Shape Analysis of Generalized Cubic Curves

Highlights

• Generalized cubic curves representation with totally positive bases.

• Construction and properties of generalized cubic curves.

• Shape analysis of spatial and planar generalized cubic curves and generalized cubic functions.

• Generalized cubic curves via Pruess spline representation.

Abstract

“Generalized cubic curves” were introduced in Costantini and Manni (2006) as a special case of the more general results in Goodman and Mazure (2001) and Costantini, Lyche and Manni (2005). In this work, the shape of these curves is analyzed in ${\mathbb{R}}^n$, $n=1,2,3$, proving that well-known theorems, which hold for cubic polynomials, hold also for this class of curves.

Introduction

The generalized cubic curves are able to represent a large variety of curves, which have been found very efficient in shape-preserving spline interpolation, playing the role of spline segments alternative to the cubic polynomial. The majority of these methods employ cubic polynomials [1], [2], [3], [4], [5], [6], [7], [8] and [9], however, there are indeed successful alternative methods employing rational polynomials [9], [10], [11] or variable degree polynomials [8], [12], [13], [14] and [15], or exponential spline segments [16], [17], [18], [19], all of which are contained in the class of generalized cubic curves. Furthermore, the tension schemes in [20] (detailed in Section 6) and the C-curves [21] are also generalized cubic curves.

As the generalized cubic curves are extensively used in shape-preserving interpolation (details are given in Section 7.1), analyzing the local shape of them is most important. In addition, the basic properties of these curves make them directly applicable to the curve-modeling component of current CAD systems (this is discussed in Section 7.2), which also requires a comprehensive shape-analysis of them. Shape analysis for real functions of a real variable is done by considering the monotonicity and convexity of the function, and by computing the roots of it. Local shape analysis of 2D curves is achieved by considering the curvature, and of 3D curves by considering curvature and torsion. In this work, analysis begins with spatial curves (Section 3), next moving to planar curves (Section 4), concluding with the generalized cubic functions (Section 5).

To begin this investigation, considering the curvature of a parametric curve $\mathbf{C}\left(s\right):[a,b]⟶{\mathbb{R}}^3$ [22, Ch. 2], one distinguishes the following two cases:

If $\mathbf{C}\left(s\right)$ is regular, i.e., ${\mathbf{C}}^{\prime }\left(s\right)\ne \mathbf{0}$, for all $s\in [a,b]$, then the unit tangent

$$\mathbf{t}\left(s\right)=\frac{{\mathbf{C}}^{\prime }\left(s\right)}{\Vert {\mathbf{C}}^{\prime }\left(s\right)\Vert },$$

(where the prime symbol refers to differentiation with respect to the parameter, $s$, of the curve) of $\mathbf{C}\left(s\right)$ is defined everywhere in $[a,b]$ and the curvature is given by:

$$\kappa \left(s\right)=\frac{\Vert {\mathbf{C}}^{\prime }\left(s\right)\times {\mathbf{C}}^{\prime \prime }\left(s\right)\Vert }{\Vert {\mathbf{C}}^{\prime }\left(s\right)\Vert }.$$

Moreover, if the curvature numerator is not zero in $[a,b]$, the binormal vector

$$\mathbf{b}\left(s\right)=\frac{{\mathbf{C}}^{\prime }\left(s\right)\times {\mathbf{C}}^{\prime \prime }\left(s\right)}{\Vert {\mathbf{C}}^{\prime }\left(s\right)\times {\mathbf{C}}^{\prime \prime }\left(s\right)\Vert }$$

and the principal normal vector

$$\mathbf{n}\left(s\right)=\mathbf{b}\left(s\right)\times \mathbf{t}\left(s\right)$$

are defined everywhere in $[a,b]$, as well as the torsion of the curve:

$$\tau \left(s\right)=\frac{\left({\mathbf{C}}^{\prime }\left(s\right)\times {\mathbf{C}}^{\prime \prime }\left(s\right)\right)\cdot {\mathbf{C}}^{\prime \prime \prime }\left(s\right)}{{\Vert {\mathbf{C}}^{\prime }\left(s\right)\times {\mathbf{C}}^{\prime \prime }\left(s\right)\Vert }^2}.$$

If $\mathbf{C}\left(s\right)$ is irregular at some point $s_0\in [a,b]$, i.e. such that ${\mathbf{C}}^{\prime }\left(s_0\right)=\mathbf{0}$, then, neither the curvature nor the binormal vector, the normal vector and the torsion can be defined at this point. At such a point the curve presents a cusp (see [22, Ch. 11.2.2]). Obviously, the curvature numerator vanishes at cusps.

In both cases (i) & (ii), the points, where the curvature numerator vanishes, are of special interest, since singularities of the curve appear at such points and the curve presents locally a straight-line behavior. In particular, regarding planar curves, one kind of singularity, which may appear, if the curve is regular but the curvature numerator vanishes at a point $s=s_0$, is that of an inflection point, where the shape changes from convex to concave or vice-versa. Thus, zeros of the curvature numerator have been the focus of work of many researchers.

Indeed, Wang [23], Stone and De Rose [24], Meek and Walton [25], Goodman [26] and Kantorowitz and Schechner [27] study zeros in the curvature numerator of planar cubic curves and determine conditions for cusps and inflection points. Sakai [28] and Chen et al. [29] do the same for planar rational cubic curve segments. Chen and Wang [30] and Saito [31] compute inflection points of planar cubic algebraic curves. Juhász [32], dealing with parametric curves (represented as linear combinations of control points and basis functions), allows one control point to vary, while the rest remain fixed, and shows that the locus of the moving control point that yields a zero curvature point on the curve is a developable surface. The works by Manocha and Canny [33] and Li and Cripps [34] deal with 3D curves having rational parameterization, and detect inflections and cusps on them. In [33], the authors show that a necessary and sufficient condition for the existence of a cusp is given by a vanishing first derivative vector, provided that the curve has proper parameterization (i.e. the curve cannot be identically described by a rational polynomial parameterization of lower degree). Then, they reduce the problem of detecting cusps in a rational curve to that of a polynomial curve, and finally, using regular parameterizations they analyze inflection points. In [34], the authors show that occurrence of cusps and inflection points on a 3D curve corresponds to zeros of one univariate polynomial, assuring efficiency and stability in computing such zeros.

In order to avoid any confusion about the terms spatial curve, non-planar curve and true 3D curve (see [35], [36, p. 429]), which are used in the sequel, the following definition clarifies the differences:

Definition 1

Spatial curve is any function, which maps the parametric interval $[a,b]\subset {\mathbb{R}}^{}$ to ${\mathbb{R}}^3$. Non-planar curve is any spatial curve, which does not lie entirely in a plane. True 3D curve is any non-planar curve, no four points of which lie on the same plane.

In this work, the definition and the basic representation of the generalized cubic curves is given in Section 2. Their Bernstein-like representation is studied in Section 2.1, followed by two sub-sections, which present two alternative ways to calculate the control points of the Bernstein-like representation: by solving a Hermite problem (Section 2.2) and a Lidstone problem (Section 2.3), respectively. Section 2.4 summarizes the most important properties of the Bernstein-like representation, which are used in shape analysis, and Section 2.5 details mathematical expressions for curvature and torsion. In Section 3, Theorem 4 provides a necessary and sufficient condition for a spatial generalized cubic curve to be planar and Theorem 5 provides a necessary and sufficient condition for a spatial generalized cubic curve to have torsion with constant sign, thus resembling the behavior of the spatial cubic polynomials. Theorem 6 proves that a non-planar generalized cubic is a true 3D curve. Finally, Theorem 7 establishes that a spatial generalized cubic curve cannot have points of vanishing curvature. In Section 4, shape analysis of a planar generalized cubic curve is performed and the same is done in Section 5 for generalized cubic functions. Theorem 8, Theorem 10 summarize the results of these two sections, respectively. Section 6 presents in-detail a way to construct basis functions for some prominent splines, which can be represented as piecewise generalized cubic curves. Section 7 presents examples demonstrating the results proved in the previous sections, and discusses application of these results in shape-preserving interpolation (Section 7.1) and in CAD methods for curve modeling (Section 7.2). Section 8 discusses the contributions of this paper and the related open issues that this research brings up.

The original contribution of this paper refers to (a) Theorem 4, Theorem 5, Theorem 6, Theorem 7 for the 3D generalized cubic curves, (b) Theorem 8, Theorem 9 for the 2D generalized cubic curves and (c) Theorem 10 on the generalized cubic functions. Moreover, the use of the Pruess representation of $C^2$-continuous splines (see Section 6) to compute generalized cubic spline bases for many prominent generalized cubics (such as the exponential spline segment and the variable degree polynomial spline segment) is also an original contribution of this work (see Theorem 11). This paper also contributes toward the calculation of control points of given Bernstein-like bases by applying Lidstone boundary conditions (see Section 2.3) and presents the most vital properties (for mathematical analysis and for CAD application) of generalized cubic curves in Section 2.4.