q -Ferguson curves

Abstract

Ferguson’s curves are widely used in airfoil design. We present a kind of Ferguson’s curves with a shape parameter by integrating the classical Ferguson’s curves with the q-derivatives, called q-Ferguson curves. This kind of curves not only preserves the interpolation properties of classical Ferguson’s curves but also has a shape parameter which provides a freedom variable to construct the desired curves satisfying the interpolation or length constraint.

Introduction

The problem of defining a curve through an array of points in space is a fundamental problem in mechanical design and computer-aided geometric design (CAGD), and several methods have been proposed. Generally, a set of functions $Y=f(X)$ is determined, each of which represents a curve segment of the composite curves. In 1963, Ferguson ¹ generalized the problem to a much broader class of permissible points and presented the resulting curve in a smooth composite of curve segments. The resolution of generalized problem is applicable to a variety of questions, such as airship aeroelasticity ² and airfoil design ³ .

In CAGD, many scholars have addressed this problem and applied for curves and surfaces reconstructions. Zhu and Wang ⁴ provided a new method for fitting $C^1$ piecewise algebraic curves to the given scattered data. Yu et al. ⁵ presented a stable and convergent method for curve interpolation with constrained length and endpoint interpolation. Bajaj and Xu ⁶ proved that the degree n A-splines can achieve in general $G^{2n-3}$ continuity by local fitting and still have degrees of freedom to achieve local data approximation. Jüttler and Felis ⁷ presented an algorithm for fitting implicitly defined algebraic spline surfaces to given scattered data. Roulier and Piper ⁸ gave the algorithms which produce a rational Bézier curve of constrained arc length. Ginnis and Kaklis ⁹ dealt with the problem of $C^2$ cubic spline interpolation with fixed the unit-tangent vector and the curvature at the endpoints of a planar point-set.

The q-derivative, or Jackson derivative, ¹⁰ is a q-analog of the ordinary derivative. It is the inverse of Jackson’s q-integration. There are many scholars who studied on q-derivative.^11–13 Simeonov and Goldman ¹⁴ integrated the q-derivatives with B-splines and proposed a kind of Quantum splines whose q-derivatives, up to some order, agree at the joins. Koekoek and Koekoek ¹⁵ proved the existence of the n-th q-derivative at the origin for every n and every function in both complex and real variable case. Zhang ¹⁶ presented a linearly parametrized set of curves, named C-curves with basis $\sin t,\cos t,t$ , and 1. C-curves are an extension of cubic curves, they depend on a parameter. Budakçı et al. ¹⁷ derived a collection of fundamental formulas for quantum B-splines analogous to known fundamental formulas for classical B-splines. Simeonov et al. ¹⁸ introduced the h-blossom and q-blossom by altering the diagonal property of the standard blossom. By applying the q-blossom, ¹⁹ Simeonov got several new identities including an explicit formula representing the monomials in terms of the q-Bernstein basis functions and a q-variant of Marsden’s identity.

The structure of the article is organized as follows. In section “Ferguson curves,” we briefly summarize the relevant definitions and results about Ferguson’s curves. Some results on q-derivatives are recalled in section “q-derivative.” In section “q-Ferguson curves,” we construct a q-Ferguson curve $R(u)$ interpolating the given points and tangent at endpoints. Moreover, the q-Ferguson curves with constrained length is presented, and using the length constraint, we obtain the exact value of q. Finally, we conclude the article in section “Conclusion.”

Ferguson’s curves

In 1963, Ferguson developed a polynomial representation of space curves

$$\mathbf{R}(u)={\mathbf{a}}_0+{\mathbf{a}}_{1}u+{\mathbf{a}}_2{u}^2+{\mathbf{a}}_3{u}^3,u\in [0,1]$$

(1)

We can express ${\mathbf{a}}_i,i=0,1,2,3$ in terms of $\mathbf{R}(0),\mathbf{R}(1),\mathbf{R}\mathbf{\prime }(0)$ , and $\mathbf{R}\mathbf{\prime }(1)$ (where $\mathbf{R}\mathbf{\prime }$ denotes derivative with respect to u)

$$\begin{array}{c}\mathbf{R}(0)={\mathbf{a}}_0\hfill \\ \mathbf{R}(1)={\mathbf{a}}_0+{\mathbf{a}}_1+{\mathbf{a}}_2+{\mathbf{a}}_3\hfill \\ \mathbf{R}\mathbf{\prime }(0)={\mathbf{a}}_1\hfill \\ \mathbf{R}\mathbf{\prime }(1)={\mathbf{a}}_1+2{\mathbf{a}}_2+3{\mathbf{a}}_3\hfill \end{array}$$

Solving the above equations yields expression for the coefficients, in terms of the geometric end conditions of the curve

$$\begin{array}{c}{\mathbf{a}}_0=\mathbf{R}(0)\hfill \\ {\mathbf{a}}_1=\mathbf{R}\mathbf{\prime }(0)\hfill \\ {\mathbf{a}}_2=3[\mathbf{R}(1)-\mathbf{R}(0)]-2\mathbf{R}\mathbf{\prime }(0)-\mathbf{R}\mathbf{\prime }(1)\hfill \\ {\mathbf{a}}_3=2[\mathbf{R}(0)-\mathbf{R}(1)]+\mathbf{R}\mathbf{\prime }(0)+\mathbf{R}\mathbf{\prime }(1)\hfill \end{array}$$

Substituting back into equation (1), we obtain

$$\mathbf{R}(u)=\mathbf{R}(0){\eta }_1(u)+\mathbf{R}(1){\eta }_2(u)+\mathbf{R}\mathbf{\prime }(0){\eta }_3(u)+\mathbf{R}\mathbf{\prime }(1){\eta }_4(u)$$

(2)

Where

$$\begin{array}{c}{\eta }_1(u)=1-3u^2+2u^3\hfill \\ {\eta }_2(u)=3u^2-2u^3\hfill \\ {\eta }_3(u)=u-2u^2+u^3\hfill \\ {\eta }_4(u)=u^3-u^2\hfill \end{array}$$

As Figure 1 shows, these basis functions, ${\eta }_i(u),i=1,2,3,4$ , are known as blending functions. ³

Figure 1.

The blending functions of cubic Ferguson curves.

q-derivative

The quantum calculus deals with discrete derivatives, and the theory of splines investigates piecewise polynomials whose derivatives, up to some order, match at the joins. Quantum splines are piecewise polynomials whose quantum derivatives, up to some order, agree at the joins. Quantum derivatives are just classical derivatives without limits. There are two common notions of quantum derivative: the h-derivative and the q-derivative.

The h-derivative of a function $f(x)$ is defined as

$${\left(\frac{d}{\mathrm{dx}}\right)}_{h}f(x)=\frac{f(x+h)-f(x)}{h}$$

The q-derivative of a function $f(x)$ is defined as

$${\left(\frac{d}{\mathrm{dx}}\right)}_{q}f(x)=\frac{f(\mathrm{qx})-f(x)}{\mathrm{qx}-x}$$

It is also often written as $D_{q}f(x)$ . Obvisously

$$\underset{h\to 0}{lim}{\left(\frac{d}{\mathrm{dx}}\right)}_{h}f(x)=\underset{q\to 1}{lim}{\left(\frac{d}{\mathrm{dx}}\right)}_{q}f(x)=\frac{d}{\mathrm{dx}}f(x)$$

The h-derivative is more familiar, but the q-derivative is more powerful. Many h-formulas are simply limits of q-formulas, ¹⁸ where we observe that the h-Bernstein basis functions are limits of the q-Bernstein basis functions. Similarly, the h-B-splines are limits of the q-B-splines.^14,20 The q-derivative is also known as the Jackson derivative. It has the analogous linear and product rules to the ordinary derivative.

• The linear rule: $D_q(f(x)+g(x))=D_{q}f(x)+D_{q}g(x)$ .

• The product rule: $D_q(f(x)g(x))=g(x)D_{q}f(x)+f(\mathrm{qx})D_{q}g(x)=g(\mathrm{qx})D_{q}f(x)+f(x)D_{q}g(x).$

Similarly, it satisfies the quotient rule

$$D_q\left(\frac{f(x)}{g(x)}\right)=\frac{g(x)D_{q}f(x)-f(x)D_{q}g(x)}{g(\mathrm{qx})g(x)},g(\mathrm{qx})g(x)\ne 0$$

Example 1

For example, the q-derivatives of the monomials are given by

$$D_q{x}^d=\frac{{(\mathrm{qx})}^d-x^d}{(q-1)x}=[d{]}_q{x}^{d-1}$$

where

$$[d{]}_q=\frac{1-q^d}{1-q}$$

q-Ferguson curves

We integrate the Ferguson’s curves with the q-derivatives and get a new kind of curves with one parameter, called q-Ferguson curves, which not only interpolates the endpoints $\mathbf{R}(0),\mathbf{R}(1)$ , but also possesses the given q-derivative slope $D_q\mathbf{R}(0),D_q\mathbf{R}(1)$ at the endpoints.

Definition 1

For the given $\mathbf{R}(0),\mathbf{R}(1),D_q\mathbf{R}(0),D_q\mathbf{R}(1)\in \mathbb{R}$ , q-Ferguson curve is a cubic polynomial such that

$$\mathbf{R}(u)=\mathbf{R}(0){\tau }_1(u)+\mathbf{R}(1){\tau }_2(u)+D_q\mathbf{R}(0){\tau }_3(u)+D_q\mathbf{R}(1){\tau }_4(u)$$

(3)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\tau }_1(u)=1-\frac{q^2+q+1}{q^2}{u}^2+\frac{q+1}{q^2}{u}^3}\\ \multicolumn{1}{c}{{\tau }_2(u)=\frac{q^2+q+1}{q^2}{u}^2-\frac{q+1}{q^2}{u}^3}\\ \multicolumn{1}{c}{{\tau }_3(u)=u-\frac{q^2+q}{q^2}{u}^2+\frac{1}{q}{u}^3}\\ \multicolumn{1}{c}{{\tau }_4(u)=-\frac{1}{q^2}{u}^2+\frac{1}{q}{u}^3}\end{array}$$

(4)

or in the matrix form

$$\mathbf{R}(u)=(\begin{array}{cccc}1\hfill & u\hfill & u^2\hfill & u^3\hfill \end{array})\left(\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{q^2+q+1}{q^2}}& \frac{q^2+q+1}{q^2}\hfill & -\frac{q^2+q}{q^2}\hfill & -\frac{1}{q^2}\hfill \\ \multicolumn{1}{c}{\frac{q+1}{q^2}}& -\frac{q+1}{q^2}\hfill & \frac{1}{q}\hfill & \frac{1}{q}\hfill \end{array}\right)\left(\begin{array}{c}\mathbf{R}(0)\hfill \\ \multicolumn{1}{c}{\mathbf{R}(1)}\\ \multicolumn{1}{c}{D_q\mathbf{R}(0)}\\ \multicolumn{1}{c}{D_q\mathbf{R}(1)}\end{array}\right)$$

In fact, without loss of generality, we assume the representation of the expect cubic polynomial curve is

$$\mathbf{R}(u)={\mathbf{a}}_0+{\mathbf{a}}_{1}u+{\mathbf{a}}_2{u}^2+{\mathbf{a}}_3{u}^3$$

(5)

where $0⩽u⩽1$ and ${\mathbf{a}}_0,{\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$ are the unknown coefficients. According to the conditions of the above definition, we get the following four equations

$$\begin{array}{c}\mathbf{R}(0)={\mathbf{a}}_0\hfill \\ D_q\mathbf{R}(0)={\mathbf{a}}_1\hfill \\ \mathbf{R}(1)={\mathbf{a}}_0+{\mathbf{a}}_1+{\mathbf{a}}_2+{\mathbf{a}}_3\hfill \\ D_q\mathbf{R}(1)={\mathbf{a}}_1+{\mathbf{a}}_2(q+1)+{\mathbf{a}}_3(q^2+q+1)\hfill \end{array}$$

Solving the above equations, we can get the following coefficients

$$\begin{array}{l}{a}_0=R(0)\\ a_1=D_{q}R(0)\\ a_2=\frac{(q^2+q+1)(R(1)-R(0))-(q^2+q)D_{q}R(0)-D_{q}R(1)}{q^2}\\ a_3=\frac{(q+1)(R(0)-R(1))+q{D}_{q}R(0)+D_{q}R(1)}{q^2}\end{array}$$

By substituting the coefficients into the above representation, we can rewrite equation (5), as shown in equation (3) of Definition 1. The blending functions ${\tau }_1(u),{\tau }_2(u),{\tau }_3(u),{\tau }_4(u)$ of q-Ferguson curves are displayed in Figure 2.

Figure 2.

The blending functions of q-Ferguson curves.

Remark 4.1

While $q=1$ , as Figure 2 shows, the blending functions ${\tau }_1(u),{\tau }_2(u),{\tau }_3(u),{\tau }_4(u)$ of q-Ferguson curves are definitely the blending functions ${\eta }_1(u),{\eta }_2(u),{\eta }_3(u),{\eta }_4(u)$ of classical Ferguson curves, and the q-Ferguson curves will degenerate into the Ferguson’s curves.

Example 4.1

Let $\mathbf{R}(0)=2,D_q\mathbf{R}(0)=2,R(1)=2,D_q\mathbf{R}(1)=2$ , the q-Ferguson curves can be given by equation (3) of Definition 1

$$\mathbf{R}(x)=2+4x+\frac{-4q^2-4q-3}{q^2}{x}^2+\frac{4q+3}{q^2}{x}^3$$

The left picture of Figure 3 depicts the q-Ferguson curves for $q=0.9,q=1,q=1.1$ , respectively. The right one shows a picture of “onion” consisted of six q-Ferguson curves with different $\mathbf{R}(0),D_q\mathbf{R}(0),\mathbf{R}(1),D_q\mathbf{R}(1)$ and q.

Figure 3.

q-Ferguson curves.

The applications of q-Ferguson curves

The Ferguson’s curves are widely used in CAGD, such as airplane wing design, car body design, and ship hull design. In this section, we present two simple applications of q-Ferguson curves: q-Ferguson curves interpolation and q-Ferguson curves with constrained length.

q-Ferguson curves interpolation

Given the endpoints $\mathbf{R}(0),\mathbf{R}(1)$ and the q-derivative $D_q\mathbf{R}(0),D_q\mathbf{R}(1)$ , we construct a q-Ferguson curve with an additional interpolation constraint. Assuming q-Ferguson curve interpolates an additional point $\mathbf{R}(t),0⩽t⩽1$ , by imposing this constraint, we get the exact value of q.

Example 5.1

Let $\mathbf{R}(0)=2,D_q\mathbf{R}(0)=-0.6,\mathbf{R}(1)=2,D_q\mathbf{R}(1)=2$ . We can get the representation of q-Ferguson curves as

$$\mathbf{R}(x)=2-0.6x+\frac{0.6q^2+0.6q+2}{q^2}{x}^2+\frac{-0.6q-2}{q^2}{x}^3$$

If imposing the additional interpolation constraint: q-Ferguson curves interpolates $\mathbf{R}(0.45)=2.3$ , we get the exact value of q. Substituting $\mathbf{R}(0.45)=2.3$ into the above equation, we get $q=0.8$ . Thus, the q-Ferguson curve interpolating point $(0.45,2.3)$ is

$$\mathbf{R}(x)=2-0.6x+4.475x^2-3.875x^3$$

The corresponding q-Ferguson curve is shown in Figure 4.

Figure 4.

q-Ferguson curves interpolation.

q-Ferguson curves with constrained length

For given two points $p_i(i=0,1)$ in $R^2$ , the corresponding tangent slope $u_i(i=0,1)$ and positive number $L⩾|p_1{p}_2|$ , our goal is to get a parametric curve $\mathbf{R}(u),0⩽u⩽1$ satisfying the following three conditions.

1. $\mathbf{R}(0)=p_0,\mathbf{R}(1)=p_1.$

2. $D_q\mathbf{R}(0)=u_0,D_q\mathbf{R}(1)=u_1.$

3. ${\int }_0^1\mathbf{R}(u)\mathrm{du}=L.$

There are many literatures addressed to constructing curves with length constraint.^9,21,22 We use q-Ferguson curves to deal with this problem. By the above first two conditions, we get the generic representation of q-Ferguson curves as

$$\mathbf{R}(u)=p_0{\tau }_1(u)+p_1{\tau }_2(u)+u_0{\tau }_3(u)+u_1{\tau }_4(u)$$

(6)

where ${\tau }_1(u),{\tau }_2(u),{\tau }_3(u),{\tau }_4(u)$ are the blending functions of q-Ferguson curves, as equation (4) shows. We impose the length constraint on the curves, that is to say, substituting equation (6) into the third condition, then the following equation can be obtained

$$\underset{0}{\overset{1}{\int }}(p_0{\tau }_1(u)+p_1{\tau }_2(u)+u_0{\tau }_3(u)+u_1{\tau }_4(u))\mathrm{du}=L$$

(7)

Since

$$\begin{array}{c}\underset{0}{\overset{1}{\int }}{\tau }_1(u)\mathrm{du}=1-\frac{q^2+q+1}{3q^2}+\frac{q+1}{4q^2}\hfill \\ \underset{0}{\overset{1}{\int }}{\tau }_2(u)\mathrm{du}=\frac{q^2+q+1}{q^2}-\frac{q+1}{q^2}\hfill \\ \underset{0}{\overset{1}{\int }}{\tau }_3(u)\mathrm{du}=\frac{1}{2}-\frac{q^2+q}{3q^2}+\frac{1}{4q}\hfill \\ \underset{0}{\overset{1}{\int }}{\tau }_4(u)\mathrm{du}=-\frac{1}{3q^2}+\frac{1}{4q}\hfill \end{array}$$

Substituting these four equations into equation (7), we can obtain the exact value of q by the following result.

Theorem 1

The length of q-Ferguson curves is L, namely ${\int }_0^{1}f(u)\mathrm{du}=L$ , if and only if the following equation holds

$$\begin{array}{c}{p}_0-\frac{1}{2}{u}_0+\frac{q^2+q+1}{3q^2}(3p_1-p_0)-\frac{(q^2+q)u_0+u_1}{3q^2}\hfill \\ +\frac{q+1}{4q^2}(p_0-4p_1)+\frac{1}{4q}(u_0+u_1)=L\hfill \end{array}$$

(8)

Example 5.2

Let the length L of q-Ferguson curve equal 2, and $p_0=2,p_1=2,u_0=1,u_1=1$ . By solving the equation in the Theorem 1, we get $q=-(1/2)$ or $q=1$ , that is, q-Ferguson curves with length 2 is $R(u)=2u^3-3u^2+u+2$ , Figure 5 shows the corresponding q-Ferguson curve.

Figure 5.

q-Ferguson’s curves with length 2.

Conclusion

The classical Ferguson’s curves deals with generalized polynomial derivatives whose derivatives, up to some order, match at the joins. In this article, we present a kind of Ferguson’s curves with q-derivative, namely q-Ferguson curves. q-Ferguson curves are parametric polynomials curves whose q-derivatives, up to some order, agree at the endpoints. This kind of curves not only preserves the interpolation properties of classical Ferguson’s curves but also has a shape parameter which provides a freedom variant to construct the desired Ferguson’s curves satisfying the interpolation or length constraint.

As well known, the Ferguson’s curves can be applied in airfoil design, materials science, civil engineering, ship manufacture, and so on. Therefore, to study the influence rule of the shape parameter and the potential applications of the q-Ferguson curves in these fields is a very interesting topic. We will focus on applications of the q-Ferguson curves in airfoil design, numerical control milling, and so on. Moreover, we will also study the q-Ferguson surfaces and the applications of q-Ferguson curves in engineering design and CAGD.