On the convexity of C 1 surfaces associated with some quadrilateral finite elements

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1 Advances in Computational Mathematics 13 (2000) On the convexity of C 1 surfaces associated with some quadrilateral finite elements J. Lorente-Pardo a, P. Sablonnière b and M.C. Serrano-Pérez a a Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain b Laboratoire L.A.N.S., I.N.S.A. de Rennes, France Received 15 April 1999; revised 29 February 2000 Communicated by Y. Xu We analyse the convexity property of two classical finite elements and of the associated piecewise polynomial C 1 surfaces. The first one is the piecewise cubic quadrilateral of Fraeijs de Veubeke. The second one is a piecewise quadratic rectangle introduced by Sibson and Thomson and generalized by Sablonnière and Zedek. In both cases, we first study the local problem and then we extend our results to the associated C 1 surfaces. Keywords: Bernstein polynomials, Bézier nets, convexity, triangular, quadrilateral and rectangular finite element AMS subject classification: 65 D 05, 65 D Introduction The use of triangular and quadrilateral finite elements is very convenient in the construction of C 1 surfaces. Moreover, building surfaces that are convexity preserving is often needed in Computer Aided Geometric Design. Recently, some authors (e.g., Li [8], Lorente-Pardo et al. [9,11], Willemans and Dierckx [15]) have studied the convexity property of various triangular finite elements. In this paper, we study the convexity of C 1 surfaces obtained by putting together either piecewise cubic quadrilateral or piecewise quadratic rectangular finite elements. The first one is called FVS (after Fraeijs de Veubeke and Sander, see, e.g., [3,4, 6]): each convex quadrilateral is decomposed into four triangles by its diagonals. The second one is called ST (after Sibson and Thomson [14], see also [1]), and has been generalized by Sablonnière and Zedek [12]. Each rectangular patch is first subdivided into four rectangles by arbitrary parallels to its vertical and horizontal edges. Then, each subrectangle is subdivided into four triangles by its diagonals: thus, the original patch is equipped with a non-uniform criss cross triangulation consisting of 16 triangles. We use a well-known and simple property of the Bézier net of a polynomial surface on a triangle: the convexity of the former implies the convexity of the latter (see, J.C. Baltzer AG, Science Publishers

2 272 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements e.g., [2,5]). Therefore, we essentially prove theorems giving necessary and sufficient conditions for the convexity of Bézier nets of finite elements and associated surfaces. Our conditions are also sufficient for the convexity of underlying functions. But, we are aware of the fact that further research has still to be done in order to connect these conditions with Hermite data defining both types of interpolants. Notations, basic results and auxiliary lemmas are introduced in section 2. In section 3, we recall the main properties of FVS finite elements and we give a geometrical characterization of their convex Bézier nets. We then extend our results to C 1 surfaces composed of finite elements of this type. A similar study is developed in section 4 for ST finite elements. Finally, in the appendix, for the sake of completeness we recall the computation of the B-coefficients of the two finite elements aforementioned as a function of Hermite data. 2. Notations and preliminary results Throughout this section, T and D denote, respectively, a nondegenerate triangle and a bounded convex polygonal domain in R Bernstein Bézier polynomials over triangles where For each n 1, the nth Bernstein polynomial of u : T R is defined by B n u(λ) = ( ) α u b n α n (λ) = c(α)b n α (λ), α =n α =n λ (λ 1, λ 2, λ 3 ) are the barycentric coordinates with respect to T ; α (α 1, α 2, α 3 ), α = α 1 + α 2 + α 3, α i N, i = 1, 2, 3; b n α (λ) = n! α 1!α 2!α 3! λα 1 1 λα 2 2 λα 3 3 (Bernstein basis of P n (T )). The elements of the set {( ) } α ϕ n = n, c(α) R 3 : α = n are called the control vertices of B n u. Let τ n (T ) be the triangulation of T whose edges are parallel to the sides of T and whose vertices are obtained by projecting ϕ n over the triangle T. The unique continuous function, piecewise affine with respect to τ n (T ) and interpolating ϕ n is called the nth Bézier net or B-net of u, and is denoted by L n u. The triangulation τ n (T ) is called the triangulation on T induced by ϕ n or by L n u. The elements of the set {c(α): α = n} aretheb-coefficients of B n u and their representation at the vertices of τ n (T ) is called the planar representation of the B-net

3 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 273 Figure 1. Planar representation of L 2u. Figure 2. A bitriangle. Figure 3. Condition (C). L n u (see figure 1 for n = 2, where c(2,0,0) = 2, c(0,2,0) = 3, c(0,0,2) = 4and c(i, j, k) = 1otherwise). Each polynomial p P n (T ), p(λ) = c(α)b n α(λ), α =n can be expressed in the form p = B n u,whereu is any function satisfying u(α/n) = c(α) for α = n. In particular, we can write p = B n v,wherev = L n u Convexity of continuous piecewise affine functions Let τ be a triangulation of D R 2 and let u C(D) be a function that is affine on each T i τ. A bitriangle of τ is any set of two adjacent triangles T 1 and T 2 of τ. Its common edge T 1 T 2 is called the diagonal of the bitriangle (see figure 2). Definition 1. We say that u satisfies condition (C) in the bitriangle T 1 T 2 if and only if s(m) r(m), where M 1, M 2, M 3, M 4 areasinfigure2,m = M 1 M 2 M 3 M 4 and r, s are the straight lines plotted in figure 3. This condition is clearly equivalent to the convexity of u in the bitriangle. Moreover, we note that if T 1 T 2 is a parallelogram, then u satisfies condition (C) in the bitriangle T 1 T 2 if and only if u(m 3 ) + u(m 4 ) u(m 1 ) + u(m 2 ).

4 274 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements Let us recall two well-known results: Theorem 2. Let v C(D) be a piecewise affine continuous function on some triangulation τ of D. Then, v is convex if and only if v satisfies condition (C) in each bitriangle of τ (see [10,13]). In particular, this occurs when v = L n u and τ = τ n (T ), for u C(T )andn 2 (see [2,5]). Theorem 3. Let u C(T ) be a function and n 1. If L n u is convex, then B n u is also convex Auxiliary lemmas In this section, we prove some technical lemmas which are used in the proofs of results given in sections 3 and 4. Lemma 4. Let D 1 and D 2 be two adjacent bitriangles such that D = D 1 D 2 is a parallelogram (as in figure 4). Let u C(D) be a piecewise affine function on the associated triangulation of D. (i) Assume that u is affine over the two segments A 1 A 4 and A 2 A 3. Then, u satisfies condition (C) in D 1 and D 2 if and only if u is affine in D. (ii) Assume that u is affine in D 1 and D 2. Then, u is convex over the segment A 1 A 4 if and only if it is convex over the parallel segment A 2 A 3. Proof. Let u i = u(a i )anda i = (A i, u i ), for 1 i 6. (i) First, we note that the points A 1, A 2, A 3, A 4 are coplanar if and only if u 1 + u 3 = u 2 + u 4 : it is enough to keeping in mind that (A 1 + A 3 )/2 = (A 2 + A 4 )/2 holds in both situations. On the other hand, if u is affine over A 1 A 4 and over A 2 A 3 then the points A 1, A 5, A 4 and the points A 2, A 6, A 3 are, respectively, colinear. Moreover, in this situation, λ (0, 1) such that u 5 = (1 λ)u 1 + λu 4 and u 6 = (1 λ)u 2 + λu 3. In consequence, we immediately deduce that u satisfies condition (C) in D 1 (respectively D 2 ) if and only if u 2 + u 4 u 1 + u 3 (respectively, u 1 + u 3 u 2 + u 4 ). Finally, we conclude the result because u is a piecewise affine function on the associated triangulation of D. (ii) Since u is affine in D 1 and D 2, we have: u 1 + u 6 = u 2 + u 5 and u 3 + u 5 = u 4 + u 6. Figure 4. Parallelogram D in lemma 4.

5 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 275 Figure 5. Notations for lemma 5. Moreover, there exists some λ (0, 1) such that A 5 = (1 λ)a 1 + λa 4 and A 6 = (1 λ)a 2 + λa 3. Therefore, the convexity of u over the segment A 1 A 4 (respectively A 2 A 3 ) is equivalent to the inequality u 5 (1 λ)u 1 + λu 4 (respectively u 6 (1 λ)u 2 + λu 3 ). Assume that the first inequality is satisfied. Using the above equations, we get successively (1 λ)u 5 = (1 λ)(u 1 + u 6 u 2 ), λu 5 = λ(u 4 + u 6 u 3 ), which, by addition and substitution in the first inequality, gives immediately the second one. The converse is similar. Lemma 5. Let u C(T ) be a piecewise affine function on τ 2 (T ) (see figure 5). Denote respectively by D 1, D 2, D 3 the three bitriangles with diagonals B 2 B 3, B 3 B 1 and B 1 B 2. Assume that u is affine in D 1 :thenu is convex in D 2 (respectively D 3 ) if and only if it is convex on the segment A 1 A 2 (respectively A 1 A 3 ). Proof. Let a i = u(a i )andb i = u(b i ), for 1 i 3. We can assume without loss of generality that a 1 = b 1 = b 2 = b 3 = 0sinceu is affine in D 1. Then u is convex in D 2 if and only if a 2 0: this is also equivalent to the convexity of u over the segment A 1 A 2. The proof is similar in the other case. Lemma 6. Let T 1 = A 1 A 2 A 3 and T 2 = A 2 A 3 A 4 be two adjacent triangles with common edge A 2 A 3 (see figure 6). Assume that Q = T 1 T 2 is a convex quadrilateral equipped with the triangulation τ = τ 2 (T 1 ) τ 2 (T 2 ). Let u C(Q) be a piecewise affine function on τ. Assume further that u is affine in each of the four bitriangles of τ whose diagonals are parallel to A 2 A 3 and that u is convex over the two edges A 1 A 3 and A 3 A 4. (i) Let (λ 1, λ 2, λ 3 ) be the barycentric coordinates of A 4 with respect to T 1 and assume that λ 3 1. Then u is convex in the bitriangle of τ 2 (T 1 ) containing A 2. (ii) In the same way, let (µ 1, µ 2, µ 3 ) be the barycentric coordinates of A 1 with respect to T 2 and assume that µ 3 1. Then u is convex in the bitriangle of τ 2 (T 2 ) containing A 2. Proof. We only give the proof of (i), the proof of (ii) being quite similar.

6 276 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements Figure 6. Case (i) of lemma 6. Figure 7. Position of A 1 and A 4. Denote a i = u(a i )for1 i 4andb i = u(b i )for1 i 5. We can assume without loss of generality that u = 0 in the bitriangle A 1 B 2 B 1 B 3, i.e., a 1 = b 1 = b 2 = b 3 = 0. Therefore the hypotheses on u are respectively equivalent to b 5 = λ 2 a 2 (u affine in B 1 B 3 A 2 B 5 ), b 4 = λ 3 a 3 (u affine in B 1 B 2 A 3 B 4 ), a 4 = b 4 + b 5 (u affine in B 1 B 4 A 4 B 5 ), a 3 0 (u convex over A 1 A 3 ), a 3 + a 4 2b 4 (u convex over A 3 A 4 ). We have to prove that a 2 0(u convex in A 2 B 1 B 2 B 3 ). But, we immediately deduce from the preceding equations and inequalities: λ 2 a 2 (λ 3 1)a 3. The result follows from the fact that a 3 0, λ 3 1andλ 2 > 0. Remark 7. Let r i be the straight line joining A 3 and A j,andlets i be the parallel line to the edge A 2 A j containing A 3, with (i, j) = (1, 4). About the previous lemma we note that the position of vertices A 1 and A 4 in (i) and (ii) has the following geometric interpretation: for i = 1, 4, the vertex A i lies inside the shaded semicone in figure 7, which contains (respectively does not contain) the boundary s i (respectively r i ). On the other hand, the lemma is not true when λ 3 < 1 in (i): it is sufficient to consider the data of figure 8, where A 4 = ( 1 2, 3 4, 3 4 ) with respect to T 1.Whenµ 3 < 1 in (ii) we have an analogous situation. Finally, we recall a result given by Grandine in [7] which is useful for n = 3in section 2: Lemma 8. Let T 1, T 2 be two triangles with common edge A 2 A 3 such that D = T 1 T 2 is a non-convex quadrilateral. Let n 1andletu C(D) be a piecewise affine function on τ = τ n (T 1 ) τ n (T 2 ). We suppose that u is affine in each bitriangle of τ whose diagonal is included in A 2 A 3. Let H D be the union of the triangles of τ

7 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 277 Figure 8. Figure 9. Domain D and region H. that have no empty intersection with A 2 A 3 (bounded by thick lines in figure 9 for n = 3). Then the following statements are equivalent: (i) condition (C) holds for all bitriangles of τ inside H, (ii) the function u is affine on H. 3. Convexity of FVS finite elements and of the associated C 1 surfaces In this section, we first study our problem on a convex quadrilateral Q of R 2 with vertices A i,1 i 4(thelocal problem). Then, we extend the obtained results to C 1 surfaces defined on a bounded convex polygonal domain D R 2 subdivided into convex quadrilaterals (the global problem) The local problem We begin by introducing some definitions and properties of FVS finite elements. Definition 9. Let τ be the triangulation of Q obtained by drawing in its two diagonals. Then, (i) A FVS finite element is a function s C 1 (Q) whose restriction to each triangle T i τ is a cubic polynomial B 3 u i (1 i 4). (ii) The Bézier net (B-net) of s is the piecewise affine continuous function that coincides with the B-net L 3 u i on each triangle T i τ. Its planar representation is given in figure 10. It is well known (see, e.g., [4,6]) that a FVS finite element is uniquely determined by the function and first derivatives values at the vertices of Q, together with first derivatives at the midpoint of each side of Q in some direction not parallel to this side. Moreover, since the differentiability of this element guarantees that the B-net is planar on each shaded region in figure 11, then the set of B-coefficients given by

8 278 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements Figure 10. Planar representation of the B-net. Figure 11. Figure 12. {a i, b i, c i, g i, i = 1, 2, 3, 4} determines all the others. For the sake of completeness, we recall in the appendix the computation of the B-coefficients in function of Hermite data. The following result characterizes the convex B-net of a FVS finite element: Theorem 10. The B-net v of a FVS finite element is convex if and only if the following statements are satisfied: (1) The control vertices of v located over each of the shaded regions in figure 12 are coplanar. (2) The restrictions of v to the diagonals of Q are convex. Proof. We denote by Ω and B i, C i, D i, E i, G i, i = 1, 2, 3, 4, the projections on Q of the control vertices corresponding, respectively, with the B-coefficients ω and b i, c i, d i, e i, g i, i = 1, 2, 3, 4, appearing in figure 10. If v is convex, then condition (2) is immediate. On the other hand, the differentiability of the FVS finite element guarantees that v is planar on each shaded region in figure 11. Then, we obtain the coplanar regions mentioned in (1) as follows. By virtue of theorem 2, v satisfies condition (C) in each bitriangle of the triangulation τ whose restriction to each T i τ is τ 3 (T i ). We consider the restriction of v to any two adjacent bitriangles of τ such that: (1) they are included in two different triangles of τ, and (2) their union is a parallelogram. Then, using (i) of lemma 4, we get the desired result.

9 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 279 (a) (b) Figure 13. Conversely, assume that conditions (1) and (2) are satisfied. From theorem 2, it is sufficient to prove that v satisfies condition (C) in each bitriangle of the triangulation on Q induced by v. Thus: (a) Of course, condition (1) implies that v satisfies condition (C) in each bitriangle situated inside each shaded region in figure 12. (b) For the remaining bitriangles, we distinguish between the bitriangles whose diagonal has a positive slope (see shaded bitriangles in figures 13(a) and (b) and those whose diagonal has a negative slope. We consider the first possibility, the proof for the second one is analogous. First, lemma 4(ii) implies that v is convex on the segments B 1 C 3 and C 1 B 3.Indeed, we consider the restriction of v to the parallelogram with vertices D 1, B 1, E 2, Ω. In virtue of condition (1), we can apply lemma 4(ii). Hence, v is convex on the segment D 1 Ω if and only if v is convex on the segment B 1 E 2. We proceed in a similar way with the restriction of v to the parallelograms D 1 C 1 E 4 Ω, D 3 B 3 E 4 Ω and D 3, C 3 E 2 Ω. Since v is convex on the segments A 1 A 3, B 1 C 3 and C 1 B 3, lemma 5 and theorem 2 imply that v satisfies condition (C) in the shaded bitriangles in figures 13(a) and (b). We only consider the bitriangle whose diagonal is B 1 D 1, the proof for the rest of bitriangles being analogous: as v is convex on the segment A 1 E 1, we can use lemma 5 for the restriction of v to the triangle T = A 1 E 1 C 2. Hence, v is convex on the bitriangle whose diagonal is B 1 D 1. Finally, we apply theorem 2 to get the result. Theorem 3 implies that conditions (1) and (2) of theorem 10 are sufficient conditions for the convexity of a FVS finite element The global problem Definition 11. Let π(d) be a partition of the domain D R 2 into convex quadrilaterals. We assume that for any pair (Q, Q ) π(d) π(d), such that Q Q, then Q Q is either a vertex or an edge of π(d). The intersections of diagonals of Q and Q are denoted respectively by Ω and Ω. (i) A function s C 1 (D), whose restriction to each Q π(d) is a FVS finite element, is called a function of FVS type on D. We denote by τ(d) the triangulation of D

10 280 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements Figure 14. Partitions π(d) and τ(d). Figure 15. Coplanar regions. obtained as the union of all triangulations τ (Q), for Q π(d) (see figure 14). Thus, s is a C 1 piecewise cubic function on τ(d). (ii) The Bézier net (B-net) of s is the continuous piecewise affine function whose restriction to each Q π(d) coincides with the B-net of the corresponding FVS finite element. It is known that a function of FVS type on D is uniquely determined by function values and gradients at the vertices of π(d), together with first derivatives at the midpoint of each edge of π(d) in some direction not parallel to this edge. In fact, the differentiability of s on the union of two adjacent quadrilaterals is equivalent to the coplanarity of the control vertices located over each of the shaded regions in figure 15. Keeping in mind theorems 2 and 10 and lemma 8, we can characterize the convexity of the Bézier net of a function of FVS type on D as follows: Theorem 12. The B-net of a function of FVS type on D is convex if and only if the following conditions hold: (i) For each Q π(d), the two conditions of theorem 10 are satisfied. (ii) For those pairs of adjacent quadrilaterals Q, Q π(d) for which the quadrilateral ΩA 3 Ω A 4 (bounded by thick lines in figure 16) is not convex, the control vertices located over the shaded region in figure 16 are coplanar. However, we note that there are some particular partitions π(d) for which condition (2) of theorem 10 can be replaced in theorem 12 by a weaker condition. This occurs, for example, in the particular case when D is a parallelogram of R 2 and π(d) is a uniform partition into convex quadrilaterals whose sides are parallels to the sides of D. Moreover, in this case, we can remove condition (ii) of theorem 12. We first define the concepts of segments of type 1 and of type 2 of the triangulation τ(d) associated with a partition π(d): Definition 13. Let D = P 1 P 2 P 3 P 4 be a parallelogram of R 2 whose diagonal with positive (respectively, negative) slope is P 1 P 3 (respectively, P 4 P 2 ). Let π(d) bea uniform partition of D into mn parallelograms Q i,j whose sides are parallel to the sides

11 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 281 Figure 16. (a) (b) Figure 17. (a) Segments of type 1 and of type 2. (b) Segments of type 1 and of type 2. of D. Assume that V i,j, V i+1,j, V i+1,j+1, V i,j+1 are the vertices of the parallelogram Q i,j,fori = 0,..., m 1, j = 0,..., n 1 (hence, P 1 = V 0,0, P 2 = V m,0, P 3 = V m,n, P 4 = V 0,n ). (1) When m = n: A segment of type 1 of τ(d) is any diagonal of D. There is no segment of type 2ofτ(D) in this case. (2) When m>n: (a) Assume that m<3n. We consider n P = Q i,j. i,j=1 Then we denote by: q 1 (respectively q 2 ), the intersection point of the line containing the side P 2 P 3 with the line that contains the point V n,n (respectively V n,0 )whichis parallel to the diagonal of P with negative (respectively positive) slope, d 1 (respectively d 2 ), the midpoint of q 1 and V n,n (respectively q 2 and V n,0 ).

12 282 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements A segment of type 1 of τ(d) is any diagonal of P. The segments of type 2 of τ(d) are the segments whose ends are the points d 1 and P 3 (respectively d 2 and P 2 ). For the sake of clarity, in figure 17(a) we show a uniform partition π(d) with m = 5, n = 2. Here, P is the parallelogram with vertices P 1, V 2,0, V 2,2, P 4 and, hence, the thick lines (respectively the dashed thick lines) are the segments of type 1 (respectively of type 2) of τ(d) associated with π(d). (b) Assume that m 3n, andletm n = 2nk + r (0 r<2n). A segment of type 1 of τ(d) is any diagonal of the parallelograms with vertices V 2in,0, V (2i+1)n,0, V (2i+1)n,n, V 2in,n,fori = 0,..., k. (b 1 ) In the case when r = 0, there is no segment of type 2 of τ(d). (b 2 ) In the case when r 0, a segment of type 2 of τ(d) is any segment of type 2 of the restriction of τ(d) to the parallelogram with vertices V 2kn,0, V m,0, V m,n, V 2kn,n. In figure 17(b) we show, as before, the segments of type 1 and the segments of type 2 of τ(d) associated with a uniform partition π(d) such that m = 8andn = 2. Here, r = 2 0. (3) When m<n, we consider the rotation α of angle π/2. We call segments of type 1 (respectively of type 2) of τ(d), the segments obtained by applying the rotation α 1 over the segments of type 1 (respectively of type 2) of the partition α(τ(d)). Theorem 14 (Uniform partitions). We consider D and π(d) as in the previous definition and let s be a function of FVS type on D. The B-net of s is convex if and only if the following conditions hold: (1) For each Q π(d), condition (1) of theorem 10 is satisfied. (2) The restrictions of the B-net to the segments of type 1 and of type 2 of τ(d) are convex. Proof. that: It is an immediate consequence of theorems 10 and 12, taking into account (a) There is no pair of adjacent parallelograms in the situation of condition (ii) of theorem 12. (b) Condition (2) is equivalent to condition (2) of theorem 10: it is sufficient to use lemma 4(ii) as in part b) of the proof of theorem 10. Finally, from theorem 3, we deduce that if the B-net of a function of FVS type satisfies conditions of theorem 12 (or of theorem 14 for uniform partitions), then this function is convex.

13 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 283 Figure 18. Triangulation of ST-type. Figure 19. Planar representation of the B-net. 4. Convexity of ST finite elements and of the associated C 1 surfaces In this section we analyse the convexity of the C 1 surfaces studied by Sablonnière and Zedek in [8] and defined on nonuniform criss cross triangulations of a rectangular domain D R 2. The monotonicity of these surfaces has been studied by Beatson and Ziegler in [1] for uniform criss cross triangulations The local problem Definition 15. A rectangle R being subdivided into four subrectangles R i,1 i 4, we denote by τ the triangulation of ST-type of R obtained by drawing the diagonals in each subrectangle (see figure 18). (i) A ST finite element is a function s C 1 (R) such that s i = s Ti is a quadratic polynomial B 2 u i in each T i τ, and whose gradient is linear on the sides of R. (ii) The Bézier net (B-net) of a ST finite element is the continuous function on R that coincides on each T i τ with the B-net L 2 u i. Its planar representation is given in figure 19. It has been proved in [12] that a ST finite element is uniquely determined by the function values and the gradients at the four vertices of R. Equivalently, it is defined by

14 284 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements Figure 20. Coplanar regions of the B-net. Figure 21. Notations. the twelve B-coefficients {a i, b i, c i, i = 1, 2, 3, 4}. Indeed, the differentiability of the element guarantees that the B-net is planar on each shaded region in figure 20: hence, these twelve B-coefficients determine all the others. For the sake of completeness, the computation of the B-coefficients of this finite element is recalled in the appendix. Let r i be the straight line passing through Ω and parallel to the diagonal M i M i 1, for 1 i 4 (see figure 21). Here and in the following, we use the convention of indices M 0 = M 4, Ω 0 = Ω 4, Ω 5 = Ω 1, r 0 = r 4, r 5 = r 1. Theorem 16. The B-net v of a ST finite element is convex if and only if the following statements are satisfied: (1) The control vertices of v located over each of the shaded regions in figure 22 are coplanar. (2) The restrictions of v to the segments ΩA i,1 i 4, are convex. (3) If there is an i {1,2,3,4} such that Ω i lies in the interior of the semicone bounded by r i 1 and r i containing M i 1, then the restriction of v to the segment Ω i 1 M i 1 is convex. (4) If there is an i {1,2,3,4} such that Ω i lies in the interior of the semicone bounded by r i and r i+1 containing M i, then the restriction of v to the segment M i Ω i+1 is convex.

15 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 285 Figure 22. Coplanar regions of the convex B-net. Figure 23. Segments of convexity for conditions (3) and (4). Before proving this theorem and for the sake of clearness, we first describe conditions (3) and (4) in each of the nine situations that can occur and that are shown in figure 23. More specifically: Case (a). The conditions of statement (3) (respectively (4)) are only satisfied for i = 1, 2 (respectively i = 2, 3) and then, the convexity of v on Ω 4 M 4 and Ω 1 M 1 (respectively M 2 Ω 3 and M 3 Ω 4 ) is needed. Case (b). The conditions of statement (3) (respectively (4)) are only satisfied for i = 2 (respectively i = 3) and then, the convexity of v on Ω 1 M 1 (respectively M 3 Ω 4 ) is needed. Case (c). The conditions of statement (3) (respectively (4)) are only satisfied for i = 2, 3 (respectively i = 3, 4) and then, the convexity of v on Ω 1 M 1 and Ω 2 M 2 (respectively M 3 Ω 4 and M 4 Ω 1 ) is needed.

16 286 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements (a) (b) Figure 24. Case (d). The conditions of statement (3) (respectively (4)) are only satisfied for i = 1 (respectively i = 2) and then, the convexity of v on Ω 4 M 4 (respectively M 2 Ω 3 ) is needed. Case (e). There is no i {1, 2, 3, 4} satisfying the conditions of statements (3) or (4). Case (f). The conditions of statement (3) (respectively (4)) are only satisfied for i = 3 (respectively i = 4) and then, the convexity of v on Ω 2 M 2 (respectively M 4 Ω 1 ) is needed. Case (g). The conditions of statement (3) (respectively (4)) are only satisfied for i = 1, 4 (respectively i = 1, 2) and then, the convexity of v on Ω 4 M 4 and Ω 3 M 3 (respectively M 1 Ω 2 and M 2 Ω 3 ) is needed. Case (h). The conditions of statement (3) (respectively (4)) are only satisfied for i = 4 (respectively i = 1) and then, the convexity of v on Ω 3 M 3 (respectively M 1 Ω 2 ) is needed. Case (i). The conditions of statement (3) (respectively (4)) are only satisfied for i = 3, 4 (respectively i = 1, 4) and then, the convexity of v on Ω 2 M 2 and Ω 3 M 3 (respectively M 1 Ω 2 and M 4 Ω 1 ) is needed. All the segments on which the convexity of v is required in conditions (3) and (4) of theorem 16 are drawn in thick line in figure 23. Proof. We denote by B i, C i, D i, E i, G i, P i, Q i, i = 1, 2, 3, 4, the projections on Q of the control vertices corresponding, respectively, with the B-coefficients b i, c i, d i, e i, g i, p i, q i, i = 1, 2, 3, 4, shown in figure 19. If the B-net v is convex then conditions (2) (4) are immediate. On the other hand, the differentiability of the ST finite element guarantees that v is planar on each shaded region in figure 20. Moreover, theorem 2 implies that v satisfies condition (C) in each bitriangle of the triangulation τ whose restriction to each T i τ is τ 2 (T i ). We consider the restriction of v to each parallelogram that can be determined by any two adjacent bitriangles of τ whose diagonal is located in the interior of two different triangles of τ. Then, we obtain the coplanar regions mentioned in (1) by using (i) of lemma 4. If conditions (1) (4) are satisfied, from theorem 2, it is sufficient to prove that v satisfies condition (C) in each bitriangle of the triangulation on R induced by v. Thus:

17 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 287 (1) Condition (1) guarantees that v satisfies condition (C) in each bitriangle located inside each shaded region of figure 22. (2) In virtue of conditions (1) and (2) we can use lemma 5 for each triangle T i τ. Hence, v satisfies condition (C) in the bitriangles whose diagonals are respectively the segments B i D i or C i D i for i = 1, 2, 3, 4. Figures 24(a) and (b) show these bitriangles (included in the shaded regions) for the case (h) of figure 23. Finally, we apply theorem 2. (3) For the remaining bitriangles, we distinguish three different cases according to the triangulation τ.leth = A 2 M 2, h = M 2 A 3, k = A 2 M 1, k = M 1 A 1 with the notations of figure 18. Case 1. h = h and k k or h h and k = k. Since the proof is purely technical, we only give it in the case h = h, k >k that corresponds to situation (h) of figure 23. The other cases, corresponding to situations (b), (d) and (f) of figure 23, are quite similar. We observe that the quadrilaterals with vertices Ω 1, Ω, Ω 4, M 4 and Ω 2, M 2, Ω 3, Ω are parallelograms. Moreover, condition (2) guarantees that v is convex on the segments ΩΩ i, i = 1,..., 4. Therefore, condition (1) being satisfied, we can use lemma 4(ii): we obtain that the B-net v is convex on the segment ΩΩ 1 (respectively, ΩΩ 3, ΩΩ 2, ΩΩ 4 ) if and only if v it is convex on the segment Ω 4 M 4 (respectively, Ω 2 M 2, Ω 3 M 2, Ω 1 M 4 ). First, we consider the restriction of v to the parallelogram with vertices ΩΩ 1 Q 4 E 4. In virtue of condition (1) we can apply lemma 4(ii). Hence, v is convex on the segment ΩΩ 1 if and only if v is convex on the segment E 4 Q 4. Then, we consider the restriction of v to the parallelogram Q 4 E 4 Ω 4 M 4. Since condition (1) is satisfied, we use lemma 4(ii). Hence, v is convex on the segment E 4 Q 4 if and only if v is convex on the segment Ω 4 M 4. Finally, we proceed in a similar way with the restriction of v to the following pairs of parallelograms: ΩΩ 3 Q 2 E 2 and Q 2 E 2 Ω 2 M 2, ΩΩ 2 P 2 E 3 and P 2 E 3 Ω 3 M 2, ΩΩ 4 P 4 E 1 and P 4 E 1 Ω 1 M 4. Since condition (1) is satisfied, we use lemma 5 for each triangle T i τ that is included in the triangles with vertices A 1 ΩA 4 or A 2 ΩA 3. Therefore, in virtue of theorem 2, v satisfies condition (C) in the shaded bitriangles in figures 25(a) and (b) whose diagonal is not included in any diagonal of the subrectangles R i, i = 1,...,4. On the other hand, using condition (1) and the convexity of v on the segments ΩΩ 1 and ΩΩ 2 (respectively ΩΩ 3 and ΩΩ 4 ), we apply lemma 6(i) (respectively (ii)) with T 1 = Ω 1 M 1 Ω and T 2 = M 1 ΩΩ 2 (respectively T 1 = Ω 3 M 3 Ω and T 2 = M 3 ΩA 4 ) and we obtain that the B-net v is convex on the segment Ω 1 M 1 (respectively Ω 4 M 3 ). Then, since condition (1) is satisfied, we use lemma 5 for each triangle T i τ that is included in the triangles with vertices A 1 M 1 Ω or A 4 M 3 Ω. Hence, from theorem 2, v satisfies condition (C) in the shaded bitriangles of figure 25(c) whose diagonal is not included in any diagonal of the subrectangles R 1 and R 4. Finally, we recall that conditions (3) and (4) in this case (situation (h) in figure 23) refers to the convexity of v on the segments Ω 3 M 3 and M 1 Ω 2. Moreover, v is convex

18 288 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements (a) (b) (c) (d) Figure 25. on the segment M 1 Ω 2 (respectively, Ω 3 M 3 ) if and only if v satisfies condition (C) in the shaded bitriangles of figure 25d whose diagonal is not included in any diagonal of the subrectangles R 2 and R 3 : it is sufficient to use condition (1) and to apply lemma 5 for each triangle T i τ that is included in the triangles with vertices M 1 A 2 Ω or M 3 A 3 Ω. Finally, theorem 2 allows us to conclude. Case 2. h h and k k. For each of the four situations corresponding to this case, namely, situations (a), (c), (g) and (i) of figure 17, we can work in a similar way to case 1. More precisely, using lemmas 5 and 6 and theorem 2 we obtain that v satisfies condition (C) in the remaining bitriangles. Case 3. h = h and k = k. We can observe (see figure 23(e)) that the lines r 1 and r 3 (respectively, r 2 and r 4 ) coincide with the diagonal of R with positive (respectively, negative) slope. As consequence, it is immediate that there is no i {1, 2, 3, 4} satisfying the conditions of statements (3) or (4) of theorem 16. Then, to prove that v satisfies condition (C) in the remaining bitriangles, we proceed as in cases 1 and 3 by using lemmas 4(ii) and 5, and theorem 2. Indeed, in the above proof we have obtained the following result for a uniform partition of R into subrectangles: Corollary 17. In the case when the M i s are the midpoints of the sides of R, then the B-net of the ST finite element is convex if and only if the statements (1) and (2) of theorem 16 are satisfied. Finally, we deduce from theorem 3 that if the B-net of a ST finite element satisfies the conditions of theorem 16 or of corollary 17, then the ST finite element defines a convex surface.

19 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 289 Figure 26. Partitions π(d) and τ(d). Figure 27. Coplanar regions The global problem Definition 18. Let π(d) be a partition of a rectangular domain D R 2 into rectangles, such that for any R, R π(d), with R R, thenr R is either a vertex or the common edge of R and R. A function of ST type on D is a function s C 1 (D) whose restriction to each R π(d) is a ST finite element. Let τ(d) the triangulation of D that, on each R π(d), coincides with the triangulation of ST-type of R (see figure 26). The piecewise affine function v C(D) whose restriction to each R π(d) coincides with the B-net of the ST finite element is called the Bézier net (B-net) of s. It has been proved in [12] that a function of ST type on D is uniquely determined by function values and gradients at the vertices of π(d). In fact, the differentiability of s is equivalent to the fact that, over each pair of adjacent rectangles, the control vertices located over each of the shaded regions in figure 27 are coplanar. As a consequence, by using theorem 16, we can characterize the convexity of the Bézier net of a function of ST type on D as follows: Theorem 19. The B-net of a function of ST type on D is convex if and only if, for all R π(d), conditions (1) (4) of theorem 16 are satisfied. As an immediate consequence of the previous theorem and of corollary 17 we obtain: Corollary 20. Assume that the restriction of τ(d) to each rectangle R π(d) isa uniform criss cross triangulation, then the B-net of a function of ST type on D is convex if and only if, for each R π(d), conditions (1) and (2) of theorem 16 are satisfied. Indeed, if τ(d) in the above corollary is a uniform triangulation, we can replace condition (2) by a weaker condition. Thus:

20 290 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements Figure 28. Segments of type 1 (thick lines) and of type 2 (dashed thick lines). Corollary 21. If τ(d), as in definition 18, is a uniform triangulation, then the B-net of a function of ST type on D is convex if and only if the following conditions hold: (1) For each R π(d), condition (1) of theorem 16 is satisfied. (2) The restrictions of the B-net to the segments of type 1 and of type 2 of τ(d) are convex (see figure 28, for π(d) with six rectangles). Proof. From corollary 20, the B-net of a function of ST type on D is convex if and only if, for each R π(d), conditions (1) and (2) of theorem 16 are satisfied. Moreover, condition (2) is equivalent to condition (2) of theorem 16: it is sufficient to use lemma 4(ii) as in case 1 of part (3) of the proof of theorem 16. Finally, we deduce from theorem 3 that conditions of theorem 19 or of corollaries 20 or 21 are sufficient conditions for the convexity of a function of ST type. Appendix. Computation of B-coefficients A.1. FVS finite element Using the notations of figure 4, we denote by p i = Ds(A i ) A i A i+1 and q i = Ds(A i ) A i A i 1 the derivatives at vertices in the directions of the edges starting from A i,for 1 i 4. We use the convention of indices A 0 = A 4, A 5 = A 1 (and also for all quantities defined below). Let M i = (A i + A i+1 )/2 andlet ν i be any direction not parallel to A i A i+1,1 i 4. For each i = 1, 2, 3, 4, there exist scalars (α i, β i ) such that νi = α iωai + β i ΩA i+1. There also exist ω 1, ω 2 (0, 1) such that Ω = (1 ω 1 )A 1 + ω 1 A 3 = (1 ω 2 )A 2 + ω 2 A 4. We first compute a i = s(a i ), b i = a i + (p i /3), c i = a i + (q i /3), for 1 i 4, from which we deduce immediately d 1 = (1 ω 2 )b 1 + ω 2 c 1, d 2 = (1 ω 1 )c 2 + ω 1 b 2, d 3 = (1 ω 2 )c 3 + ω 2 b 3, d 4 = (1 ω 1 )b 4 + ω 1 c 4.

21 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements 291 The computation of g i is a little bit more intricate. For sake of clarity, we set { γi = α i (a i d i ) + β i (b i d i ), δ i = α i (b i g i ) + β i (c i+1 g i ), ε i = α i (c i+1 d i+1 ) + β i (a i+1 d i+1 ). Therefore, we have the directional derivatives at the vertices: Ds(A i ) ν i = 3γ i, Ds(A i+1 ) ν i = 3ε i. Now, using the fact that Ds() ν i is a univariate quadratic polynomial on the edge A i A i+1 (with B-coefficients γ i, δ i, ε i ), we obtain the equation 4Ds(M i ) ν i = 3(γ i + 2δ i + ε i ) from which we deduce the value of g i,for1 i 4. Finally, we easily obtain the remaining B-coefficients: A.2. ST finite element e 1 = (1 ω 2 )g 1 + ω 2 g 4, e 2 = (1 ω 1 )g 1 + ω 1 g 2, e 3 = (1 ω 2 )g 2 + ω 2 g 3, e 4 = (1 ω 1 )g 4 + ω 1 g 3, ω = (1 ω 1 )e 1 + ω 1 e 3 = (1 ω 2 )e 2 + ω 2 e 4. Using the notations of figures 12 and 13, let M i = (1 α i )A i +α i A i+1,1 i 4, with α 1, α 2 (0, 1), α 3 = 1 α 1, α 4 = 1 α 2 (we adopt the same convention on indices as for the FVS finite element). We denote by γ i = Ds(A i ) A i A i+1 and δ i = Ds(A i ) A i A i 1 the two partial derivatives at the vertex A i of R, inthe directions of the sides starting from A i,1 i 4. Thus, for each i = 1, 2, 3, 4, we obtain succesively a i = s(a i ), b i = a i (1 α i)γ i, c i = a i α i 1γ i, d i = 1 2 (b i + c i ), g i = m i (c i a i + b i+1 a i+1 ), p i = 1 2 (b i + g i ), q i = 1 2 (c i+1 + g i ), e i = 1 2 (g i 1 + g i ), ω i = 1 2 (d i + e i ) = 1 2 (p i + q i 1 ), ω = (1 α 1 )g 4 + α 1 g 2 = (1 α 2 )g 1 + α 2 g 3. References [1] R.K. Beatson and Z. Ziegler, Monotonicity preserving surface interpolation, SIAM J. Numer. Anal. 22 (1985) [2] G.Z. Chang and P.J. Davis, The convexity of Bernstein polynomials over triangles, J. Approx. Theory 40 (1984)

22 292 J. Lorente-Pardo et al. / On the convexity of C 1 surfaces associated with finite elements [3] P.G. Ciarlet, Basic error estimates for elliptic problems, in: Handbook of Numerical Analysis, Vol.II (North-Holland, Amsterdam, 1991) pp [4] J.F. Ciavaldini and J.C. Nédélec, Sur l élément de Fraeijs de Veubeke et Sander, Rev. Française Automat. Informat. Rech. Opérationnelle, Sér. Rouge Anal. Numér. R-2 (1974) [5] W. Dahmen and C.A. Micchelli, Convexity of multivariate Bernstein polynomials and box spline surfaces, Studia Sci. Math. Hungar. 23 (1988) [6] G. Fraeijs de Veubeke, Bending and stretching of plates, in: Conference on Matrix Methods in Structural Mechanics, Wright Patterson AFB, OH (1965). [7] T.A. Grandine, On convexity of piecewise polynomial functions on triangulations, Comput. Aided Geom. Design 6 (1989) [8] A. Li, Convexity preserving interpolation, Comput. Aided Geom. Design 16 (1999) [9] J. Lorente-Pardo, P. Sablonnière and M.C. Serrano-Pérez, On the convexity of Powell Sabin finite elements, in: Advances in Computational Mathematics, eds. Z. Chen, Y. Li, C.A. Micchelli and Y. Xu (Marcel Dekker, New York, 1998) pp [10] J. Lorente-Pardo, P. Sablonnière and M.C. Serrano-Pérez, Subharmonicity and convexity properties of Bernstein polynomials and Bézier nets on triangles, Comput. Aided Geom. Design 16 (1999) [11] J. Lorente-Pardo, P. Sablonnière and M.C. Serrano-Pérez, On the convexity of Bézier nets of quadratic Powell-Sabin splines on 12-fold refined triangulations, J. Comput. Appl. Math. 115 (2000) [12] P. Sablonnière and F. Zedek, Hermite and Lagrange interpolation by quadratic splines on nonuniform criss-cross triangulations, in: Wavelets, Images and Surfaces Fitting, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (A.K. Peters, 1994) pp [13] M.C. Serrano-Pérez, Sobre subarmonía y diversos tipos de convexidad de ciertas funciones polinomiales a trozos, Doctoral Dissertation, University of Granada, Spain (1997). [14] R. Sibson and G.D. Thomson, A seamed quadratic element for contouring, Comput. J. 24 (1981) [15] K. Willemans and P. Dierckx, Surface fitting using convex Powell Sabin splines, J. Comput. Appl. Math. 56 (1994)