Subharmonicity and convexity properties of Bernstein polynomials and ...
Computer Aided Geometric Design 16 (1999) 287–300
Subharmonicity and convexity properties of Bernstein polynomials and Bézier nets on triangles J. Lorente-Pardo a , P. Sablonnière b , M.C. Serrano-Pérez a,∗ a Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain b I.N.S.A. de Rennes, 35043 Rennes Cedex, France
Received December 1997; revised September 1998
Abstract This paper is devoted to the comparison of various shape properties of triangular Bézier surfaces and of their Bézier nets, such as polyhedral convexity, axial convexity and subharmonicity. In order to better compare these properties, the different notations used by different authors are unified. It also includes counterexamples for the results that are not true. 1999 Elsevier Science B.V. All rights reserved. Keywords: Bernstein polynomials; Bézier nets; Convexity; Subharmonicity; Axial convexity; Polyhedral convexity
1. Introduction This paper compares several shape properties of Bernstein approximants of functions and of their associated Bézier nets on triangular domains. It also unifies the notations used by different authors (Chang and Davis, 1984; Sauer, 1991; Be´ska, 1992) and takes special care of counterexamples when a property is not satisfied. We are aware of the fact that this paper is far from being a complete survey of all researches in this field. We restrict our study to a stronger notion than convexity, namely, polyhedral convexity, and to weak forms of convexity, namely, axial convexity, subharmonicity and w-subharmonicity, for which there may exist some interesting relationships between a function u, its Bernstein approximants Bn u and its Bézier nets Ln u. Axial convexity and polyhedral convexity were introduced by Schmid (1975) and Sauer (1991), respectively. A slightly different form of w-subharmonicity has been given in (Be´ska, 1992) under the name of weak axial convexity. ∗ Corresponding author. E-mail: [email protected].
0167-8396/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 3 9 6 ( 9 8 ) 0 0 0 5 0 - 8
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Most of these properties coincide with classical convexity in the univariate case, i.e., for Bézier curves. The problem lies in the fact that there is a lot of possible extensions of classical convexity to the bivariate case, each of them having some independent interest with respect to the corresponding properties of sequences {Bn u} or {Ln u}. At the present time, there is no evidence at all whether the properties that we compare are directly applicable to problems in CAGD. However, we think that this comparison of close and delicate notions with counterexamples can be useful. Clearly, the most useful one is the fact that the convexity of Bézier nets implies that of polynomial surfaces. Applications to finite elements have been given recently in (Carnicer and Dahmen, 1992; Lorente-Pardo et al., 1997, 1998, 1999; Serrano-Pérez, 1997, Chapter 5). Notations are introduced in Section 2, while Section 3 is devoted to convexities and subharmonicities. In Sections 4 and 5, we study the shape properties of Bn u and Ln u. Finally, in Section 6, we use the preceding results to compare the polyhedral convexities (respectively, axial convexities or subharmonicities) of a function u, of its n-th Bernstein polynomial Bn u and of its n-th Bézier net Ln u.
2. Notations Throughout the paper, T denotes a nondegenerate triangle in R2 with vertices Ai , i = 1, 2, 3. Given M ∈ R2 there exists λ = (λ1 , λ2 , λ3 ) ∈ R3 uniquely determined by the conditions M = λ1 A1 + λ2 A2 + λ3 A3 and |λ| = λ1 + λ2 + λ3 = 1. These real numbers are called the barycentric coordinates of M with respect to T . Thus, for i = 1, 2, 3 the barycentric coordinates of Ai with respect to T are ei = (δij )3j =1 where, δii = 1 and, −−−→ −−−→ −−−→ δij = 0 for j 6= i. We denote by γ1 = A2 A3 , γ2 = A3 A1 , γ3 = A1 A2 the directions of the sides of T (see Fig. 1) and by li the length of γi , for i = 1, 2, 3. Let u : T → R be a function and let n > 1. Bernstein polynomials and Bézier nets. The n-th Bernstein polynomial of u is defined by X α X u c (α) bαn (λ), Bn u(λ) = bαn (λ) = n |α|=n
|α|=n
Fig. 1. Notations for the triangle T .
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where: α ≡ (α1 , α2 , α3 ), |α| = α1 + α2 + α3 , αi ∈ N, ∀i = 1, 2, 3 n! α λα1 λα2 λ 3 (Bernstein basis). bαn (λ) = α1 !α2 !α3 ! 1 2 3 The elements of the set {c(α): |α| = n} are called B-coefficients of Bn u. The triangulation of T whose edges are parallel to the sides of T , and whose vertices are obtained by projecting the set of points ϕn = {(α/n, c(α)) ∈ R3 : |α| = n} over the triangle T , is called the triangulation on T induced by ϕn , and is denoted by τn (T ). The unique continuous function which interpolates ϕn and is affine on every Ti ∈ τn (T ) is called the n-th Bézier net or B-net of u, and is denoted by Ln u. Directional derivatives. For each i = 1, 2, 3, let Di (u) = ∂u/∂λi . For an arbitrary direction ω = ω1 A1 + ω2 A2 + ω3 A3 with coefficients satisfying ω1 + ω2 + ω3 = 0, the derivative of u in the direction ω is defined as: X ωi Di u(P ), ∀P ∈ T . Dω u(P ) = 16i63
u ∈ C 2 (T ),
For we define Dω2 u = Dω Dω u and Dγ2 ω u = Dγ Dω u for any directions γ and ω. In particular, we have: Dγi u = (Di−1 − Di+1 )u,
for i = 1, 2, 3,
with the usual convention of indices, say, D0 = D3 and D4 = D1 . Moreover, a straightforward computation using barycentric coordinates as the unique solution of the linear system λ A + λ A + λ A = M, 1 1 2 2 3 3 λ +λ +λ =1 1
2
3
leads to the following expressions of the Laplacian operator 1 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 : 1 X 2 2 li Dγi−1 γi+1 , 4a 2 3
1=−
i=1
1=
1 4a 2
3 X
li−1 li+1 cos θi Dγ2i ,
i=1
where a = area(T ) 6= 0 and θi is the (interior) angle at vertex Ai , for i = 1, 2, 3. 3. Convexity and subharmonicity properties Definitions. Let u : T → R be a function, we say that: (i) u is subharmonic in T if and only if u ∈ C(T ) and it satisfies: 1 u(x0 , y0 ) 6 2π
Z2π u(x0 + r cos θ, y0 + r sin θ) dθ 0
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for all interior point P = (x0 , y0 ) ∈ T and for all disk with center P and radius r included in T . For u ∈ C 2 (T ), this is equivalent to 1u > 0 on T (Montel, 1928), and therefore (by using the above expressions of 1 on T ) to any of the two following conditions: 3 X
li2 Dγ2i−1 γi+1 u 6 0,
i=1 3 X
li−1 li+1 cos θi Dγ2i u > 0.
i=1
(ii) u is axially convex (Sauer, 1991) if u is convex with respect to the directions γ1 , γ2 and γ3 . For u ∈ C 2 (T ), this is obviously equivalent to Dγ2i u > 0,
for i = 1, 2, 3.
(iii) u is polyhedrally convex (Sauer, 1991) if and only Ln u is convex for all n > 1. For u ∈ C 2 (T ), this is equivalent to Dγ2i−1 γi+1 u 6 0,
for i = 1, 2, 3
(a proof is given in Proposition 10 of (Sauer, 1991)). (iv) u is w-subharmonic if and only if Ln u satisfies the following geometric property for all n > 1: For each interior vertex G = (α1 /n, α2 /n, α3 /n) of τn (T ), the point (G, u(G)) is under of the plane that contains the points (Gi , Ln u(Gi )), i = 1, 2, 3, where Gi is the point of T whose barycentric coordinates with respect to Ti are (α1 /n, α2 /n, α3 /n), for i = 1, 2, 3 (see Fig. 2 for n = 3). Or equivalently, if and only if, for all n > 1 and all α with |α| = n holds, 3 X α 2 06 αi αi+1 δγi−1 u , n i=1
where γ0 = γ3 , α4 = α1 and, for each i = 1, 2, 3, ( α−γ i α u( n i ) − 2u( αn ) + u( α+γ 2 n ), δγ i u = n 0,
◦
if G = ( αn ) ∈ T , otherwise.
Fig. 2. Geometric interpretation of w-subharmonicity.
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For u ∈ C 2 (T ), this implies that Au > 0, where A is a second order differential operator which can be expressed in the two following equivalent forms: Au(λ) =
3 X
λi λi+1 Dγ2i−1 u(λ),
i=1
Au(λ) = −
3 X
λi (1 − λi )Dγ2i−1 γi+1 u(λ).
i=1
The proof of this implication is immediate since, for each n > 1, ( 3 ) X X α 2 ABn u(λ) = αi αi+1 δγi−1 u bαn (λ) n |α|=n
i=1
and {ABn u}n∈N converges to Au when u ∈ C 2 (T ). Remarks. (1) The last property implies, for C 2 functions, the property of weak axial convexity introduced by Be´ska (1992), and it can be considered as a subharmonicity property of each Bézier net, Ln u, in the following sense: the value of Ln u at each interior vertex G of τn (T ) is less than or equal to the weighted average of the values of the Bézier net at the vertices of the boundary of the hexagon determined by the triangles Ti , i = 1, 2, 3. As it seems closer to a notion of subharmonicity, we have chosen the name of wsubharmonicity. Moreover, the w also refers to Woronovskaya, since we have 1 lim n(u − Bn u) = Au 2
n→+∞
for C 2 functions. As we shall see later, w-subharmonicity is more “natural” than subharmonicity on triangular domains. (2) For bivariate continuous functions on triangles, it is known (Sauer, 1991) that polyhedral convexity implies convexity, and convexity implies axial convexity and subharmonicity (Montel, 1928). But the relationships between axial convexity and subharmonicity can depend on the geometry of the triangle. More specifically, from the above definitions, we can deduce easily: (i) If T is not obtuse angled (cos θi > 0, for i = 1, 2, 3), then axial convexity implies subharmonicity at least for u ∈ C 2 (T ) (see Section 5 for u ∈ C(T )). (ii) If T is obtuse angled (there exists i ∈ {1, 2, 3} with cos θi < 0), then there is no relation, a priori, between axial convexity and subharmonicity. It is enough to consider the functions u1 : T1 → R and u2 : T2 → R given by u1 (x, y) = −y(x + y) and u2 (x, y) = 4x 2 − y 2 , respectively, where T1 is the triangle with vertices A = (0, 0), B = (1, 0) and C1 = (−1, 1), and T2 is the triangle with vertices A, B and C2 = (−1, −3). Then, u1 is axially convex but not subharmonic. However, u2 is subharmonic though not axially convex.
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(3) Concerning w-subharmonicity, things are clearer: obviously, axial convexity always implies w-subharmonicity whatever be the shape of T . This is an argument in favor of w-subharmonicity versus classical subharmonicity, at least on triangular domains.
4. Convexity and subharmonicity properties of Ln u Let us begin with some general considerations on piecewise affine and continuous functions on triangulations. Let D be a convex polygonal domain of R2 and let τ = {Ti : i = 1, . . . , n} be a finite triangulation of D, such that for any Ti , Tj ∈ τ , with Ti ∩ Tj 6= ∅, then Ti ∩ Tj is either a vertex or a common edge of Ti and Tj . For each quadrilateral Q which is the union of two adjacent triangles Ti and Tj of τ , Ti ∩ Tj is called the diagonal of Q. Let u ∈ C(D) be a function which is affine over each Ti , i = 1, . . . , n. Since convexity is a local property, it is clear that u is convex if and only if it is convex over any pair of adjacent triangles of τ . Similarly, over two adjacent triangles, u is subharmonic if and only if u is convex, hence, the subharmonicity of u is equivalent to the convexity of u. In fact, these properties have a clear geometric interpretation (see Fig. 3), namely, that over each pair of adjacent triangles Ti and Tj either r intersects s, or r is over s, where s is the straight line containing u(Ti ∩ Tj ) and r is the straight line which contains u(A) and u(B) with A and B the vertices of Ti ∪ Tj such that A, B ∈ / T i ∩ Tj . Moreover, for the particular case in which Ti ∪ Tj is a parallelogram this geometrical condition can be written as u(A) + u(B) − u(C) − u(D) > 0,
(1)
where C and D are the vertices on the diagonal of Ti ∪ Tj . In particular, if D = T , both convexity and subharmonicity are equivalent to axial convexity since a piecewise linear function is convex if and only if it is convex in at least
Fig. 3. Geometric interpretation of convexity property of u over Ti ∪ Tj .
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any two directions. However, there are convex and continuous piecewise linear function over T which are not polyhedrally convex: Example 4.1. We consider T with vertices A1 , A2 , A3 . For M = (A1 + A2 )/2, let τ (T ) = {Ti : i = 1, 2} be a triangulation of T where T1 and T2 are the triangles with vertices A1 , A3 , M and A2 , A3 , M respectively. Let u : T → R be a function which is affine over each triangle in τ (T ) and satisfies u(Ai ) = 0, i = 1, 2, 3, and u(M) = −1/2. We see that u is convex, but that L2 u is not convex. We will show that for uniform triangulations τn (T ) and associated Bézier nets Ln u, this problem does not occur. Lemma 4.2. (a) Let q ∈ N be a divisor of n, if Ln u is convex then Lq (Ln u) is convex. (b) Let p ∈ N be a multiple of n, then Lp (Ln u) ≡ Ln u. Proof. (a) Let τq (T ) be the triangulation of T induced by Lq (Ln u), and let R be any parallelogram with vertices A, B, C, D, which is the union of two adjacent triangles of τq (T ). We observe that since q divides n then the triangulation τn (T ) restricted to the parallelogram R constitutes a triangulation of R, denoted by τr (R), with 2r 2 triangles, where r is such that qr = n. (See Fig. 4 for case n = 4, q = 2.) The convexity of Ln u allows us to consider for the function Ln u the analytical condition (1) over each parallelogram of τr (R) whose diagonal is parallel to the diagonal of R. If we add all these inequalities, and since the values of Ln u on the vertices located on the diagonal of these parallelograms appear with minus sign, and the values of Ln u at the vertices out of the diagonal of these parallelograms appear with plus sign, we obtain that the only terms that remain in the sum are those associated with the vertices of R, namely, we get Ln u(A) + Ln u(B) − Ln u(C) − Ln u(D) > 0. Hence, Lq (Ln u) is convex on R, since Lq (Ln u) interpolates Ln u at the vertices of R. (b) Let τp (T ) be the triangulation of T induced by Lp (Ln u), it is enough to observe that, since p is a multiple of n, each triangle of τn (T ) contains r 2 triangles of τp (T ), with p = nr (see Fig. 5 for case n = 2, p = 4). 2
Fig. 4. Triangulations τ4 (T ) and τ2 (T ) and parallelogram R with vertices A, B, C, D.
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Fig. 5. Triangulations τ2 (T ) and τ4 (T ).
Theorem 4.3. The following statements are equivalent: (i) Ln u is convex. (ii) Ln u is subharmonic. (iii) Ln u is axially convex. (iv) Ln u is polyhedrally convex. Proof. Of course, the first three statements are equivalent. It remains to prove, e.g., that (i) is equivalent to (iv). Thus, (i) ⇒ (iv): Let m ∈ N∗ and p = l.c.m.(m, n) (least common multiple), then there exists r ∈ N such that p = nr. Thus, from Lemma 4.2(b), we have Lp (Ln u) ≡ Ln u, and therefore Lm (Lp (Ln u)) ≡ Lm (Ln u). The conclusion follows by applying Lemma 4.2(a), taking into account that m is a divisor of p and Ln u is convex. (iv) ⇒ (i): It is immediate since polyhedral convexity implies convexity. 2 5. Convexity and subharmonicity properties of Bernstein polynomials Bn u Since B1 u is the linear interpolant of u at the vertices of T , it is clear that it is axially and polyhedrally convex, subharmonic and w-subharmonic. For n > 2, we begin by introducing some definitions about specific conditions on the B-nets Ln u which are used below to give sufficient conditions on convexities and subharmonicity of Bn u. Thus, we say that: (i) Condition (PC) is satisfied if and only if for all β such that |β| = n − 2, and for i = 1, 2, 3, mi (β) > 0, where: mi (β) = c(β + 2ei ) − c(β + ei + ei−1 ) − c(β + ei + ei+1 ) + c(β + ei−1 + ei+1 ) and e0 = e3 , e4 = e1 (see Fig. 6 for n = i = 2). (ii) Condition (AC) is satisfied if and only if for all β such that |β| = n − 2, and for 1 6 j < k 6 3, aj k (β) > 0, where: aj k (β) = mj (β) + mk (β) = c(β + 2ej ) − 2c(β + ej + ek ) + c(β + 2ek ). (iii) Condition (S) is satisfied if and only if for all β such that |β| = n − 2, s(β) > 0, where s(β) = l12 m1 (β) + l22 m2 (β) + l32 m3 (β).
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Fig. 6. Geometric interpretation of condition m2 (β) > 0 for n = 2.
Firstly, we give necessary and sufficient conditions for the convexities and subharmonicity of the Bernstein polynomial Bn u. Theorem 5.1. For n > 2, the following characterizations hold: (i) Bn u is polyhedrally convex if and only if for each i ∈ {1, 2, 3} there holds: X mi (β)bβn−2 (λ) > 0. |β|=n−2
(ii) Bn u is axially convex if and only if for 1 6 j < k 6 3 there holds: X aj k (β)bβn−2 (λ) > 0. |β|=n−2
(iii) Bn u is subharmonic if and only if there holds: X s(β)bβn−2 (λ) > 0. |β|=n−2
Proof. Since Bn u ∈ C 2 (T ), each characterization is directly obtained from the definitions of Section 3 and from the following formula for 1 6 i, j 6 3: X Dγ2i γj Bn u(λ) = n(n − 1) bβn−2 (λ) c(β + ei−1 + ej −1 ) − c(β + ei−1 + ej +1 ) |β|=n−2
− c(β + ei+1 + ej −1 ) + c(β + ei+1 + ej +1 ) . 2 Corollary 5.2. For n > 2, if condition (PC) (respectively, (AC) or (S)) is satisfied then Bn u is polyhedrally convex (respectively, axially convex or subharmonic). Remarks. (1) For n = 2 and n = 3, condition (PC) (respectively, (AC) or (S)) is also necessary for the polyhedral convexity (respectively, axial convexity or subharmonicity) of Bn u. (2) Using the nonnegativity condition for bivariate quadratic polynomials given in (Micchelli and Pinkus, 1989), we deduce criteria of polyhedral convexity (respectively,
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axial convexity and subharmonicity) for B4 u (see Serrano-Pérez, 1997, Chapter 2, for details). (3) We note that, for n > 2, condition (PC) is also a sufficient condition for the convexity of Bn u (Chang and Davis, 1984) and, moreover, it is a condition that is equivalent to the convexity of Ln u (Dahmen and Micchelli, 1988). Finally, in relation with Remark 2 of Section 3, we obtain the following property: Proposition 5.3. If T is not obtuse-angled, then axial convexity implies subharmonicity. Proof. Let u : T → R be a continuous and axially convex function, then for each n > 2 condition (AC) is satisfied. On the other hand, for each β with |β| = n − 2, we can rewrite s(β) as s(β) = l1 l2 cos(θ3 )a12(β) + l1l3 cos(θ2 )a13(β) + l2 l3 cos(θ1 )a23(β), where θi is the (interior) angle at vertex Ai , i = 1, 2, 3. Therefore, condition (S) is satisfied, since li > 0 and cos(θi ) > 0, i = 1, 2, 3, and then, Bn u is subharmonic for all n > 2. Finally, u is subharmonic because it is a uniform limit of subharmonic functions. 2 6. Relationships between the shape properties of u, Bn u and Ln u In this section (P) will denote any property of u, given in Section 3. The results of the previous sections allow us to conclude the following: Theorem 6.1. For u ∈ C(T ), we consider the following statements: (1) u satisfies property (P). (2) For all n ∈ N∗ , Bn u satisfies property (P). (3) For all n ∈ N∗ , Ln u satisfies property (P). Then: (i) When (P) is the polyhedral convexity, the three above statements are equivalent. (ii) When (P) is the axial convexity, then (1) and (2) are equivalent, (3) implies (1), but (1) does not imply (3). (iii) When (P) is the subharmonicity, then (2) implies (1), (3) implies (1), but the converses are not always true. Proof. (i) The proof of (1) ⇒ (2) can be seen in page 475 of (Sauer, 1991). For (2) ⇒ (1), we consider for each m ∈ N∗ the operator Lm : C(T ) → C(T ) given by Lm (f ) := Lm f , for all f ∈ C(T ). From (2), we know that for each n ∈ N∗ the function (Lm ◦ Bn )u is convex. The continuity of Lm , together with the property of uniform approximation of the sequence of Bernstein polynomials, guarantee that Lm u is convex, and hence, that u is polyhedrally convex. The equivalence between (1) and (3) is immediate from Theorem 4.3. (ii) The proof of (1) ⇒ (2) can be seen in page 470 of (Sauer, 1991). The converse is deduced from the property of uniform approximation of the sequence of Bernstein polynomials.
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Fig. 7. B-coefficients over the parallelogram R of τ4 (T ).
On the other hand, if Ln u is axially convex then it is convex (Theorem 4.3), and hence, Bn u is convex. Finally, because of the property of uniform approximation of u by {Bn u}n∈N∗ and since convexity implies axial convexity, it can be deduced that (3) implies (1). To prove that the converse is not always true, it is enough to consider Example 4.1 in which u is axially convex but L2 u is not. (iii) Since u is a uniform limit of subharmonic functions, it is immediate that (2) implies (1). For the converse we consider the triangle T in R2 with vertices A1 = (0, 0), A2 = (1, 0), A3 = (−1, 1). Now, let τ (T ) = {Ti : i = 1, 2} be the triangulation of T where T1 and T2 are the triangles with vertices A1 , A2 , M and A1 , A3 , M, respectively, where M = (0, 1/2) is the midpoint of A2 A3 . Let u : T → R be the continuous function which is affine over each triangle of τ (T ) and satisfies u(A1 ) = u(A2 ) = u(M) = 0, u(A3) = 1. The function u is subharmonic because it is convex and, however, B3 u is not subharmonic since 1B3 u(A1 ) = −4 < 0. Finally, the proof of (3) ⇒ (1) follows from Theorem 4.3, part (i) of this theorem and from the fact that polyhedral convexity implies convexity, which further implies subharmonicity. For the converse, it is enough to consider T with vertices A1 = (1, 0), A2 = (0, 1) and A3 = (0, 0), and u : T → R given by u(λ1 , λ2 , λ3 ) = 2λ21 − λ22 . Then c(α) = (2α12 − α22 )/n2 for each α with |α| = n. Since 1u = 2 > 0, u is subharmonic, but there is no n ∈ N, n > 2 for which Ln u is subharmonic. This is due to the fact that over the parallelogram R of τn (T ) whose vertices, given in barycentric coordinates with respect 1 1 n−1 1 n−2 1 to T , are (0, 1, 0), (0, n−1 n , n ), ( n , n , 0), ( n , n , n ), condition (PC) is not satisfied (see Fig. 7 for n = 4). 2 Theorem 6.2. (i) If Ln u satisfies property (P), then Bn u satisfies property (P), where property (P) is either polyhedral convexity, axial convexity or subharmonicity. (ii) If (P) is the polyhedral convexity property, then the reverse statement of (i) is only true for n = 2, 3.
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(iii) If property (P) is either axial convexity or subharmonicity, then the converse statement of (i) is not always true. Proof. (i) From Theorem 4.3, Ln u is convex, namely, condition (PC) is satisfied and, consequently, conditions (AC) and (S); then, Theorem 5.1 implies (i). (ii) Follows again from Theorem 5.1 together with the following example: we consider n = 4 and c(0, 4, 0) = c(0, 0, 4) = c(0, 2, 2) = 2, c(0, 1, 3) = c(0, 3, 1) = 1, c(i, j, k) = 0, otherwise, then, it is immediate (see the nonnegativity condition for polynomials in P2 [x, y] given by Micchelli and Pinkus (1989)), that B4 u is polyhedrally convex. However, L4 u is not convex and, hence, is not polyhedrally convex. For (iii), if (P) is axial convexity, it is sufficient to consider B2 u with c(1, 1, 0) = −1/2 and c(i, j, k) = 0 otherwise, and to use Theorem 5.1 and Theorem 4.3. If (P) is subharmonicity, we consider the example given in the proof of assertion (1) ; (3) for (iii) of the previous theorem. Then, Bn u is subharmonic since condition (S) is satisfied for all n > 2 (see Theorem 5.1) though Ln u is not subharmonic. 2 Theorem 6.3. Let property (P) be either polyhedral convexity or axial convexity, and u ∈ C(T ), then: (i) If u satisfies property (P), then Bn u > Bn+1 u, for all n ∈ N∗ . (ii) If u satisfies property (P), then Bn u > u, for all n ∈ N∗ . (iii) The converses of (i) and (ii) are not always true. Proof. (i) If (P) is polyhedral convexity, the result follows since polyhedral convexity implies convexity and, therefore, {Bn u}n∈N∗ is decreasing (cf. (Chang and Davis, 1984)). If (P) is axial convexity, see page 472 of (Sauer, 1991). (ii) is immediate from (i) in virtue of the property of uniform approximation of u by {Bn u}n∈N∗ . To prove (iii) we consider the triangle T with vertices A1 = (−1, 0), A2 = (1, 0) and A3 = (0, 1) and u : T → R given by u(x, y) = −xy. We observe that: (a) u is not polyhedrally convex since L2 u is not convex; (b) u is not axially convex because it is not convex in the direction of side A1 A3 ; and however, Bn u > Bn+1 u and Bn u > u, for all n ∈ N∗ , because Bn u(x, y) = −(1 − (1/n))xy. 2 Theorem 6.4. If u ∈ C(T ) then there is no relation between the subharmonicity of u and either the decrease of the sequence {Bn u}n∈N∗ or the nonnegativity of the functions of the sequence {Bn u − u}n∈N∗ . More precisely: (i) u subharmonic does not imply that Bn u(λ) > u(λ) for all n ∈ N∗ , and hence, does not imply that Bn u(λ) > Bn+1 u(λ) for all n ∈ N∗ .
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(ii) Bn u(λ) > Bn+1 u(λ), for all n ∈ N∗ , and also, Bn u(λ) > u(λ) for all n ∈ N∗ , does not imply that u is subharmonic. Proof. (i) We consider again the last example of the proof of Theorem 6.1. There u is 1 3 7 , 4 , 32 ). subharmonic, but B1 u(λ) < u(λ) for λ = ( 32 (ii) It is sufficient to consider T the triangle with vertices A1 = (1, 0), A2 = (0, 1), A3 = (0, 0) and u : T → R given by u(λ1 , λ2 , λ3 ) = −λ1 λ2 − λ21 λ22 . Then, u is not subharmonic and, however, for each n ∈ N∗ , Bn u(λ) > Bn+1 u(λ) because: Bn u(λ1 , λ2 , λ3 ) =
−n3 + n2 − n + 1 −n2 + 3n − 2 λ λ + λ1 λ2 (λ1 + λ2 ) 1 2 n3 n3 −n3 + 6n2 − 11n + 6 2 2 + λ1 λ2 . 2 n3
Acknowledgements The authors would like to thank Professor Jesús Carnicer for his helpful suggestions and the referees for their thorough review and their comments.
References Be´ska, M. (1992), Geometric properties on Bernstein polynomials over simplices, in: Lyche, T. and Schumaker, L.L., eds., Mathematical Methods in CAGD and Image Processing, Academic Press, Boston, 1–8. Carnicer, J.M. and Dahmen, W. (1992), Convexity preserving interpolation and Powell–Sabin elements, Computer Aided Geometric Design 9, 279–289. Carnicer, J.M., Floater, M.S. and Peña, J.M. (1997), Linear convexity conditions for rectangular and triangular Bernstein–Bézier surfaces, Computer Aided Geometric Design 15, 27–38. Chang, G.Z. and Davis, P.J. (1984), The convexity of Bernstein polynomials over triangles, J. Approx. Theory 40, 11–28. Chang, G.Z. and Zhang, J.Z. (1990), Converse theorems of convexity for Bernstein polynomials over triangles, J. Approx. Theory 61, 265–278. Dahmen, W. (1991), Convexity and Bernstein–Bézier polynomials, in: Curves and Surfaces, Laurent, P.J., Le Méhauté, A. and Schumaker, L.L., eds., Academic Press, Boston, MA, 107–134. Dahmen, W. and Micchelli, C.A. (1988), Convexity of multivariate Bernstein polynomials and box spline surfaces, Studia Sci. Math. Hungar. 23, 265–287. Feng, Y.Y., Chen, F.L. and Zhou, H.L. (1994), The invariance of weak convexity conditions of B-nets with respect to subdivision, Computer Aided Geometric Design 11, 97–107. Floater, M. (1997), A counterexample to a theorem on Powell–Sabin elements, Computer Aided Geometric Design 14, 383–385. Goodman, T.N.T. (1989), Shape preserving representations, in: Mathematical Methods in Computer Aided Geometric Design, Lyche, T. and Schumaker, L.L., eds., Academic Press, Boston, MA, 333–351. He, T.-X. (1995), Shape criteria of Bernstein–Bézier polynomials over simplexes, Comput. Math. Appl. 30, 317–333.
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Lorente-Pardo, J., Sablonnière, P. and Serrano-Pérez, M.C. (1997), On convexity and subharmonicity of some functions on triangles, in: Curves and Surfaces with Applications in CAGD, Le Méhauté, A., Rabut, C. and Schumaker, L.L., eds., Vanderbilt University Press, 271–278. Lorente-Pardo, J., Sablonnière, P. and Serrano-Pérez, M.C. (1998), On the convexity of Powell–Sabin finite elements, in: Advances in Computational Mathematics, Chen, Z., Li, Y., Micchelli, C.A. and Xu, Y., eds., Marcel Dekker, Inc., 395–404. Lorente-Pardo, J., Sablonnière, P. and Serrano-Pérez, M.C. (1999), On the convexity of some piecewise quadratic Powell–Sabin surfaces, submitted to Journal C.A.M. Micchelli, C.A. and Pinkus, A. (1989), Some remarks on nonnegative polynomials on polyhedra, in: Prob. Stat. and Math. Papers in Honor of Samuel Karlin, Anderson, T., Athreya, K. and Iglebert, D., eds., Academic Press, New York, 163–186. Montel, P. (1928), Sur les fonctions convexes et les fonctions sousharmoniques. J. Math. Pures Appl., Series 9, Vol. 7, 29–60. Prautzsch, H. (1992), On convex Bézier triangles, Modélisation Math. Anal. Numer. 26, 23–36. Sauer, T. (1991), Multivariate Bernstein polynomials and convexity, Computer Aided Geometric Design 8, 465–478. Schmid, H.J. (1975), Bernsteinpolynome, Unpublished manuscript. Serrano-Pérez, M.C. (1997), Sobre subarmonía y diversos tipos de convexidad de ciertas funciones polinomiales a trozos, Doctoral Dissertation, University of Granada, Spain.
Subharmonicity and convexity properties of Bernstein polynomials and Bézier nets on triangles J. Lorente-Pardo a , P. Sablonnière b , M.C. Serrano-Pérez a,∗ a Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain b I.N.S.A. de Rennes, 35043 Rennes Cedex, France
Received December 1997; revised September 1998
Abstract This paper is devoted to the comparison of various shape properties of triangular Bézier surfaces and of their Bézier nets, such as polyhedral convexity, axial convexity and subharmonicity. In order to better compare these properties, the different notations used by different authors are unified. It also includes counterexamples for the results that are not true. 1999 Elsevier Science B.V. All rights reserved. Keywords: Bernstein polynomials; Bézier nets; Convexity; Subharmonicity; Axial convexity; Polyhedral convexity
1. Introduction This paper compares several shape properties of Bernstein approximants of functions and of their associated Bézier nets on triangular domains. It also unifies the notations used by different authors (Chang and Davis, 1984; Sauer, 1991; Be´ska, 1992) and takes special care of counterexamples when a property is not satisfied. We are aware of the fact that this paper is far from being a complete survey of all researches in this field. We restrict our study to a stronger notion than convexity, namely, polyhedral convexity, and to weak forms of convexity, namely, axial convexity, subharmonicity and w-subharmonicity, for which there may exist some interesting relationships between a function u, its Bernstein approximants Bn u and its Bézier nets Ln u. Axial convexity and polyhedral convexity were introduced by Schmid (1975) and Sauer (1991), respectively. A slightly different form of w-subharmonicity has been given in (Be´ska, 1992) under the name of weak axial convexity. ∗ Corresponding author. E-mail: [email protected].
0167-8396/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 3 9 6 ( 9 8 ) 0 0 0 5 0 - 8
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Most of these properties coincide with classical convexity in the univariate case, i.e., for Bézier curves. The problem lies in the fact that there is a lot of possible extensions of classical convexity to the bivariate case, each of them having some independent interest with respect to the corresponding properties of sequences {Bn u} or {Ln u}. At the present time, there is no evidence at all whether the properties that we compare are directly applicable to problems in CAGD. However, we think that this comparison of close and delicate notions with counterexamples can be useful. Clearly, the most useful one is the fact that the convexity of Bézier nets implies that of polynomial surfaces. Applications to finite elements have been given recently in (Carnicer and Dahmen, 1992; Lorente-Pardo et al., 1997, 1998, 1999; Serrano-Pérez, 1997, Chapter 5). Notations are introduced in Section 2, while Section 3 is devoted to convexities and subharmonicities. In Sections 4 and 5, we study the shape properties of Bn u and Ln u. Finally, in Section 6, we use the preceding results to compare the polyhedral convexities (respectively, axial convexities or subharmonicities) of a function u, of its n-th Bernstein polynomial Bn u and of its n-th Bézier net Ln u.
2. Notations Throughout the paper, T denotes a nondegenerate triangle in R2 with vertices Ai , i = 1, 2, 3. Given M ∈ R2 there exists λ = (λ1 , λ2 , λ3 ) ∈ R3 uniquely determined by the conditions M = λ1 A1 + λ2 A2 + λ3 A3 and |λ| = λ1 + λ2 + λ3 = 1. These real numbers are called the barycentric coordinates of M with respect to T . Thus, for i = 1, 2, 3 the barycentric coordinates of Ai with respect to T are ei = (δij )3j =1 where, δii = 1 and, −−−→ −−−→ −−−→ δij = 0 for j 6= i. We denote by γ1 = A2 A3 , γ2 = A3 A1 , γ3 = A1 A2 the directions of the sides of T (see Fig. 1) and by li the length of γi , for i = 1, 2, 3. Let u : T → R be a function and let n > 1. Bernstein polynomials and Bézier nets. The n-th Bernstein polynomial of u is defined by X α X u c (α) bαn (λ), Bn u(λ) = bαn (λ) = n |α|=n
|α|=n
Fig. 1. Notations for the triangle T .
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where: α ≡ (α1 , α2 , α3 ), |α| = α1 + α2 + α3 , αi ∈ N, ∀i = 1, 2, 3 n! α λα1 λα2 λ 3 (Bernstein basis). bαn (λ) = α1 !α2 !α3 ! 1 2 3 The elements of the set {c(α): |α| = n} are called B-coefficients of Bn u. The triangulation of T whose edges are parallel to the sides of T , and whose vertices are obtained by projecting the set of points ϕn = {(α/n, c(α)) ∈ R3 : |α| = n} over the triangle T , is called the triangulation on T induced by ϕn , and is denoted by τn (T ). The unique continuous function which interpolates ϕn and is affine on every Ti ∈ τn (T ) is called the n-th Bézier net or B-net of u, and is denoted by Ln u. Directional derivatives. For each i = 1, 2, 3, let Di (u) = ∂u/∂λi . For an arbitrary direction ω = ω1 A1 + ω2 A2 + ω3 A3 with coefficients satisfying ω1 + ω2 + ω3 = 0, the derivative of u in the direction ω is defined as: X ωi Di u(P ), ∀P ∈ T . Dω u(P ) = 16i63
u ∈ C 2 (T ),
For we define Dω2 u = Dω Dω u and Dγ2 ω u = Dγ Dω u for any directions γ and ω. In particular, we have: Dγi u = (Di−1 − Di+1 )u,
for i = 1, 2, 3,
with the usual convention of indices, say, D0 = D3 and D4 = D1 . Moreover, a straightforward computation using barycentric coordinates as the unique solution of the linear system λ A + λ A + λ A = M, 1 1 2 2 3 3 λ +λ +λ =1 1
2
3
leads to the following expressions of the Laplacian operator 1 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 : 1 X 2 2 li Dγi−1 γi+1 , 4a 2 3
1=−
i=1
1=
1 4a 2
3 X
li−1 li+1 cos θi Dγ2i ,
i=1
where a = area(T ) 6= 0 and θi is the (interior) angle at vertex Ai , for i = 1, 2, 3. 3. Convexity and subharmonicity properties Definitions. Let u : T → R be a function, we say that: (i) u is subharmonic in T if and only if u ∈ C(T ) and it satisfies: 1 u(x0 , y0 ) 6 2π
Z2π u(x0 + r cos θ, y0 + r sin θ) dθ 0
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for all interior point P = (x0 , y0 ) ∈ T and for all disk with center P and radius r included in T . For u ∈ C 2 (T ), this is equivalent to 1u > 0 on T (Montel, 1928), and therefore (by using the above expressions of 1 on T ) to any of the two following conditions: 3 X
li2 Dγ2i−1 γi+1 u 6 0,
i=1 3 X
li−1 li+1 cos θi Dγ2i u > 0.
i=1
(ii) u is axially convex (Sauer, 1991) if u is convex with respect to the directions γ1 , γ2 and γ3 . For u ∈ C 2 (T ), this is obviously equivalent to Dγ2i u > 0,
for i = 1, 2, 3.
(iii) u is polyhedrally convex (Sauer, 1991) if and only Ln u is convex for all n > 1. For u ∈ C 2 (T ), this is equivalent to Dγ2i−1 γi+1 u 6 0,
for i = 1, 2, 3
(a proof is given in Proposition 10 of (Sauer, 1991)). (iv) u is w-subharmonic if and only if Ln u satisfies the following geometric property for all n > 1: For each interior vertex G = (α1 /n, α2 /n, α3 /n) of τn (T ), the point (G, u(G)) is under of the plane that contains the points (Gi , Ln u(Gi )), i = 1, 2, 3, where Gi is the point of T whose barycentric coordinates with respect to Ti are (α1 /n, α2 /n, α3 /n), for i = 1, 2, 3 (see Fig. 2 for n = 3). Or equivalently, if and only if, for all n > 1 and all α with |α| = n holds, 3 X α 2 06 αi αi+1 δγi−1 u , n i=1
where γ0 = γ3 , α4 = α1 and, for each i = 1, 2, 3, ( α−γ i α u( n i ) − 2u( αn ) + u( α+γ 2 n ), δγ i u = n 0,
◦
if G = ( αn ) ∈ T , otherwise.
Fig. 2. Geometric interpretation of w-subharmonicity.
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For u ∈ C 2 (T ), this implies that Au > 0, where A is a second order differential operator which can be expressed in the two following equivalent forms: Au(λ) =
3 X
λi λi+1 Dγ2i−1 u(λ),
i=1
Au(λ) = −
3 X
λi (1 − λi )Dγ2i−1 γi+1 u(λ).
i=1
The proof of this implication is immediate since, for each n > 1, ( 3 ) X X α 2 ABn u(λ) = αi αi+1 δγi−1 u bαn (λ) n |α|=n
i=1
and {ABn u}n∈N converges to Au when u ∈ C 2 (T ). Remarks. (1) The last property implies, for C 2 functions, the property of weak axial convexity introduced by Be´ska (1992), and it can be considered as a subharmonicity property of each Bézier net, Ln u, in the following sense: the value of Ln u at each interior vertex G of τn (T ) is less than or equal to the weighted average of the values of the Bézier net at the vertices of the boundary of the hexagon determined by the triangles Ti , i = 1, 2, 3. As it seems closer to a notion of subharmonicity, we have chosen the name of wsubharmonicity. Moreover, the w also refers to Woronovskaya, since we have 1 lim n(u − Bn u) = Au 2
n→+∞
for C 2 functions. As we shall see later, w-subharmonicity is more “natural” than subharmonicity on triangular domains. (2) For bivariate continuous functions on triangles, it is known (Sauer, 1991) that polyhedral convexity implies convexity, and convexity implies axial convexity and subharmonicity (Montel, 1928). But the relationships between axial convexity and subharmonicity can depend on the geometry of the triangle. More specifically, from the above definitions, we can deduce easily: (i) If T is not obtuse angled (cos θi > 0, for i = 1, 2, 3), then axial convexity implies subharmonicity at least for u ∈ C 2 (T ) (see Section 5 for u ∈ C(T )). (ii) If T is obtuse angled (there exists i ∈ {1, 2, 3} with cos θi < 0), then there is no relation, a priori, between axial convexity and subharmonicity. It is enough to consider the functions u1 : T1 → R and u2 : T2 → R given by u1 (x, y) = −y(x + y) and u2 (x, y) = 4x 2 − y 2 , respectively, where T1 is the triangle with vertices A = (0, 0), B = (1, 0) and C1 = (−1, 1), and T2 is the triangle with vertices A, B and C2 = (−1, −3). Then, u1 is axially convex but not subharmonic. However, u2 is subharmonic though not axially convex.
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(3) Concerning w-subharmonicity, things are clearer: obviously, axial convexity always implies w-subharmonicity whatever be the shape of T . This is an argument in favor of w-subharmonicity versus classical subharmonicity, at least on triangular domains.
4. Convexity and subharmonicity properties of Ln u Let us begin with some general considerations on piecewise affine and continuous functions on triangulations. Let D be a convex polygonal domain of R2 and let τ = {Ti : i = 1, . . . , n} be a finite triangulation of D, such that for any Ti , Tj ∈ τ , with Ti ∩ Tj 6= ∅, then Ti ∩ Tj is either a vertex or a common edge of Ti and Tj . For each quadrilateral Q which is the union of two adjacent triangles Ti and Tj of τ , Ti ∩ Tj is called the diagonal of Q. Let u ∈ C(D) be a function which is affine over each Ti , i = 1, . . . , n. Since convexity is a local property, it is clear that u is convex if and only if it is convex over any pair of adjacent triangles of τ . Similarly, over two adjacent triangles, u is subharmonic if and only if u is convex, hence, the subharmonicity of u is equivalent to the convexity of u. In fact, these properties have a clear geometric interpretation (see Fig. 3), namely, that over each pair of adjacent triangles Ti and Tj either r intersects s, or r is over s, where s is the straight line containing u(Ti ∩ Tj ) and r is the straight line which contains u(A) and u(B) with A and B the vertices of Ti ∪ Tj such that A, B ∈ / T i ∩ Tj . Moreover, for the particular case in which Ti ∪ Tj is a parallelogram this geometrical condition can be written as u(A) + u(B) − u(C) − u(D) > 0,
(1)
where C and D are the vertices on the diagonal of Ti ∪ Tj . In particular, if D = T , both convexity and subharmonicity are equivalent to axial convexity since a piecewise linear function is convex if and only if it is convex in at least
Fig. 3. Geometric interpretation of convexity property of u over Ti ∪ Tj .
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any two directions. However, there are convex and continuous piecewise linear function over T which are not polyhedrally convex: Example 4.1. We consider T with vertices A1 , A2 , A3 . For M = (A1 + A2 )/2, let τ (T ) = {Ti : i = 1, 2} be a triangulation of T where T1 and T2 are the triangles with vertices A1 , A3 , M and A2 , A3 , M respectively. Let u : T → R be a function which is affine over each triangle in τ (T ) and satisfies u(Ai ) = 0, i = 1, 2, 3, and u(M) = −1/2. We see that u is convex, but that L2 u is not convex. We will show that for uniform triangulations τn (T ) and associated Bézier nets Ln u, this problem does not occur. Lemma 4.2. (a) Let q ∈ N be a divisor of n, if Ln u is convex then Lq (Ln u) is convex. (b) Let p ∈ N be a multiple of n, then Lp (Ln u) ≡ Ln u. Proof. (a) Let τq (T ) be the triangulation of T induced by Lq (Ln u), and let R be any parallelogram with vertices A, B, C, D, which is the union of two adjacent triangles of τq (T ). We observe that since q divides n then the triangulation τn (T ) restricted to the parallelogram R constitutes a triangulation of R, denoted by τr (R), with 2r 2 triangles, where r is such that qr = n. (See Fig. 4 for case n = 4, q = 2.) The convexity of Ln u allows us to consider for the function Ln u the analytical condition (1) over each parallelogram of τr (R) whose diagonal is parallel to the diagonal of R. If we add all these inequalities, and since the values of Ln u on the vertices located on the diagonal of these parallelograms appear with minus sign, and the values of Ln u at the vertices out of the diagonal of these parallelograms appear with plus sign, we obtain that the only terms that remain in the sum are those associated with the vertices of R, namely, we get Ln u(A) + Ln u(B) − Ln u(C) − Ln u(D) > 0. Hence, Lq (Ln u) is convex on R, since Lq (Ln u) interpolates Ln u at the vertices of R. (b) Let τp (T ) be the triangulation of T induced by Lp (Ln u), it is enough to observe that, since p is a multiple of n, each triangle of τn (T ) contains r 2 triangles of τp (T ), with p = nr (see Fig. 5 for case n = 2, p = 4). 2
Fig. 4. Triangulations τ4 (T ) and τ2 (T ) and parallelogram R with vertices A, B, C, D.
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Fig. 5. Triangulations τ2 (T ) and τ4 (T ).
Theorem 4.3. The following statements are equivalent: (i) Ln u is convex. (ii) Ln u is subharmonic. (iii) Ln u is axially convex. (iv) Ln u is polyhedrally convex. Proof. Of course, the first three statements are equivalent. It remains to prove, e.g., that (i) is equivalent to (iv). Thus, (i) ⇒ (iv): Let m ∈ N∗ and p = l.c.m.(m, n) (least common multiple), then there exists r ∈ N such that p = nr. Thus, from Lemma 4.2(b), we have Lp (Ln u) ≡ Ln u, and therefore Lm (Lp (Ln u)) ≡ Lm (Ln u). The conclusion follows by applying Lemma 4.2(a), taking into account that m is a divisor of p and Ln u is convex. (iv) ⇒ (i): It is immediate since polyhedral convexity implies convexity. 2 5. Convexity and subharmonicity properties of Bernstein polynomials Bn u Since B1 u is the linear interpolant of u at the vertices of T , it is clear that it is axially and polyhedrally convex, subharmonic and w-subharmonic. For n > 2, we begin by introducing some definitions about specific conditions on the B-nets Ln u which are used below to give sufficient conditions on convexities and subharmonicity of Bn u. Thus, we say that: (i) Condition (PC) is satisfied if and only if for all β such that |β| = n − 2, and for i = 1, 2, 3, mi (β) > 0, where: mi (β) = c(β + 2ei ) − c(β + ei + ei−1 ) − c(β + ei + ei+1 ) + c(β + ei−1 + ei+1 ) and e0 = e3 , e4 = e1 (see Fig. 6 for n = i = 2). (ii) Condition (AC) is satisfied if and only if for all β such that |β| = n − 2, and for 1 6 j < k 6 3, aj k (β) > 0, where: aj k (β) = mj (β) + mk (β) = c(β + 2ej ) − 2c(β + ej + ek ) + c(β + 2ek ). (iii) Condition (S) is satisfied if and only if for all β such that |β| = n − 2, s(β) > 0, where s(β) = l12 m1 (β) + l22 m2 (β) + l32 m3 (β).
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Fig. 6. Geometric interpretation of condition m2 (β) > 0 for n = 2.
Firstly, we give necessary and sufficient conditions for the convexities and subharmonicity of the Bernstein polynomial Bn u. Theorem 5.1. For n > 2, the following characterizations hold: (i) Bn u is polyhedrally convex if and only if for each i ∈ {1, 2, 3} there holds: X mi (β)bβn−2 (λ) > 0. |β|=n−2
(ii) Bn u is axially convex if and only if for 1 6 j < k 6 3 there holds: X aj k (β)bβn−2 (λ) > 0. |β|=n−2
(iii) Bn u is subharmonic if and only if there holds: X s(β)bβn−2 (λ) > 0. |β|=n−2
Proof. Since Bn u ∈ C 2 (T ), each characterization is directly obtained from the definitions of Section 3 and from the following formula for 1 6 i, j 6 3: X Dγ2i γj Bn u(λ) = n(n − 1) bβn−2 (λ) c(β + ei−1 + ej −1 ) − c(β + ei−1 + ej +1 ) |β|=n−2
− c(β + ei+1 + ej −1 ) + c(β + ei+1 + ej +1 ) . 2 Corollary 5.2. For n > 2, if condition (PC) (respectively, (AC) or (S)) is satisfied then Bn u is polyhedrally convex (respectively, axially convex or subharmonic). Remarks. (1) For n = 2 and n = 3, condition (PC) (respectively, (AC) or (S)) is also necessary for the polyhedral convexity (respectively, axial convexity or subharmonicity) of Bn u. (2) Using the nonnegativity condition for bivariate quadratic polynomials given in (Micchelli and Pinkus, 1989), we deduce criteria of polyhedral convexity (respectively,
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axial convexity and subharmonicity) for B4 u (see Serrano-Pérez, 1997, Chapter 2, for details). (3) We note that, for n > 2, condition (PC) is also a sufficient condition for the convexity of Bn u (Chang and Davis, 1984) and, moreover, it is a condition that is equivalent to the convexity of Ln u (Dahmen and Micchelli, 1988). Finally, in relation with Remark 2 of Section 3, we obtain the following property: Proposition 5.3. If T is not obtuse-angled, then axial convexity implies subharmonicity. Proof. Let u : T → R be a continuous and axially convex function, then for each n > 2 condition (AC) is satisfied. On the other hand, for each β with |β| = n − 2, we can rewrite s(β) as s(β) = l1 l2 cos(θ3 )a12(β) + l1l3 cos(θ2 )a13(β) + l2 l3 cos(θ1 )a23(β), where θi is the (interior) angle at vertex Ai , i = 1, 2, 3. Therefore, condition (S) is satisfied, since li > 0 and cos(θi ) > 0, i = 1, 2, 3, and then, Bn u is subharmonic for all n > 2. Finally, u is subharmonic because it is a uniform limit of subharmonic functions. 2 6. Relationships between the shape properties of u, Bn u and Ln u In this section (P) will denote any property of u, given in Section 3. The results of the previous sections allow us to conclude the following: Theorem 6.1. For u ∈ C(T ), we consider the following statements: (1) u satisfies property (P). (2) For all n ∈ N∗ , Bn u satisfies property (P). (3) For all n ∈ N∗ , Ln u satisfies property (P). Then: (i) When (P) is the polyhedral convexity, the three above statements are equivalent. (ii) When (P) is the axial convexity, then (1) and (2) are equivalent, (3) implies (1), but (1) does not imply (3). (iii) When (P) is the subharmonicity, then (2) implies (1), (3) implies (1), but the converses are not always true. Proof. (i) The proof of (1) ⇒ (2) can be seen in page 475 of (Sauer, 1991). For (2) ⇒ (1), we consider for each m ∈ N∗ the operator Lm : C(T ) → C(T ) given by Lm (f ) := Lm f , for all f ∈ C(T ). From (2), we know that for each n ∈ N∗ the function (Lm ◦ Bn )u is convex. The continuity of Lm , together with the property of uniform approximation of the sequence of Bernstein polynomials, guarantee that Lm u is convex, and hence, that u is polyhedrally convex. The equivalence between (1) and (3) is immediate from Theorem 4.3. (ii) The proof of (1) ⇒ (2) can be seen in page 470 of (Sauer, 1991). The converse is deduced from the property of uniform approximation of the sequence of Bernstein polynomials.
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Fig. 7. B-coefficients over the parallelogram R of τ4 (T ).
On the other hand, if Ln u is axially convex then it is convex (Theorem 4.3), and hence, Bn u is convex. Finally, because of the property of uniform approximation of u by {Bn u}n∈N∗ and since convexity implies axial convexity, it can be deduced that (3) implies (1). To prove that the converse is not always true, it is enough to consider Example 4.1 in which u is axially convex but L2 u is not. (iii) Since u is a uniform limit of subharmonic functions, it is immediate that (2) implies (1). For the converse we consider the triangle T in R2 with vertices A1 = (0, 0), A2 = (1, 0), A3 = (−1, 1). Now, let τ (T ) = {Ti : i = 1, 2} be the triangulation of T where T1 and T2 are the triangles with vertices A1 , A2 , M and A1 , A3 , M, respectively, where M = (0, 1/2) is the midpoint of A2 A3 . Let u : T → R be the continuous function which is affine over each triangle of τ (T ) and satisfies u(A1 ) = u(A2 ) = u(M) = 0, u(A3) = 1. The function u is subharmonic because it is convex and, however, B3 u is not subharmonic since 1B3 u(A1 ) = −4 < 0. Finally, the proof of (3) ⇒ (1) follows from Theorem 4.3, part (i) of this theorem and from the fact that polyhedral convexity implies convexity, which further implies subharmonicity. For the converse, it is enough to consider T with vertices A1 = (1, 0), A2 = (0, 1) and A3 = (0, 0), and u : T → R given by u(λ1 , λ2 , λ3 ) = 2λ21 − λ22 . Then c(α) = (2α12 − α22 )/n2 for each α with |α| = n. Since 1u = 2 > 0, u is subharmonic, but there is no n ∈ N, n > 2 for which Ln u is subharmonic. This is due to the fact that over the parallelogram R of τn (T ) whose vertices, given in barycentric coordinates with respect 1 1 n−1 1 n−2 1 to T , are (0, 1, 0), (0, n−1 n , n ), ( n , n , 0), ( n , n , n ), condition (PC) is not satisfied (see Fig. 7 for n = 4). 2 Theorem 6.2. (i) If Ln u satisfies property (P), then Bn u satisfies property (P), where property (P) is either polyhedral convexity, axial convexity or subharmonicity. (ii) If (P) is the polyhedral convexity property, then the reverse statement of (i) is only true for n = 2, 3.
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(iii) If property (P) is either axial convexity or subharmonicity, then the converse statement of (i) is not always true. Proof. (i) From Theorem 4.3, Ln u is convex, namely, condition (PC) is satisfied and, consequently, conditions (AC) and (S); then, Theorem 5.1 implies (i). (ii) Follows again from Theorem 5.1 together with the following example: we consider n = 4 and c(0, 4, 0) = c(0, 0, 4) = c(0, 2, 2) = 2, c(0, 1, 3) = c(0, 3, 1) = 1, c(i, j, k) = 0, otherwise, then, it is immediate (see the nonnegativity condition for polynomials in P2 [x, y] given by Micchelli and Pinkus (1989)), that B4 u is polyhedrally convex. However, L4 u is not convex and, hence, is not polyhedrally convex. For (iii), if (P) is axial convexity, it is sufficient to consider B2 u with c(1, 1, 0) = −1/2 and c(i, j, k) = 0 otherwise, and to use Theorem 5.1 and Theorem 4.3. If (P) is subharmonicity, we consider the example given in the proof of assertion (1) ; (3) for (iii) of the previous theorem. Then, Bn u is subharmonic since condition (S) is satisfied for all n > 2 (see Theorem 5.1) though Ln u is not subharmonic. 2 Theorem 6.3. Let property (P) be either polyhedral convexity or axial convexity, and u ∈ C(T ), then: (i) If u satisfies property (P), then Bn u > Bn+1 u, for all n ∈ N∗ . (ii) If u satisfies property (P), then Bn u > u, for all n ∈ N∗ . (iii) The converses of (i) and (ii) are not always true. Proof. (i) If (P) is polyhedral convexity, the result follows since polyhedral convexity implies convexity and, therefore, {Bn u}n∈N∗ is decreasing (cf. (Chang and Davis, 1984)). If (P) is axial convexity, see page 472 of (Sauer, 1991). (ii) is immediate from (i) in virtue of the property of uniform approximation of u by {Bn u}n∈N∗ . To prove (iii) we consider the triangle T with vertices A1 = (−1, 0), A2 = (1, 0) and A3 = (0, 1) and u : T → R given by u(x, y) = −xy. We observe that: (a) u is not polyhedrally convex since L2 u is not convex; (b) u is not axially convex because it is not convex in the direction of side A1 A3 ; and however, Bn u > Bn+1 u and Bn u > u, for all n ∈ N∗ , because Bn u(x, y) = −(1 − (1/n))xy. 2 Theorem 6.4. If u ∈ C(T ) then there is no relation between the subharmonicity of u and either the decrease of the sequence {Bn u}n∈N∗ or the nonnegativity of the functions of the sequence {Bn u − u}n∈N∗ . More precisely: (i) u subharmonic does not imply that Bn u(λ) > u(λ) for all n ∈ N∗ , and hence, does not imply that Bn u(λ) > Bn+1 u(λ) for all n ∈ N∗ .
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(ii) Bn u(λ) > Bn+1 u(λ), for all n ∈ N∗ , and also, Bn u(λ) > u(λ) for all n ∈ N∗ , does not imply that u is subharmonic. Proof. (i) We consider again the last example of the proof of Theorem 6.1. There u is 1 3 7 , 4 , 32 ). subharmonic, but B1 u(λ) < u(λ) for λ = ( 32 (ii) It is sufficient to consider T the triangle with vertices A1 = (1, 0), A2 = (0, 1), A3 = (0, 0) and u : T → R given by u(λ1 , λ2 , λ3 ) = −λ1 λ2 − λ21 λ22 . Then, u is not subharmonic and, however, for each n ∈ N∗ , Bn u(λ) > Bn+1 u(λ) because: Bn u(λ1 , λ2 , λ3 ) =
−n3 + n2 − n + 1 −n2 + 3n − 2 λ λ + λ1 λ2 (λ1 + λ2 ) 1 2 n3 n3 −n3 + 6n2 − 11n + 6 2 2 + λ1 λ2 . 2 n3
Acknowledgements The authors would like to thank Professor Jesús Carnicer for his helpful suggestions and the referees for their thorough review and their comments.
References Be´ska, M. (1992), Geometric properties on Bernstein polynomials over simplices, in: Lyche, T. and Schumaker, L.L., eds., Mathematical Methods in CAGD and Image Processing, Academic Press, Boston, 1–8. Carnicer, J.M. and Dahmen, W. (1992), Convexity preserving interpolation and Powell–Sabin elements, Computer Aided Geometric Design 9, 279–289. Carnicer, J.M., Floater, M.S. and Peña, J.M. (1997), Linear convexity conditions for rectangular and triangular Bernstein–Bézier surfaces, Computer Aided Geometric Design 15, 27–38. Chang, G.Z. and Davis, P.J. (1984), The convexity of Bernstein polynomials over triangles, J. Approx. Theory 40, 11–28. Chang, G.Z. and Zhang, J.Z. (1990), Converse theorems of convexity for Bernstein polynomials over triangles, J. Approx. Theory 61, 265–278. Dahmen, W. (1991), Convexity and Bernstein–Bézier polynomials, in: Curves and Surfaces, Laurent, P.J., Le Méhauté, A. and Schumaker, L.L., eds., Academic Press, Boston, MA, 107–134. Dahmen, W. and Micchelli, C.A. (1988), Convexity of multivariate Bernstein polynomials and box spline surfaces, Studia Sci. Math. Hungar. 23, 265–287. Feng, Y.Y., Chen, F.L. and Zhou, H.L. (1994), The invariance of weak convexity conditions of B-nets with respect to subdivision, Computer Aided Geometric Design 11, 97–107. Floater, M. (1997), A counterexample to a theorem on Powell–Sabin elements, Computer Aided Geometric Design 14, 383–385. Goodman, T.N.T. (1989), Shape preserving representations, in: Mathematical Methods in Computer Aided Geometric Design, Lyche, T. and Schumaker, L.L., eds., Academic Press, Boston, MA, 333–351. He, T.-X. (1995), Shape criteria of Bernstein–Bézier polynomials over simplexes, Comput. Math. Appl. 30, 317–333.
300
J. Lorente-Pardo et al. / Computer Aided Geometric Design 16 (1999) 287–300
Lorente-Pardo, J., Sablonnière, P. and Serrano-Pérez, M.C. (1997), On convexity and subharmonicity of some functions on triangles, in: Curves and Surfaces with Applications in CAGD, Le Méhauté, A., Rabut, C. and Schumaker, L.L., eds., Vanderbilt University Press, 271–278. Lorente-Pardo, J., Sablonnière, P. and Serrano-Pérez, M.C. (1998), On the convexity of Powell–Sabin finite elements, in: Advances in Computational Mathematics, Chen, Z., Li, Y., Micchelli, C.A. and Xu, Y., eds., Marcel Dekker, Inc., 395–404. Lorente-Pardo, J., Sablonnière, P. and Serrano-Pérez, M.C. (1999), On the convexity of some piecewise quadratic Powell–Sabin surfaces, submitted to Journal C.A.M. Micchelli, C.A. and Pinkus, A. (1989), Some remarks on nonnegative polynomials on polyhedra, in: Prob. Stat. and Math. Papers in Honor of Samuel Karlin, Anderson, T., Athreya, K. and Iglebert, D., eds., Academic Press, New York, 163–186. Montel, P. (1928), Sur les fonctions convexes et les fonctions sousharmoniques. J. Math. Pures Appl., Series 9, Vol. 7, 29–60. Prautzsch, H. (1992), On convex Bézier triangles, Modélisation Math. Anal. Numer. 26, 23–36. Sauer, T. (1991), Multivariate Bernstein polynomials and convexity, Computer Aided Geometric Design 8, 465–478. Schmid, H.J. (1975), Bernsteinpolynome, Unpublished manuscript. Serrano-Pérez, M.C. (1997), Sobre subarmonía y diversos tipos de convexidad de ciertas funciones polinomiales a trozos, Doctoral Dissertation, University of Granada, Spain.
