## 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 [7,8,10–12]. 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 (*u*_{0},...,*u _{n}*) a basis of

*𝒰*. If a control polygon

*P*

_{0}···

*P*is given then we define a parametric curve

_{n}In CAGD the functions *u*_{0},...,*u _{n}* are usually nonnegative and

*u*

_{0},...,

*u*) is

_{n}*normalized*) and in this case we say that (

*u*

_{0},...,

*u*) is a

_{n}*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 (

*u*

_{0},...,

*u*) 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.

_{n}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 *P*_{0} and finishing in *P _{n}*, and the curve

*γ*(

*t*) taking all values of

*t*starting from

*a*and finishing at

*b*. Let

*P*

_{0},··· ,

*P*be control points in ℝ

_{n}*and let*

^{k}*γ*be the curve defined by (1). A surjective affine mapping

*T*: ℝ

^{k}→ 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*(

*P*

_{0}) ≤ ··· ≤

*T*(

*P*), then the projection of the corresponding curve

_{n}*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:

A system of functions (*u*_{0},...,*u _{n}*) is

*monotonicity preserving*(resp.,

*strictly monotonicity preserving*) if for any

*α*

_{0}≤

*α*

_{1}≤ ... ≤

*α*(resp.,

_{n}*α*

_{0}<

*α*

_{1}< ... <

*α*) in ℝ, the function

_{n}Some properties and applications of monotonicity preserving systems can be seen in [2–4]. A proof of the following result, which characterizes monotonicity preserving systems, appears in Proposition 2.3 of [4].

*Let* (*u*_{0},...,*u _{n}*)

*be a system of functions defined on an interval*[

*a*,

*b*].

*Let*

*for i*∈ {0,1,...,

*n*}.

*Then:*

- (
*u*_{0},...,*u*)_{n}*is monotonicity preserving if and only if v*_{0}*is a constant function and the functions v*= 1,...,_{i}are increasing for i*n*. - (
*u*_{0},...,*u*)_{n}*is strictly monotonicity preserving if and only if*(*u*_{0},...,*u*)_{n}*is monotonicity preserving and the function*$\sum}_{j=1}^{n}{v}_{j$ *is 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

for all *i* = 0,1,...,*n*. The system *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].

*The Bernstein basis**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* ∈ ℕ_{0} and a sequence of nondecreasing real numbers *d* + 2 elements. The *j*-th B-spline of degree *d* with nodes x is defined by

for all *x* ∈ ℝ, with

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].

*The B-spline basis**associated to a sequence of nodes**preserves 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* = (*u*_{0},...,*u _{n}*) defined in [

*a*,

*b*], it is usual to construct a rational system of functions (

*r*

_{0},...,

*r*) from a sequence of positive weights

_{n}The weights act as shape parameters.

*Rational Bernstein bases* arise taking *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].

*Let us consider a rational Bézier curve g given by*

*with* (*w _{i}*)

_{0≤i≤n}

*a sequence of positive weights and P*

_{0}···

*P*

_{n}the control polygon. Then we have*where* ∆*P _{k}* :=

*P*

_{k+1}−

*P*.

_{k}*Proof.* Taking into account that

where

Since

we can write (5) in the following way

From the previous formula, since ∆(*P _{i} w_{i}*) =

*w*

_{i+1}(∆

*P*) + (∆

_{i}*w*)

_{i}*P*, we deduce

_{i}Applying the univariate de Casteljau algorithm to the right-hand of the previous formula we derive

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

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

By performing an index change and reordering, we have

Extending the sum on *k* and operating, we deduce that

From the previous formula we can derive in a straightforward way

Then, arranging the indexes and operating, we have

So the result follows from the last expression.

As a consequence, we can derive the following result.

* A rational Bézier basis* (*r*_{0},...,*r _{n}*)

*defined by*

*with* (*w _{i}*)

_{0≤i≤n}

*a sequence of positive weights, is strictly monotonicity preserving.*

*Proof.* Let us consider the function

with *α*_{0} < *α*_{1} < ··· < *α _{n}*. Differentiating the previous formula we have

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 *T*_{0}, *T*_{1} and *T*_{2}:

If *t ∈ 𝒯* , then *τ _{i}* ≥ 0,

*i*= 0,1,2. Let

**i**= (

*i*

_{0},

*i*

_{1},

*i*

_{2}) denote a multi-index where

*i*

_{0},

*i*

_{1},

*i*

_{2}∈ ℕ

_{0}= {0,1,2,...} and let us denote by |

*i*| the sum

*i*

_{0}+

*i*

_{1}+

*i*

_{2}.

Given *n* ≥ 1, let us consider for each i such that |*i*| = *n* a function *ϕ*_{i} : *𝒯* → ℝ. We shall refer to them as a system and write (*ϕ*_{i})_{|i|=n}. Then, given (*c*_{i})_{|i|=n} a sequence of coefficients in ℝ, we can define a function *F* : *𝒯* → ℝ, as

Let us consider the following points **i**| = *n*. Then we define the control net of *F* as the function

which is linear on each subtriangle of *𝒯* and satisfies

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.

A system (*ϕ*_{i})_{|i|=n} is *axially monotonicity preserving* (AMP) if the function *F* is increasing in the direction *T*_{1} − *T*_{0}, *T*_{2} − *T*_{1} or *T*_{0} − *T*_{2} 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,

are AMP and even satisfy stronger monotonicity preserving properties.

Now let us consider the rational Bernstein basis of order *n* (*ϕ*_{i})_{|i|=n} given by *w*_{i})_{|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.

*(see Theorem 2 of [6]) If a rational Bernstein basis on a triangle with positive weights is AMP, then w*_{i} = *w*_{j}*for all***i**,**j***such that* |**i**| = |**j**| = *n*.

### 3.2 Monotonicity preservation of nonrational surfaces defined in rectangular patches

Given a normalized nonnegative system of bivariate functions *a*_{1} , **b**_{1}] × [*a*_{2} , *b*_{2}] and a sequence of values in ℝ,

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

we define the control net

to be the unique function which satisfies the interpolation conditions

and is bilinear on each rectangle

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* = (*d*_{1}, *d*_{2}) ∈ ℝ^{2}, if

In particular, the control net *p* can be increasing in a direction *d*. In [16] Floater and Peña characterized this situation as follows:

*The control net p is increasing in the direction d* = (*d*_{1},*d*_{2}) *in* ℝ^{2}*if and only if for i* = 0,1,...,*m* − 1 *and j* = 0,1,...,*n* − 1,

*where* ∆_{1}*c _{i}j* := (

*c*

_{i+1,j}−

*c*)/(

_{ij}*α*).

_{i+1}−*α*)_{i}*and*∆_{2}*c*:= (_{i}j*c*_{i,j+1}−*c*)/(_{ij}*β*_{j+1}−*β*_{j}Given a sequence _{1}*c _{ij}* :=

*c*

_{i+1,j}−

*c*for

_{ij}*i*= 0,1,...,

*m*− 1 and

*j*= 0,1,...,

*n*, and Λ

_{2}

*c*:=

_{ij}*c*

_{i,j+1}−

*c*for

_{ij}*i*= 0,1,...,

*m*and

*j*= 0,1,...,

*n*− 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, Λ_{1}*c _{ij}* ≥ 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, Λ

_{2}

*c*≥ 0

_{ij}In [16] several concepts of monotonicity preservation for rectangular patches 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 (8) is increasing in any direction*d*in ℝ^{2}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 *a*_{1},*b*_{1}] and [*a*_{2},*b*_{2}], respectively. Then we consider the system of tensor-product functions

defined on the rectangle [*a*_{1},*b*_{1}] × [*a*_{2},*b*_{2}]. If the systems *U*^{1} and *U*^{2} are blending then the system *U* is also blending. Given *c _{ij}* ∈ ℝ and taking

*i*∈ {0,1,...,

*m*} and

*j*∈ {0,1,...,

*n*}, the system

*U*generates the following parametric function:

The next result characterizes axial monotonicity preservation:

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

*are increasing.*

As a consequence, we can derive the following result.

*If the blending univariate systems U*^{1}*and U*^{2}*are 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.

*The tensor product of Bernstein bases*$${({b}_{i}^{m}(x)\cdot {b}_{j}^{n}(y))}_{0\le i\le m}^{0\le j\le n}$$ *preserves monotonicity axially.**Given two sequence of nodes*$x={({x}_{i})}_{i=1}^{m+{d}_{1}+1}$ *and* ,$y={({y}_{j})}_{j=1}^{n+{d}_{2}+1}$ *the tensor product of the corresponding B-spline bases*$${({N}_{i,{d}_{1},x}(x)\cdot {N}_{j,{d}_{2},y}(y))}_{1\le i\le m}^{1\le j\le n}$$ *preserves monotonicity axially.*

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

Let *F* be a *rational Bézier surface* defined as

where *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

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

where *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

Now we present our conjecture:

*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

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.

*Let us consider the system*

*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

where

By Remark 1, a control net *p* is increasing in the X-axis direction (1,0) if and only if Λ_{1}*c _{ij}* ≥ 0 for

*i*= 0,1,...,

*m*−1 and

*j*= 0,1,...,

*n*−1. Taking

*c*

_{00}=

*c*

_{10}=

*c*≠ = 0 and

*c*

_{01}=

*c*

_{11}= 2

*c*we have that Λ

_{1}

*c*

_{00}= Λ

_{1}

*c*

_{01}= 0 and so the control net is increasing in the X-axis direction. With the previous choice of the coefficients we have that

*F*

_{1}(

*y*) = 0 and

*F*is increasing in the X-axis direction if and only if

*x*,

*y*) ∈ [0,1]

^{2}. Since

*c*can be any real number

*F*

_{2}(

*y*) ≥ 0 if and only if

*w*

_{00}

*w*

_{11}−

*w*

_{01}

*w*

_{10}= 0, which is equivalent to rank

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:

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

*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* = *U* ⊗ *Ū*, *where*

^{}

Communicated by Juan L.G. Guirao

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.

## References

- [1]
R.H. Cushman and L. Bates, (1997), Global Aspects of Classical Integrable Systems, Birkhäuser Verlag, Basel.

- [2]↑
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.

- [3]
J.M. Carnicer, M. García-Esnaola and J.M. Peña, (1996), Generalized convexity preserving transformations, Computer Aided Geometric Design 13, No 2, 179-197.

- [4]↑
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.

- [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.

- [6]↑
J. Delgado and J.M. Peña, (2007), Are rational Bézier surfaces monotonicity preserving?, Computer Aided Geometric Design 24, No 5, 303-306.

- [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.

- [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.

- [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.

- [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.

- [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.

- [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.

- [15]↑
M.S. Floater, (1992), Derivatives of rational Bézier curves, Computer Aided Geometric Design 9, No 3, 161-174.

- [16]↑
M.S. Floater and J.M. Peña, (1998), Tensor-product monotonicity preservation, Advances in Computational Mathematics 9, No 3, 353-362.

- [17]↑
M.S. Floater and J.M. Peña, (2000), Monotonicity preservation on triangles, Mathematics of Computation 69, No 232, 1505-1519.

- [18]↑
Y. Huang and H. Su, (2006), The bound on derivatives of rational Bézier curves, Computer Aided Geometric Design 23, No 9, 698-702.

- [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.

- [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.

- [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.

- [23]↑
L. Shi, J. Wang and H. Pottmann, (2014), Smooth surfaces from rational bilinear patches, Computer Aided Geometric Design 31, No 1, 1-12.

- [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.