, 4α 3, 6α 3 α, 4α 3, 6α 3 (3.3) U,u = α α α 3 α α α 3 α α 3 α α,3 3α 3 α,3 α, 3α α 3, α 3, 3α α, α,3 3α 3α 3 α, 3α α,3 α, 3α 3 3α α 3, α 3, 3α α 3 α α and (3.4) U k,u = U,u (k, g k ), U k,u = U,u (g k, g k )., For a spline f d := s T d c = n d j= c js j,d n d (3.5) D u f d = ds T d c [] = d c [] j S j,d, where the c [] j are given in Table. Moreover, for k =,..., (3.6) D u c j Sj,d k = d c [] j Sk j,d, d =,, j= j G k d j G k d Proof. Let k and x, y R. Recall from (3.3) that R k d(x)ψ k d(y) = ( x T y)ψ k d (y), x = [x x ] T, y = [y y ] T R.

21 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT Table. The coefficients c [] j in (3.5). Here α i,j := α i α j, c i,j := c i c j. j c [] j, d = c[] j, d = α 3, c 6, + 4α c 7, α c, + α 3 c, 4α 3 c 7, + α,3 c 4, α c 4,5 + α 3 c 6,5 3 α,3 c 4, + 4α 3 c 8, α c 8,9 + α c,9 4 4α c 8, + α 3, c 5, α,3 c,3 + α,3 c 4,3 5 α, c 5,3 + 4α c 9,3 α, c 6,7 + α 3, c 8,7 6 4α c 9,3 + α, c 6,3 α 3, c, + α, c, 7 α 3, c 6, + 4α,3 c 7, /α,3 c,3 + α,3 c 3, + /α, c, 8 4α, c 7, + α,3 c 4, /α,3 c 4,3 + /α, c 6,7 + α 3, c 7,3 9 α,3 c 4, + 4α, c 8, /α 3, c 8,7 + /α 3, c, + α, c,7 4α,3 c 8, + α 3, c 5, /3α 3, c 7,3 + /3α,3 c 3, + /3α, c,7 α, c 5, + 4α 3, c 9, 4α 3, c 9, + α, c 6, Let r {, }. Differentiation gives (3.7) But, then x r R k d(x)ψ k d(y) = y r ψ k d (y). x r s k d(x) T ψ k d(y) (3.5) = x r ( x T y) d = dy r ( x T y) d (3.5) = dy r s k d (x) T ψ k d (y) (3.7) = ds k d (x) T x r R k d(x)ψ k d(y). Since the elements of ψ k d are linearly independent it follows that s k x d(x) T = ds k d (x) T R k r x d(x), r =, r and (3.) with U k d,u = D u R k d(x) follows by linearity of the directional derivative. Moreover, (3.), with U d,u = D u R d (x), follows immediately from (3.). Let x have barycentric coordinates β with respect to. For t R the barycentric coordinates of x + tu are β + tα, where the α is determined from (3.9). But then, U d,u (x) = D u R d (x) = d dt R d((β + tα )p + (β + tα )p + (β 3 + tα 3 )p 3 ) t= R d R d = α + α β β =: α T β R d. + α 3 R d β 3 R d has elements like γ j = β j and β i,j = β i β j. Therefore, γ j β i,j = δ k,j and = δ k,i δ k,j, β k β k and the formulas for U,u and U,u follow.

22 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD To prove (3.5) we note that (3.) implies that c [] = U d,u c. The entries in Table are then obtained by eliminating one α j using α + α + α 3 = and regrouping terms. For d = k = (3.6) takes the form D u (c S, + c 6 S 6, + c 7 S 7, ) = α 3, c 6, + 4α c 7, or in terms of control points D u (c S, + c 6 S 6, + c 7 S 7, ) = α 3, c 6, + 4α c 7,, where c j = (p j, c j ) and c i,j = c i c j. This involves two differences of control points pointing from c to c 6 and from c to c 7. Similarly, on an inner triangle, say k = 7, (3.6) will involve differences of control points pointing from c to control points c 6 and c 7 of 7. Only those two types of differences are involved in computing c []. For d =, to each component c [] i of c [] there is a corresponding triangular or quadrilateral region. The differences of control points used are along directed edges of that region. There is also a simple differentiation formula for the second derivative of a quadratic S-spline. Theorem 3.. For direction vectors u, v R and any x not on a knot line of, we have (3.8) (3.9) D v D u s k (x) T = U k,vu k,u = U k,uu k,v, D v D u s (x) T = s (x) T U,v U,u = s (x) T U,u U,v, where the U matrices are defined according to the conventions in Theorem 3.9. Proof. Applying D v to (3.) and since U k,u := D u R k (x) is independent of x, D v D u s k (x) T = D v (s k (x) T D u R k ) = D v R k D u R k = U k,vu k,u. The commutativity of differentiation follows from Lemma Evaluation Algorithms Consider computing an S-spline f d (x) = s d (x) T c. If x k, then by Corollary.8, d f d (x) = s k d(x)c = R k i (x)c, c := c(g k d), d =,. i= We also consider computing directional derivatives D u f d and D v D u f d of order r = and r =, respectively. For d = and x k we start with c = c(g k ). Then for r =, Duf (x) = f (x) = R k (x)c, while Duf (x) = U k,uc for r =. In the quadratic case, d =, and x k we start with c = c(g k ). Then for r =, c = R k (x)c and Duf (x) = f (x) = R k (x)c and for r =, Duf (x) = c = R k (x)c, where c = U k,uc. Finally for r =, D v D u f (x) = c = U k,vc, while c = U k,uc. We give two algorithms. Going from right to left, we obtain Duf r d (x), and from left to right, we can compute the S-spline elements Dus r k d (x) that can be nonzero at x. Algorithm 4.. Given x, r {,, }, u R if r >, and coefficients c. () Determine the barycentric coordinates of x with respect to and k such that x k using Algorithm.. () c = c(g k d ).

23 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT 3 Figure 8. Evaluating f (x) using Algorithm 4. for (a) x, and (b) x 5 (3) for i =,..., r c i+ = (d i)u k d i,uc i. (4) for i = r,..., d c i+ = R k d i(x)c i. (5) D r uf d (x) = c d. In Figure 8, the algorithm is illustrated for d =, r =, and values of k. Table gives the relevant values of g k and g k. For k = 5, the 6 coefficients with indices 6, 7, 8, 9,, are combined to give c j, j = 3, 5, 9, and these 3, in turn, give rise to f (x) = c 5. The c j s needed for x have indices 3, 6, 7, 8,, giving c j, j = 5, 9,, and these 3 give rise to f (x) = c. We see that c 9 requires 4 c j s, while the other two level one coefficients combine only 3 c j s. Consider next an algorithm for computing Dus r k d (x)t for some x. If x k, then s k d (x)t = d i= Rk i (x), and this leads to, Algorithm 4.. Given x, r {,, }, and u R if r >. () Determine the barycentric coordinates of x with respect to and k such that x k using Algorithm.. () s k =. (3) for i =,..., d r s k i = sk i Rk i (x). (4) for i = d r +,..., d s k i = isk i U k i,u. (5) D r us k d (x)t = s k d. Since the elements in R k (x) and R k (x) are nonnegative for x k the algorithms for r = are quite stable.

24 4 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Figure 9. A triangle divided into 48 triangles 5. Subdivision Suppose we divide the triangle in Figure uniformly into 4 triangles δ, δ, δ3, δ4, and on each of these triangles we use the PS-split. Thus, we develop a () triangulation P S of comprised of 48 triangles (Figure 9). On δi we have nd linearly independent S-splines Wnd (i )+j,d, j =,..., nd for i =,..., 4, for a total of, Id := 4nd. () () Since Sd ( P S ) Sd ( P S ) and W d spans Sd ( P S ), the S-spline Sj,d is a linear combination of the Wj,d, (5.) Sj,d = Id X i= αj,d (i)wi,d, j =,..., nd.

25 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT 5 In analogy to B-splines, the numbers α j,d (i) are discrete S-splines of degree d relating P S to () P S, and the matrix, (5.) A d = [α j,d (i)] I d,n d i=,j= = E,d E,d E 3,d E 4,d RI d,n d, where E j,d R n d,n d, is called the knot insertion matrix of degree d taking P S to () P S. In succinct vector form (5.) is expressed by, (5.3) s T d = w T d A d, where If then s T d := [S,d,..., S nd,d], n d f = c j S j,d = s T d c = w T d ω = j= (5.4) ω = A d c. w T d := [W,d,..., W Id,d]. I d i= ω i W i,d, To compute the coefficients of the subdivided surface we can either: i) compute the knot insertion matrix and do a matrix vector multiplication or use a modified approach that directly averages the original coefficients. We will present both and start by deriving the knot insertion matrix. 5.. Computing the knot insertion matrix. To determine the matrix A d we start with a lemma involving the dual polynomials ψ j,d of S j,d and φ i,d of W i,d. For fixed i with i I d we let x,..., x d be the dual points of W i,d, so that, (5.5) φ i,d (y) = d ( x T r y). r= Recall that the triangles of the PS-split of are denoted by k, for k =,...,. It is easy to see that the d dual points of W i,d are located in (at least) one closed triangle ki, the closure of ki. Lemma 5.. For fixed i, i I d and d =,, we get n d (5.6) φ i,d = α j,d (i)ψ j,d = α j,d (i)ψ j,d j= j L i,d and (5.7) {j : α j,d (i) } L i,d, where (5.8) L i,d := {j : supp(w i,d ) supp(s j,d )}. Moreover, L i,d Gd k for all k such that k supp(w i,d ). In particular, the number of elements in L i,d is at most ν d.

26 6 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Proof. Fix i. By linear independence of the W i,d it follows that α j,d (i) only if the support of W i,d is contained in the support of S j,d. Therefore, if α j,d (i), then j L i,d. Using (3.4) and (5.), for x and any y R, ( x T y) d (3.4) = (5.) = = n d j= n d j= I d ( n d i= S j,d (x)ψ j,d (y) ( I d α j,d (i)w i,d (x) ) ψ j,d (y) i= α j,d (i)ψ j,d (y) ) W i,d (x). j= On the other hand, by adding together the Marsden like identities on δ r for r =,..., 4, we also have, ( x T y) d = I d i= φ i,d (y)w i,d (x). Equation (5.6) follows from the linear independence of the I d S-splines W i,d on and (5.7). Suppose k supp(w i,d ). Then, for all j L i,d, k supp(s j,d ), and simplex spline properties assure k supp(s j,d ). Therefore, j Gd k, so L i,d Gd k. Noting Gd k has exactly ν d elements independent of k, the result follows. A recurrence relation for the α j,d s in the ith row of A d can be determined from the following theorem. Theorem 5.. Suppose, for fixed d {, } and i I d, that x,..., x d are the dual points of W i,d. Also, let k be such that x,..., x d k. Then d (5.9) α k d(i) T = A d (i, g k d) = R k r(x r ) and r= (5.) α d (i) T = [α,d (i),..., α nd,d(i)] = e T k d R r (x r ). Moreover, α j,d (i) = for j / G k d. Here R (x) is given by (.5); R (x), by (.6); and R k d, in Definition.7. Proof. By Lemma 5., By (3.6) r= φ i,d (y) = α j,d (i)ψ j,d (y) = α k d(i) T ψ k d(y). φ i,d (y) = j G k d d ( x T r y) = r= d R k r(x r )ψ k d(y). Theorem 3.6 shows that the set given by the vector ψ k d is linearly independent, so (5.9) follows. From (5.7) the remaining α j,d s in row i of A d are. Thus, (5.) follows (cf. Lemma 5.). r=

27 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT 7 The blocks E r, of the quadratic subdivision matrix (5.) are as follows: E, = , and E, = Q E, Q T, Q = [e 5, e 4, e 3, e, e, e, e, e, e 9, e 8, e 7, e 6 ], E 3, = Q 3 E, Q T 3, Q 3 = [e 9, e 8, e 7, e 6, e 5, e 4, e 3, e, e, e, e, e ], E 4, = Coefficient averaging algorithm. For any n N and real numbers r,..., r n we define, n (5.) µ(r,..., r n ) := r j /n. As shorthand we write for some [c,..., c ] T and i < < i m, j= (5.) µ i,i,...,i m = µ(c i, c i,..., c im ). Continuing as defined above, we state the following: Theorem 5.3. If then 48 f = c j S j, = ω i W i,, j= (5.3) ω (r )+j = ξ j,r, r =,, 3, 4, j =,...,, i=

28 8 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD Figure. Subdivided domain mesh and surfaces. Left: one level of subdivision; center: two levels of subdivision; right: three levels of subdivision. where Ξ = [ξ j,r ] R,4 is the following, c µ,4 µ, µ 6,8 µ, µ(µ,4, c 4 ) µ(µ,, c ) µ(µ 7,8, c 7 ) µ(µ,,, µ,3 ) µ(µ 4,5,6, µ 3,4 ) µ(µ 7,8, µ, ) µ 7, µ(c, µ,4 ) µ 4,5 µ(µ 7,8, c 7 ) µ(µ,, c ) µ,4 c 5 µ 6,8 µ, (5.4) Ξ = µ(µ,3, c 3 ) µ 5,6 µ(µ 6,8, c 8 ) µ(µ,, c ) µ(µ,3, µ, ) µ(µ 4,5,6, µ 6,7 ) µ(µ 8,9,, µ 7,8 ) µ 3,. µ(c, µ, ) µ(c 6, µ 6,8 ) µ 8,9 µ(c 3, µ,3 ) µ, µ 6,8 c 9 µ,4 µ(µ,, c ) µ(µ 6,7, c 7 ) µ 9, µ(µ 3,4, c 3 ) µ(µ,,, µ, ) µ(µ 3,4, µ 6,7 ) µ(µ 8,9,, µ, ) µ 3,7 µ, µ(µ 3,4, c 3 ) µ(µ,, c ) µ(µ 6,7, c 7 ) Proof. This is straightforward using the explicit form of the matrices E,,..., E 4,. Figure illustrates a geometric interpretation of the subdivision algorithm. 6. Stability of the S-Basis and the Distance to the Control Points 6.. Linear and quadratic quasi-interpolant. We recall from (3.7) that the domain points m j,d are specified as n d x = m j,d S j,d (x), for d =, and x. j=

29 SIMPLEX SPLINES FOR THE POWELL-SABIN -SPLIT 9 Of the many possible quasi-interpolants to consider, we examine only Q d : C( ) S d given by (6.) Q f := (λ j, f)s j,, where λ j, f := f(m j, ), j= Q f := (λ j, f)s j,, where λ j, f := f(m j, ) f(p j,,) f(p j,,), j= where m j,d, p j,, and p j,, are given in (3.8) and (3.3). We next show that the λ j,d s are dual functionals with respect to the S-spline basis. Lemma 6.. For d =,, i =,...,, and j =,..., n d, λ i,d S j,d = δ ij. Proof. The case d = follows trivially because the corresponding basis functions are hat functions. Now, consider the case d =, for which the value of the jth quadratic S-spline at the ith linear domain point is Υ (i, j) in the matrix below: Υ := [ S j, (m i, ) ] = Analogously, Υ (i, j) below describes the value of the jth quadratic S-spline at the ith quadratic domain point: (6.) Υ := [ S j, (m i, ) ] 5 = The value of [λ i, S,,..., λ i, S, ] is found by combining row i of Υ with suitable rows of the matrices Υ and Υ. For example, Υ (, :) (Υ (, :) + Υ (4, :))/ = [,,,,,,,,,,, ], Υ (3, :) (Υ (4, :) + Υ (, :))/ = [,,,,,,,,,,, ], showing that λ, S j, = δ,j, and λ 3, S j, = δ 3,j.

30 3 ELAINE COHEN, TOM LYCHE, AND RICHARD F. RIESENFELD 6.. Stability of the quadratic S-spline basis. For each j we let Ω j,d := supp(s j,d ) be the support of S j,d. The following theorem shows the S-spline basis achieves stability in the L norm. Theorem 6.. For any f d = s T d c S d, (6.3) K d c f d L ( ) c, d =,,, where K = K = and K = 3. Furthermore, the constants K i are the best possible. Proof. Since the S-splines form a nonnegative partition of unity the upper bound is elementary. Also the values for K and K follow easily. Using Lemma 6. we see that c j = f(m j, ) f(p j,, ) f(p j,, ) for all j. Since p j,,, p j,, and m j, all belong to Ω j,d we obtain for each j, (6.4) c j 3 f d L (Ω j,d ) 3 f d L ( ), thus establishing the lower bound. Equality follows by choosing f so that it reduces to the quadratic Chebyshev polynomial, on say, the edge [p, p 4 ]. Consider next the L q norm. The next theorem and corollary show that a scaled S-basis is stable in the L q norm. Theorem 6.3. For any f d = s T d c S d and q there is a constant C d depending only on d such that, (6.5) C d c q,σ f d Lq( ) c q,σ, d =,,, where (6.6) c q,σ := ( j c j q σ j,d ) /q, σ j,d := S j,d = v(ω j,d) ν d. Proof. We first show that there is a constant κ depending only on d such that for any g S d, (6.7) g(x) κ g(z) dz, x Ωj,d, j =,..., n d. v(ω j,d ) Ω j,d To show this, observe that for all k, j, v(ω j,d ) 3 4 v( ) 3 4 4v( k) = 8v( k ). If x Ω j,d, then x k for some k. Now g is a polynomial on k and by equivalence of norms there is a constant C depending only on d such that g(x) C g(z) dz 8C g(z) dz, v( k ) k v(ω j,d ) Ω j,d which proves (6.7). The rest of the proof is based on standard L q gymnastics. For the upper inequality, using the relation /q + /q =, and Hölder s inequality for sums f q L = ( q( ) c j S /q ) j S /q q j c j q S j,d = c q q,σ. j j