Monotonicity preserving representations of curves and surfaces

Jorge Delgado
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  • Departamento de Matemática Aplicada, Universidad de Zaragoza, Escuela Universitaria Politécnica de Teruel, 44003, Teruel, Spain
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and Juan Manuel Peña
  • Departamento de Matemática Aplicada, Universidad de Zaragoza, Escuela Universitaria Politécnica de Teruel, 44003, Teruel, Spain
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Abstract

In this paper we revisit the problem of monotonicity preservation of curves and surfaces and we provide some new proofs and open problems. In particular, we prove a new formula for the derivation of rational Bézier curves. We also deal with the rational monotonicity preservation of rational Bézier surfaces and a related conjecture is presented.

1 Introduction

Recently there have been important advances on the stability and accuracy of algorithms in Computer-Aided Geometric Design (CAGD), as can be seen in [7,8,1012]. For the computation of curves and surfaces in CAGD, another very important topic is shape preservation. One of the simpler shape properties is monotonicity, to which we devote this paper and for which important problems are still open, as will be shown. These open problem arise in particular for surface design. In contrast to the variation diminishing properties of Bézier curves, tensor product Bézier surfaces and Bézier triangles do not satisfy simple extensions of these properties, as shown in [21]. Several definitions of monotonicity preserving properties for these surfaces were introduced in [16] and [17], including the simplest of them: axial monotonicity preservation. As for monotonicity preserving properties for rational Bézier surfaces, it has been proved only the axial monotonicity preservation of surfaces generated by the tensor product of two univariate rational Bernstein bases (see [5, 6, 16]).

Monotonicity preservation for curves has been deeply studied and we recall in Section 2 some basic results. In Section 2, we also include a new formula for the derivation of rational Bézier curves, which has its own interest and will be used here to give a direct proof of the monotonicity preservation of these curves. Section 3 is devoted to the monotonicity preservation for surfaces, where there are still open problems, as we had announced before and we shall comment at the end of the paper. In Subsection 3.1 we consider triangular patches and in Subsections 3.2 and 3.3 rectangular patches, considering the rational case in Subsection 3.3. In Subsection 3.3 we conjecture that there are no more axially monotonicity preserving surfaces, that is, a rational bilinear patch is axially monotonicity preserving if and only if it corresponds to the tensor product of two univariate rational Bernstein bases.

2 Monotonicity preservation of curves

In this section we revisit the main results on the monotonicity preserving representation of curves and we also include a new formula for the derivation of rational Bézier curves, which has its own interest and further potential applications in addition to its application in this section.

Let 𝒰 be a vector space of real functions defined on [a,b] ⊆ ℝ and (u0,...,un) a basis of 𝒰 . If a control polygon P0 ···Pn is given then we define a parametric curve

γ(t)=i=0nPiui(t),t[a,b].

In CAGD the functions u0,...,un are usually nonnegative and i=0nui(t)=1t[a,b] (i.e. the system (u0,...,un) is normalized) and in this case we say that (u0,...,un) is a blending system. The convex hull property is an important property for curve design: for any control polygon, the curve always lies in the convex hull of the control polygon. The convex hull property holds if and only if (u0,...,un) is a blending system. In interactive design we want that the shape of a parametrically defined curve mimics the shape of its control polygon; thus we can predict or manipulate the shape of the curve by choosing or changing the control polygon suitably. One of the simplest shape properties is monotonicity, which will be now described.

In the design of curves it is required that the sense of the path tracing of the curve and the polygon agree. Let us draw the control polygon starting from P0 and finishing in Pn, and the curve γ(t) taking all values of t starting from a and finishing at b. Let P0,··· ,Pn be control points in ℝk and let γ be the curve defined by (1). A surjective affine mapping T : ℝk → ℝ can be interpreted as the projection of the space onto some line. In the case that the projection of the control polygon onto this line is increasing, that is, T (P0) ≤ ··· ≤ T (Pn), then the projection of the corresponding curve γ(t) onto that line T (γ(t)) must also be increasing. So, we see that this shape preserving property is essentially 1-dimensional. We can see a graphical example of this property in Figure 1. A system satisfying this property is said to be a monotonicity preserving system. This shape preserving property can be formalized as follows:

Fig. 1
Fig. 1

Monotonicity preservation of a representation of curves

Citation: Applied Mathematics and Nonlinear Sciences 1, 2; 10.21042/AMNS.2016.2.00041

Definition 1

A system of functions (u0,...,un) is monotonicity preserving (resp., strictly monotonicity preserving) if for any α0α1 ≤ ... ≤ αn (resp., α0 < α1 < ... < αn) in ℝ, the function i=0nαiui is increasing (resp., strictly increasing).

Some properties and applications of monotonicity preserving systems can be seen in [24]. A proof of the following result, which characterizes monotonicity preserving systems, appears in Proposition 2.3 of [4].

Proposition 1

Let (u0,...,un) be a system of functions defined on an interval [a,b]. Letvi:=j=inujfor i ∈ {0,1,...,n}. Then:

  1. (u0,...,un) is monotonicity preserving if and only if v0is a constant function and the functions vi are increasing for i = 1,...,n.
  2. (u0,...,un) is strictly monotonicity preserving if and only if (u0,...,un) is monotonicity preserving and the functionj=1nvjis strictly increasing.

2.1 Nonrational curves

Bézier curves provide the most usual representation of curves in CAGD. These curves are represented in terms of Bernstein polynomials. The Bernstein polynomials of degree n are defined by

bin(x)=(ni)xi(1x)ni,x[0,1],

for all i = 0,1,...,n. The system (b0n,,bnn) forms a basis of the space of polynomials of degree at most n, ∏n. For more details in Bernstein polynomials and applications see [13] and [14]. The following result is well known and an argument for its proof can be found in p. 381 of [4].

Proposition 2

The Bernstein basis(b0n,,bnn)preserves monotonicity strictly.

Bézier curves are polynomial curves. But when working with polynomials some problems arise. For example, polynomial curves have a global behavior, that is, if one modifies, even slightly, one of its control points, then the modification affects to the whole function. In addition, polynomial curves with high degree can present big oscillations. Piecewise polynomials are the solution to avoid these poblems in diverse fields of mathematics including CAGD. In this field, the role played by Bernstein polynomials in Béizer cuves is played by B-splines in the case of piecewise polyomial curves. Let us consider d ∈ ℕ0 and a sequence of nondecreasing real numbers x=(xj)j=1n+d+1 with at least d + 2 elements. The j-th B-spline of degree d with nodes x is defined by

Nj,d,x(x)=xxjxj+dxjNj,d1,x(x)+xj+d+1xxj+d+1xj+1Nj+1,d1,x(x),

for all x ∈ ℝ, with

Nj,0,x(x)={1,if xjx<xj+1,0,in other case.

For more details in B-splines and applications see [13] and [20]. The following result is also well known and an argument for its proof can be found in pp. 381–382 of [4].

Proposition 3

The B-spline basis(Ni,d,x)i=1nassociated to a sequence of nodesx=(xj)j=1n+d+1preserves monotonicity.

2.2 Rational curves

In this subsection, we include a new formula for the derivation of rational Bézier curves, which has its own interest and will be used here to give a direct proof of the monotonicity preservation of these curves.

In CAGD given a system of functions U = (u0,...,un) defined in [a,b], it is usual to construct a rational system of functions (r0,...,rn) from a sequence of positive weights (wi)i=0n defined by

ri(t)=wiui(t)i=0nwiui(t),t[a,b].

The weights act as shape parameters.

Rational Bernstein bases arise taking U=(b0n,,bnn) in (4) and the corresponding curve is called rational Bézier curve. It is known, from the results of [4], that rational bases constructed from the Bernstein basis of ∏n are also monotonicity preserving. Now let us prove this result from a new and different approach. We will use a new formula for the derivative of a rational Bézier curve presented in the folowing result. This formula is not only important for proving that rational Bernstein bases are monotonicity preserving, but it can be also useful for the problems studied in [15, 18, 22, 24].

Proposition 4

Let us consider a rational Bézier curve g given by

γ(t)=i=0nPiwibin(t)i=0nwibin(t),t[0,1],

with (wi)0≤ina sequence of positive weights and P0 ···Pn the control polygon. Then we have

γ(t)=ni=0n1j=in1(j+1i)n(j+1)(ni)(k=ij(ΔPk))bin1(t)bjn1(t)(i=0nwibin(t))2,

wherePk := Pk+1Pk.

Proof. Taking into account that

γ(t)=N(t)D(t),

where N(t):=i=0nPiwibin(t) and D(t):=i=0nwibin(t), we deduce

γ(t)=N(t)D(t)N(t)D(t)(D(t))2.

Since

N(t)=ni=0n1Δ(Piwi)bin1(t),D(t)=ni=0n1(Δwi)bin1(t),

we can write (5) in the following way

γ(t)=ni=0n1j=0n[Δ(Piwi)wjPjwj(Δwi)]bin1(t)bjn(t)(D(t))2.

From the previous formula, since ∆(Pi wi) = wi+1 (∆Pi) + (∆wi)Pi, we deduce

γ(t)(D(t))2=ni=0n1j=0n[wi+1wj(ΔPi)+(Δwi)wjPi(Δwi)wjPj]bin1(t)bjn(t).

Applying the univariate de Casteljau algorithm to the right-hand of the previous formula we derive

γ(t)(D(t))2=i=0n1j=0n1k=01[wi+1wj+k(ΔPi)+(Δwi)wj+kPi(Δwi)wj+kPj+k]bin1(t)bjn1(t)bk1(t).

Now, rearranging the right-hand of the previous formula, we have

γ(t)(D(t))2=i=0n1j=0i1k=01[wi+1wj+k(ΔPi)+(Δwi)wj+kPi(Δwi)wj+kPj+k]bin1(t)bjn1(t)bk1(t)+i=0n1k=01[wi+1wi+k(ΔPi)+(Δwi)wi+kPi(Δwi)wi+kPi+k]bin1(t)bin1(t)bk1(t)+i=0n1j=i+1n1k=01[wi+1wj+k(ΔPi)+(Δwi)wj+kPi(Δwi)wj+kPj+k]bin1(t)bjn1(t)bk1(t)

Then we can deduce that the previous expression can be written as

γ(t)(D(t))2=Σi=0n1Σj=0i1Σk=01[wi+1wj+kΣl=j+ki(ΔPl)wiwj+kΣl=j+ki1(ΔPl)]bin1(t)bjn1(t)bk1(t)+Σi=0n1Σk=01wiwi+1(ΔPi)bin1(t)bin1(t)bk1(t)+Σi=0n1Σj=i+1n1Σk=01[wi+1wj+kΣl=i+1j+k1(ΔPl)+wiwj+kΣl=ij+k1(ΔPl)]bin1(t)bjn1(t)bk1(t).

By performing an index change and reordering, we have

γ(t)(D(t))2=i=0n1j=i+1n1k=01[wj+1wi+kl=i+kj(ΔPl)wjwi+kl=i+kj1(ΔPl)]bin1(t)bjn1(t)bk1(t)+i=0n1k=01wiwi+1(ΔPi)bin1(t)bin1(t)bk1(t)+i=0n1j=i+1n1k=01[wi+1wj+kl=i+1j+k1(ΔPl)+wiwj+kl=ij+k1(ΔPl)]bin1(t)bjn1(t)bk1(t).

Extending the sum on k and operating, we deduce that

γ(t)(D(t))2=i=0n1j=i+1n1[wiwj+1l=ij(ΔPl)wi+1wjl=i+1j1(ΔPl)]bin1(t)bjn1(t)+i=0n1wiwi+1(ΔPi)bin1(t)bin1(t).

From the previous formula we can derive in a straightforward way

γ(t)(D(t))2=i=0n1j=in1[wiwj+1l=ij(ΔPl)]bin1(t)bjn1(t)i=0n1j=i+1n1[wi+1wj+1l=i+1j1(ΔPl)]bin1(t)bjn1(t).

Then, arranging the indexes and operating, we have

γ(t)(D(t))2=i=0n1j=in1[wiwj+1l=ij(ΔPl)](bin1(t)bjn1(t)bi1n1(t)bj+1n1(t)).

So the result follows from the last expression.

As a consequence, we can derive the following result.

Corollary 5

A rational Bézier basis (r0,...,rn) defined by

ri(t)=wibin(t)i=0nwibin(t),

with (wi)0≤ina sequence of positive weights, is strictly monotonicity preserving.

Proof. Let us consider the function

γ(t)=i=0nαiwibin(t)i=0nwibin(t)

with α0 < α1 < ··· < αn. Differentiating the previous formula we have

γ(t)=ni=0n1j=in1(j+1i)n(j+1)(ni)(k=ij(Δαk))bin1(t)bjn1(t)(i=0nwibin(t))2>0,t(0,1),

and so γ is strictly increasing. Therefore, the rational basis is strictly monotonicity preserving.

3 Monotonicity preservation of surfaces

In [16] and [17], some generalizations of the monotonicity preservation of curves have been extended for surfaces defined on rectangular and triangular patches, respectively. Subsection 3.1 considers triangular patches, and rectangular patches are analyzed in the remaining subsections. In Subsection 3.3 we present a conjecture on axially monotonicity preserving rational Bézier surfaces defined on rectangular patches.

3.1 Monotonicity preservation of surfaces defined in triangular patches

Any point τ in a plane can be expressed in terms of its barycentric coordinates with respect to any nonde-generate triangle 𝒯 in that plane with vertices T0, T1 and T2:

τ=i=02τiTi,i=02τi=1.

If t ∈ 𝒯 , then τi ≥ 0, i = 0,1,2. Let i = (i0,i1,i2) denote a multi-index where i0,i1,i2 ∈ ℕ0 = {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 F : 𝒯 → ℝ, as

F(τ)=|i|=nciϕi(τ),τT.

Let us consider the following points xi=i0nT0+i1nT1+i2nT2, with |i| = n. Then we define the control net of F as the function

p:T,

which is linear on each subtriangle of 𝒯 and satisfies

p(xi)=ci,|i|=n.

The control net is important in interactive design because it is a mesh of points used to control the shape of the surface. So, in [17] Peña and Floater provided several generalizations of the concept of monotonicity preservation of curves to surfaces.

Definition 2

A system (ϕi)|i|=n is axially monotonicity preserving (AMP) if the function F is increasing in the direction T1T0, T2T1 or T0T2 whenever the control net p is increasing in the same direction.

In [17] it was proved that the Bernstein polynomials of degree n on a triangle, (bin)|i|=n defined by

bin(τ)=n!i0!i1!i2!τ0i0τ1i1τ2i2,|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=wibin|i|=nwibin, where (wi)|i|=n is a sequence of positive weights. In [6] it was proved that the Bernstein basis on a triangle is the unique rational Bernstein basis which is AMP.

Theorem 6

(see Theorem 2 of [6]) If a rational Bernstein basis on a triangle with positive weights is AMP, then wi = wjfor alli,jsuch that |i| = |j| = n.

3.2 Monotonicity preservation of nonrational surfaces defined in rectangular patches

Given a normalized nonnegative system of bivariate functions U=(uij(x,y))0im0jn defined on [a1 , b1] × [a2 , b2] and a sequence of values in ℝ, (cij)0im0jn, let us consider the corresponding generated bivariate function

F(x,y)=i=0mj=0ncijuij(x,y),(x,y)[a1,b1]×[a2,b2].

Now we shall associate a control net p with the function F. Given two strictly increasing sequences of abscissae

α=(α0,α1,,αm)andβ=(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)=cijfor all i=0,1,m and j=0,1,,n,

and is bilinear on each rectangle

Rij=[αi,αi+1]×[βj,βj+1].

As in the triangular case, the control net is used to control the shape of the surface in interactive design. 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. In [16] Floater and Peña characterized this situation as follows:

Lemma 7

The control net p is increasing in the direction d = (d1,d2) in2if and only if for i = 0,1,...,m − 1 and j = 0,1,...,n − 1,

d1Δ1ci,j+l+d2Δ2ci+k,j0,k,l{0,1},

where1cij := (ci+1,jcij)/(αi+1αi) and2cij := (ci,j+1cij)/(βj+1βj).

Given a sequence (cij)0im0jn, Λ1cij := ci+1,jcij for i = 0,1,...,m − 1 and j = 0,1,...,n, and Λ2cij := ci,j+1cij for i = 0,1,...,m and j = 0,1,...,n − 1.

Remark 1

As a consequence of Lemma 7 we have that the control net p is increasing in the X-axis direction d = (1,0) if and only if for i = 0,1,...,m − 1 and j = 0,1,...,n − 1, Λ1cij ≥ 0. Analogously, the control net p is increasing in the Y-axis direction d = (0,1) if and only if for i = 0,1,...,m − 1 and j = 0,1,...,n − 1, Λ2cij ≥ 0

In [16] several concepts of monotonicity preservation for rectangular patches were introduced.

Definition 3

  • The system U preserves monotonicity with respect to the abscissae α and β if when the control net p of the function F in (8) 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.

Now let us consider the particular case of tensor product surfaces. So let us consider two systems of univariate functions U1=(u01,u11,,um1) and U2=(u02,u12,,un2) defined on [a1,b1] and [a2,b2], respectively. Then we consider the system of tensor-product functions

U=U1U2=(ui1uj2)i=0,1,,mj=0,1,,n

defined on the rectangle [a1,b1] × [a2,b2]. If the systems U1 and U2 are blending then the system U is also blending. Given cij ∈ ℝ and taking uij=ui1uj2 in (8), i ∈ {0,1,...,m} and j ∈ {0,1,...,n}, the system U generates the following parametric function:

F(x,y)=i=0mj=0ncijui1(x)uj2(y),(x,y)[a1,b1]×[a2,b2].

The next result characterizes axial monotonicity preservation:

Proposition 8

(Proposition 2.3 of [16]) The blending system U in (9) preserves axial monotonicity if and only if the functions

vi1:=k=imuk1,i=1,,m,andvj2:=k=jnuk2,j=1,,n,

are increasing.

As a consequence, we can derive the following result.

Corollary 9

If the blending univariate systems U1and U2are monotonicity preserving, then the blending bivariate system U preserves axial monotonicity.

Taking into account that systems of Bernstein polynomials and of B-splines are monotonicity preserving its corresponding tensor products are axially monotonicity preserving.

Proposition 10

  • The tensor product of Bernstein bases
    (bim(x)bjn(y))0im0jn
    preserves monotonicity axially.
  • Given two sequence of nodesx=(xi)i=1m+d1+1andy=(yj)j=1n+d2+1, the tensor product of the corresponding B-spline bases
    (Ni,d1,x(x)Nj,d2,y(y))1im1jn
    preserves monotonicity axially.

3.3 Monotonicity preservation of rational Bézier surfaces defined in rectangular patches.

Let F be a rational Bézier surface defined as

F(x,y)=i=0mj=0ncijwijbim(x)bjn(y)i=0mj=0nwijbim(x)bjn(y),(x,y)[0,1]×[0,1],

where (wij)0im0jn is a sequence of positive weights and bik(t), i = 0,1,...,k, are the Bernstein polynomials of degree k. In [6] it was proved that rational Bézier surfaces are not, in general, even axially monotonicity preserving. In addition, in that paper a particular case of rational Bézier surfaces was considered, the surfaces

F(x,y)=i=0mj=0ncijwibim(x)i=0mwibim(x)w¯jbjn(y)j=0nw¯jbjn(y),(x,y)[0,1]×[0,1],

generated by the bases formed by the tensor product of univariate rational Bernstein bases

(wibim(x)i=0mwibim(x))i=0m(w¯jbjn(y)j=0nw¯jbjn(y))j=0n,

where (wi)i=0m and (w¯j)j=0n are two sequences of strictly positive weights. Let us observe that taking wij=wiw¯j, i = 0,1...,m and j = 0,1...,n, in (10) we obtain the surface in (11). By Corollary 9 and Corollary 5 these rational bases preserve monotonicity axially as it was pointed in [5]. We conjecture that the converse also holds. A particular case of this problem has been considered in [9]. Let us consider the system

B=(wijbim(x)bjn(y)i=0mj=0nwijbimx)bjn(y))0im0jn

Now we present our conjecture:

Conjecture 11

The system B of (13) is axially monotonicity preserving if and only if it corresponds to the tensor product of two univariate rational Bernstein systems.

If m = n = 1, the surface (10) is called a rational bilinear patch (see [23]) and we shall see that the conjecture holds in this case. Rational bilinear patches have been considered recently in [19] and in [23]. The quotient

W:=w00w11w01w10

is called in [23] shape parameter of the rational bilinear patch. In general, it is desirable that W to be sufficiently close to 1, as remarked in page 3 of [23]. If W = 1, it corresponds to the case when the patch exhibits minimal curvature along a diagonal of the control net.

Now we characterize in several ways for the case m = n = 1 the rational bases preserving the axial monotonicity.

Theorem 12

Let us consider the system

B=(wijbi1(x)bj1(y)i=01j=01wijbi1(x)bj1(y))0i10j1

and the rational bilinear patch given by (10) with m = n = 1. Then the following properties are equivalent:

  • (i) B is axially monotonicity preserving.
  • (ii) The optimal shape parameter W of (14) is 1.
  • (iii) B can be expressed as in (12) for m = n = 1.

Proof. Let us consider a rational Bézier surface (10) with m = n = 1.

(i) ⇔ (ii). Differentiating the surface respect to x we obtain

F(x,y)x=F1(y)+F2(y)(i=01j=01wijbi1(x)bj1(y))2,

where

F1(y)=w00w10(Λ1c00)b02(y)+12[w00w11(Λ1c01)+w01w10(Λ1c00)]b12(y)+w01w11(Λ1c01)b22(y),F2(y)=(w00w11w01w10)(Λ2c00)b12(y).

By Remark 1, a control net p is increasing in the X-axis direction (1,0) if and only if Λ1cij ≥ 0 for i = 0,1,...,m−1 and j = 0,1,...,n−1. Taking c00 = c10 = c ≠ = 0 and c01 = c11 = 2c we have that Λ1c00 = Λ1c01 = 0 and so the control net is increasing in the X-axis direction. With the previous choice of the coefficients we have that F1(y) = 0 and F2(y)=c(w00w11w01w10)b11(y). Taking into account that (i=01j=01wijbi1(x)bj1(y))2>0, F is increasing in the X-axis direction if and only if F2(y)=c(w00w11w01w10)b11(y)0 for all (x,y) ∈ [0,1]2. Since c can be any real number F2(y) ≥ 0 if and only if w00w11w01w10 = 0, which is equivalent to rank

rank(w00w01w10w11)=1.

Analogously, it can be proved that monotonicity preservation in the Y-axis direction is also equivalent to (16). Finally, (16) is clearly equivalent to (ii).

(ii) ⇔ (iii). Since (ii) is clearly equivalent to (16), it is sufficient to observe that a rank one positive matrix can be written as the product of a positive column vector and a positive row vector:

(w00w01w10w11)=(w0w1)(w¯0w¯1)

The following corollary is a reformulation of the equivalence of (i) and (iii) in the previous theorem.

Corollary 13

The system B of (15) is axially monotonicity preserving if and only if it corresponds to the tensor product of two univariate rational Bernstein systems, that is, if and only if B = UŪ, where

U=(w0b01(x)i=01wibi1(x),w1b11(x)i=01wibi1(x))andU¯=(w¯0b01(y)i=01w¯ibi1(y),w¯1b11(y)i=01w¯ibi1(y)).

Communicated by Juan L.G. Guirao

Acknowledgements

This work has been partially supported by the Spanish Research Grant MTM2015-65433-P (MINECO/FEDER) and by Gobierno de Aragón and Fondo Social Europeo.

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    J. Delgado and J.M. Peña, (2015), Axially Monotonicity Preserving Curves and Surfaces, in: Proceedings of the 3rd International Conference on Mathematical, Computational and Statistical Sciences (MCSS’15), Mathematics and Computers in Science and Engineering Series 40 (N. E. Mastorakis, A. Ding, M. V. Shitikova, Eds.), 28-32.

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    J. Delgado and J.M. Peña, (2015), Accurate computations with collocation matrices of q−Bernstein polynomials, SIAM Journal on Matrix Analysis and its Applications 36, No 2, 880-893.

    • Crossref
    • Export Citation
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    J. Delgado and J.M. Peña, (2015), Accurate evaluation of Bézier curves and surfaces and the Bernstein-Fourier algorithm, Applied Mathematics and Computation 271, 113-122.

    • Crossref
    • Export Citation
  • [12]

    J. Delgado and J.M. Peña, (2016), Algorithm 960: POLYNOMIAL: An Object-Oriented Matlab Library of Fast and Efficient Algorithms for Polynomials, ACM Transactions on Mathematical Software 42, No 3, 23:1-23:19.

    • Crossref
    • Export Citation
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    G. Farin, (2002), Curves and Surfaces for CAGD: A Practical Guide, Morgan-Kaufmann Publishers, San Francisco, CA.

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    R. T. Farouki, (2012), The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design 29, No 6, 379-419.

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    M.S. Floater, (1992), Derivatives of rational Bézier curves, Computer Aided Geometric Design 9, No 3, 161-174.

    • Crossref
    • Export Citation
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    M.S. Floater and J.M. Peña, (1998), Tensor-product monotonicity preservation, Advances in Computational Mathematics 9, No 3, 353-362.

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    • Export Citation
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    M.S. Floater and J.M. Peña, (2000), Monotonicity preservation on triangles, Mathematics of Computation 69, No 232, 1505-1519.

    • Crossref
    • Export Citation
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    Y. Huang and H. Su, (2006), The bound on derivatives of rational Bézier curves, Computer Aided Geometric Design 23, No 9, 698-702.

    • Crossref
    • Export Citation
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    F. Käferböck and H. Pottmann, (2013), Smooth surfaces from bilinear patches: Discrete affine minimal surfaces, Computer Aided Geometric Design 30, No 5, 476-489.

    • Crossref
    • Export Citation
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    L. Piegl and W. Tiller, (1997), The NURBS Book, Springer-Verlag, Berlin-Heidelberg.

    • Crossref
    • Export Citation
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    H. Prautzsch and T. Gallagher, (1992) Is there a geometric variation diminishing property for B−spline or Bézier surfaces?, Computer Aided Geometric Design 9, No 2, 119-124.

    • Crossref
    • Export Citation
  • [22]

    I. Selimovic, (2005), New bounds on the magnitude of the derivative of rational Bézier curves and surfaces, Computer Aided Geometric Design 22, No 4, 321-326.

    • Crossref
    • Export Citation
  • [23]

    L. Shi, J. Wang and H. Pottmann, (2014), Smooth surfaces from rational bilinear patches, Computer Aided Geometric Design 31, No 1, 1-12.

    • Crossref
    • Export Citation
  • [24]

    R.-J. Zhang and W. Ma, (2006), Some improvements on the derivative bounds of rational Bézier curves and surfaces, Computer Aided Geometric Design 23, No 7, 563-572.

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

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    J.M. Carnicer, M. García-Esnaola and J.M. Peña, (1994), Monotonicity Preserving Representations, in: Curves and Surfaces in Geometric Design (P. J. Laurent, A. Le Méhauté, and L. L. Schumaker, eds.), 83-90.

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    J.M. Carnicer, M. García-Esnaola and J.M. Peña, (1996), Generalized convexity preserving transformations, Computer Aided Geometric Design 13, No 2, 179-197.

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    J.M. Carnicer, M. García-Esnaola and J.M. Peña, (1996), Convexity of rational curves and total positivity, Journal of Computational and Applied Mathematics 71, No 2, 365-382.

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  • [5]

    J. Delgado and J.M. Peña, (2005), On efficient algorithms for the evaluation of rational tensor product surfaces, in: Mathematical methods for curves and surfaces: Tromso 2004, Modern Methods in Mathematics, Nashboro Press, Brentwood, TN, 115-124.

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    J. Delgado and J.M. Peña, (2007), Are rational Bézier surfaces monotonicity preserving?, Computer Aided Geometric Design 24, No 5, 303-306.

    • Crossref
    • Export Citation
  • [7]

    J. Delgado and J.M. Peña, (2013), Accurate computations with collocation matrices of rational bases, Applied Mathematics and Computation 219, No 9, 4354-4364.

    • Crossref
    • Export Citation
  • [8]

    J. Delgado and J.M. Peña, (2013), On the evaluation of rational triangular Bézier surfaces and the optimal stability of the basis, Advances in Computational Mathematics 38, No 4, 701-721.

    • Crossref
    • Export Citation
  • [9]

    J. Delgado and J.M. Peña, (2015), Axially Monotonicity Preserving Curves and Surfaces, in: Proceedings of the 3rd International Conference on Mathematical, Computational and Statistical Sciences (MCSS’15), Mathematics and Computers in Science and Engineering Series 40 (N. E. Mastorakis, A. Ding, M. V. Shitikova, Eds.), 28-32.

  • [10]

    J. Delgado and J.M. Peña, (2015), Accurate computations with collocation matrices of q−Bernstein polynomials, SIAM Journal on Matrix Analysis and its Applications 36, No 2, 880-893.

    • Crossref
    • Export Citation
  • [11]

    J. Delgado and J.M. Peña, (2015), Accurate evaluation of Bézier curves and surfaces and the Bernstein-Fourier algorithm, Applied Mathematics and Computation 271, 113-122.

    • Crossref
    • Export Citation
  • [12]

    J. Delgado and J.M. Peña, (2016), Algorithm 960: POLYNOMIAL: An Object-Oriented Matlab Library of Fast and Efficient Algorithms for Polynomials, ACM Transactions on Mathematical Software 42, No 3, 23:1-23:19.

    • Crossref
    • Export Citation
  • [13]

    G. Farin, (2002), Curves and Surfaces for CAGD: A Practical Guide, Morgan-Kaufmann Publishers, San Francisco, CA.

  • [14]

    R. T. Farouki, (2012), The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design 29, No 6, 379-419.

    • Crossref
    • Export Citation
  • [15]

    M.S. Floater, (1992), Derivatives of rational Bézier curves, Computer Aided Geometric Design 9, No 3, 161-174.

    • Crossref
    • Export Citation
  • [16]

    M.S. Floater and J.M. Peña, (1998), Tensor-product monotonicity preservation, Advances in Computational Mathematics 9, No 3, 353-362.

    • Crossref
    • Export Citation
  • [17]

    M.S. Floater and J.M. Peña, (2000), Monotonicity preservation on triangles, Mathematics of Computation 69, No 232, 1505-1519.

    • Crossref
    • Export Citation
  • [18]

    Y. Huang and H. Su, (2006), The bound on derivatives of rational Bézier curves, Computer Aided Geometric Design 23, No 9, 698-702.

    • Crossref
    • Export Citation
  • [19]

    F. Käferböck and H. Pottmann, (2013), Smooth surfaces from bilinear patches: Discrete affine minimal surfaces, Computer Aided Geometric Design 30, No 5, 476-489.

    • Crossref
    • Export Citation
  • [20]

    L. Piegl and W. Tiller, (1997), The NURBS Book, Springer-Verlag, Berlin-Heidelberg.

    • Crossref
    • Export Citation
  • [21]

    H. Prautzsch and T. Gallagher, (1992) Is there a geometric variation diminishing property for B−spline or Bézier surfaces?, Computer Aided Geometric Design 9, No 2, 119-124.

    • Crossref
    • Export Citation
  • [22]

    I. Selimovic, (2005), New bounds on the magnitude of the derivative of rational Bézier curves and surfaces, Computer Aided Geometric Design 22, No 4, 321-326.

    • Crossref
    • Export Citation
  • [23]

    L. Shi, J. Wang and H. Pottmann, (2014), Smooth surfaces from rational bilinear patches, Computer Aided Geometric Design 31, No 1, 1-12.

    • Crossref
    • Export Citation
  • [24]

    R.-J. Zhang and W. Ma, (2006), Some improvements on the derivative bounds of rational Bézier curves and surfaces, Computer Aided Geometric Design 23, No 7, 563-572.

    • Crossref
    • Export Citation
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