Are rational Bézier surfaces monotonicity preserving?

Computer Aided Geometric Design 24 (2007) 303–306 www.elsevier.com/locate/cagd

Are rational Bézier surfaces monotonicity preserving? J. Delgado a,∗,1 , J.M. Peña b,1 a Departamento de Matemáticas, Universidad de Oviedo, Spain b Departamento de Matemática Aplicada, Universidad de Zaragoza, Spain

Received 22 December 2006; received in revised form 26 March 2007; accepted 29 March 2007 Available online 3 April 2007

Abstract We present examples which show that rational Bézier surfaces are not axially monotonicity preserving and that surfaces generated by the tensor product of rational bases are not monotonicity preserving. Besides, we prove that surfaces generated by rational functions through Bernstein basis on a triangle are not axially monotonicity preserving. © 2007 Published by Elsevier B.V. Keywords: Monotonicity preservation; Rational surfaces

1. Introduction Bézier curves present variation diminishing and even optimal shape preserving properties (cf. Farin, 2002; Peña, 1999). In (Prautzsch and Gallagher, 1992) it was shown that simple extensions of a geometric variation diminishing property do not work even for tensor product Bézier surfaces and Bézier triangles. One of the simplest shape properties is monotonicity. In (Floater and Peña, 1998, 2000) the concept of monotonicity preservation was extended to surfaces and it was proved that the tensor product Bernstein basis and the Bernstein basis on a triangle are axially monotonicity preserving and even monotonicity preserving. In this paper we analyze the monotonicity preserving properties of the rational Bézier surfaces in both cases, four-sided and three-sided Bézier patches. In Section 2, we present examples which show that rational Bézier surfaces are not axially monotonicity preserving and that surfaces generated by the tensor product of rational bases do not satisfy the stronger property of monotonicity preservation, although they are axially monotonicity preserving. In Section 3, we prove that surfaces generated by rational functions with positive weights through Bernstein basis on a triangle are axially monotonicity preserving only in the trivial case of all weights equal to 1, that is, in the case of Bézier triangles.

* Corresponding author.

E-mail address: [email protected] (J. Delgado). 1 Partially supported by the Spanish Research Grant MTM2006-03388 and by Gobierno de Aragón and Fondo Social Europeo.

0167-8396/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.cagd.2007.03.006

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2. Rational Bézier surfaces are not monotonicity preserving nLet (u0 , . . . , un ) be a system of functions defined on Ω. In CAGD this system is usually normalized, i.e., i=0 ui (x) = 1 for all x ∈ Ω, and nonnegative, i.e., ui (x)  0 for all x ∈ Ω and i ∈ {0, 1, . . . , n}. From now on, we will say that such a system is normalized nonnegative. Given a normalized nonnegative system of bivariate func0j n 0j n tions U = (uij (x, y))0im defined on [a1 , b1 ] × [a2 , b2 ] and a sequence of values in R, (cij )0im , let us consider the corresponding generated bivariate function F (x, y) =

m  n 

cij uij (x, y),

(x, y) ∈ [a1 , b1 ] × [a2 , b2 ].

(1)

i=0 j =0

Now we shall associate a control net p with the function F . Given two strictly increasing sequences of abscissae α = (α0 , α1 , . . . , αm ) and β = (β0 , β1 , . . . , βn ), we define the control net p: [α0 , αm ] × [β0 , βn ] to be the unique function which satisfies the interpolation conditions p(αi , βj ) = cij , for all i = 0, 1, . . . , m, and j = 0, 1, . . . , n, and is bilinear on each rectangle Rij = [αi , αi+1 ] × [βj , βj +1 ]. A bivariate function g is increasing in a direction d = (d1 , d2 ) ∈ R2 , if g(x + λd1 , y + λd2 )  g(x, y), λ > 0. In particular, the control net p can be increasing in a direction d. Floater and Peña (1998) characterized this situation in the following way: Lemma 1. The control net p is increasing in the direction d = (d1 , d2 ) in R2 if and only if for i = 0, 1, . . . , m − 1 and j = 0, 1, . . . , n − 1, d1 Δ1 ci,j +l + d2 Δ2 ci+k,j  0,

k, l ∈ {0, 1},

where Δ1 cij := (ci+1,j − cij )/(αi+1 − αi ) and Δ2 cij := (ci,j +1 − cij )/(βj +1 − βj ). 0j n

Given a sequence (cij )0im , Λ1 cij := ci+1,j − cij for i = 0, 1, . . . , m − 1 and j = 0, 1, . . . , n, and Λ2 cij := ci,j +1 − cij for i = 0, 1, . . . , m and j = 0, 1, . . . , n − 1. In (Floater and Peña, 1998) two concepts of monotonicity preservation for surfaces were introduced. The system U preserves monotonicity with respect to the abscissae α and β if when the control net p of the function F in (1) is increasing in any direction d in R2 then so is F . The system U preserves axial monotonicity if, for any abscissae α and β, when p is increasing in the direction d = (1, 0) or d = (0, 1) then so is F . Let F be a rational Bézier surface defined in the following way: F (x, y) =

m  n  i=0 j =0

wij bim (x)bjn (y) , cij m n m n i=0 j =0 wij bi (x)bj (y)

(x, y) ∈ [0, 1]2 ,

(2)

 0j n where (wij )0im is a sequence of positive weights and bik (t) = ki t i (1 − t)k−i , i = 0, 1, . . . , k, are the Bernstein polynomials of degree k. Now let us consider a rational Bézier surface (2) with m = n = 1 and, differentiating the previous expression respect to x, we get 2 ci b2 (y) ∂F (x, y) =   i=0 i 2 , 1 1 1 1 ∂x i=0 j =0 wij bi (x)bj (y) where c0 = w00 w10 (Λ1 c00 ), c2 = w01 w11 (Λ1 c01 ) and  1 w00 w11 (Λ1 c01 ) + w01 w10 (Λ1 c00 ) + (w00 w11 − w01 w10 )(Λ2 c00 ) . 2 Then, for the particular case where w00 = w11 = 2, w01 = w10 = 1, c00 = 10, c01 = 5, c10 = 11 and c11 = 6, the previous formulas become c1 =

2b02 (y) − 5b12 (y) + 2b22 (y) ∂F (x, y) = . 1 1 ∂x (2b0 (x)b0 (y) + b01 (x)b11 (y) + b11 (x)b01 (y) + 2b11 (x)b11 (y))2

(3)

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305

Fig. 1. Plot of (3) on [0, 1] × [0, 1].

Obviously, with this choice we have Δ1 c00 = Δ1 c01 > 0 for any strictly increasing sequence α. So, by Lemma 1, the control net of F is increasing in the direction (1, 0). However, ∂F (1/2,1/2) = −2/3 < 0 (see also Fig. 1). Hence, in ∂x general, rational Bézier surfaces are not even axially monotonicity preserving. Let us now consider the following particular case of rational Bézier surfaces: F (x, y) =

m  n  i=0 j =0

wj bjn (y) wi bm (x) n , cij m i m n i=0 wi bi (x) j =0 w j bj (y)

(x, y) ∈ [0, 1]2 ,

(4)

with (wi )0im and (w j )0j n two sequences of positive weights. The rational surfaces defined in (4) possess a tensor product structure, in contrast to the more general rational surfaces of (2). In Theorem 1 of (Delgado and Peña, 2005), we showed that the surfaces (4) are axially monotonicity preserving. So, the following question arises: are the surfaces (4) monotonicity preserving? The following example shows that the answer is negative even for m = n = 1. y Let us consider a normalized nonnegative system (uxi ⊗ uj )0i,j 1 , where uxi =

wi bi1 (x) w0 b01 (x) + w1 b11 (x)

,

wj bj1 (y)

y

uj =

w0 b01 (y) + w1 b11 (y)

,

(5)

with wi , wj positive weights. Let us suppose that this system is monotonicity preserving with respect to the abscissae α = (α0 , α1 ) and β = (β0 , β1 ). Proposition 5.1 of (Floater and Peña, 1998) states that, if a normalized nonnegative y 0j n system (uxi ⊗ uj )0im is differentiable and monotonicity preserving with respect to the abscissae α and β, then m n y x i=0 αi ui (x) and j =0 βj uj (y) are linear functions of x and y, respectively. Therefore, if the rational system y (uxi ⊗ uj )0i,j 1 with positive weights given by (5) is monotonicity preserving with respect to the abscissae α = (α0 , α1 ) and β = (β0 , β1 ), we have that 1  i=0

αi

wi bi1 (x) w0 b01 (x) + w1 b11 (x)

and

1  j =0

βj

wj bj1 (y) w 0 b01 (y) + w1 b11 (y)

0 (1−x)+α1 w1 x must be linear functions of x and y, respectively. Then α0 w = cx for all x and some constant real number w0 (1−x)+w1 x c > 0. The last formula is equivalent to α0 w0 + (α1 w1 − α0 w0 )x = cw0 x + c(w1 − w0 )x 2 , which implies w0 = w1 . Analogously, we can deduce that w0 = w1 .

3. Triangular rational Bézier surfaces are not axially monotonicity preserving Let us recall that any point τ in a plane can be expressed in terms of its barycentric coordinates with respect to any nondegenerate triangle T in that plane with vertices T0 , T1 and T2 : τ = 2i=0 τi Ti ( 2i=0 τi = 1). If τ ∈ T , then τi  0, i = 0, 1, 2. Let i = (i0 , i1 , i2 ) be a multi-index where i0 , i1 , i2 ∈ Z+ = {0, 1, 2, . . .} and let us denote by |i| the sum i0 + i1 + i2 . Given n  1, let us consider for each i such that |i| = n a function φi : T → R. We shall refer to them as a system and write (φi )|i|=n . Then, given (ci )|i|=n a sequence of coefficients in R, we can define a function

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 F : T → R, as F (τ ) = |i|=n ci φi (τ ), τ ∈ T . Let us consider the following points xi = in0 T0 + in1 T1 + in2 T2 , |i| = n. Then we define the control net of F as the function p : T → R, which is linear on each subtriangle of T and satisfies p(xi ) = ci , |i| = n. Floater and Peña (2000) provided several generalizations of the concept of monotonicity preservation for curves. A system (φi )|i|=n is axially monotonicity preserving (AMP) if the function F is increasing in the direction T1 − T0 , T2 − T1 or T0 − T2 whenever the control net p is increasing in the same direction. In (Floater and Peña, 2000) it was i proved that the Bernstein polynomials of degree n on a triangle, (bin )|i|=n defined by bin (τ ) = i0 !in!1 !i2 ! τ00 τ1i1 τ2i2 , |i| = n, are AMP and even satisfy stronger monotonicity preserving properties.  Now let us consider the rational Bernstein basis of order n (φi )|i|=n given by φi = wi bin /( |i|=n wi bin ), where (wi )|i|=n is a sequence of positive weights. The following result shows that the Bernstein basis on a triangle is the unique rational Bernstein basis which is AMP. Theorem 2. If a rational Bernstein basis on a triangle with positive weights is AMP, then wi = wj for all i, j such that |i| = |j| = n. Proof.According to Proposition 2.9 of  (Floater and Peña, 2000), if a system(Ψi )|i|=n is AMP then the function sums |j|=n,j0 =i0 Ψj (i0 = 0, 1, . . . , n), |j|=n,j1 =i1 Ψj (i1 = 0, 1, . . . , n) and |j|=n,j2 =i2 Ψj (i2 = 0, 1, . . . , n) are constants in the directions T2 − T1 , T0 − T2 and T1 − T0 , respectively. So, if the rational Bernstein basis (φi )|i|=n on a triangle with positive weights is AMP, in particular the functions φ(n,0,0) , φ(0,n,0) and φ(0,0,n) are constants in the directions T2 − T1 , T0 − T2 and  T1 − T0 , respectively. So, taking into account that φ(n,0,0) is constant in the direction T2 − T1 and denoting W := |i|=n φi , we can deduce that W (τ ) =

n w(n,0,0) b(n,0,0) (τ )

K0 (τ0 )

,

for all τ0 ∈ (0, 1) and for all τ1 , τ2  0 such that τ1 + τ2 = 1 − τ0 , where K0 is a function such that 0 < K0 (τ0 ) < 1 for all τ0 ∈ (0, 1). Analogously, we can deduce that W (τ ) =

n w(0,n,0) b(0,n,0) (τ )

K1 (τ1 )

and W (τ ) =

n w(0,0,n) b(0,0,n) (τ )

K2 (τ2 )

,

for all τ1 ∈ (0, 1) and τ0 , τ2  0 such that τ0 + τ2 = 1 − τ1 , and for all τ2 ∈ (0, 1) and τ0 , τ1  0 such that τ0 + τ1 = 1 − τ2 , respectively, where Ki (i = 1, 2) is a function such that 0 < Ki (τi ) < 1 for all τi ∈ (0, 1). Taking into account n n n (τ ), b(0,n,0) (τ ) and b(0,0,n) (τ ) only depends on τ0 , τ1 the three different ways of expressing W (τ ) and that b(n,0,0) and τ2 , respectively, we can conclude that W (τ ) ≡ c, for some positive constant c, for all τ in the interior of T (i.e., τ ∈ T such that τ0 , τ1 , τ2 > 0 and τ0 + τ1 + τ2 = 1). Since W is a continuous function on all τ ∈ T we have W (τ ) ≡ c for all τ ∈ T . Finally, since (bin )|i|=n is a basis and W is a linear combination of their basis functions,there exists a unique sequence of weights (wi )|i|=n such that W (τ ) = c for all τ ∈ T and, taking into account that |i|=n bin = 1, we deduce that the unique possibility is wi = c for all i with |i| = n. 2 References Delgado, J., Peña, J.M., 2005. On efficient algorithms for the evaluation of rational tensor product surfaces. Mathematical methods for curves and surfaces. In: Modern Methods in Mathematics. Tromso 2004. Nashboro Press, Brentwood, TN, pp. 115–124. Farin, G., 2002. Curves and Surfaces for Computer Aided Geometric Design, fifth ed. Academic Press, Inc., San Diego, CA. Floater, M.S., Peña, J.M., 1998. Tensor-product monotonicity preservation. Adv. Comput. Math. 9, 353–362. Floater, M.S., Peña, J.M., 2000. Monotonicity preservation on triangles. Math. Comp. 69, 1505–1519. Peña, J.M. (Ed.), 1999. Shape Preserving Representations in Computer Aided-Geometric Design. Nova Science Publishers, Commack, NY. Prautzsch, H., Gallagher, T., 1992. Is there a geometric variation diminishing property for B-spline or Bézier surfaces? Comput. Aided Geom. Design 9, 119–124.