Shape Analysis of Generalized Cubic Curves☆
Graphical abstract
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 [22, Ch. 2], one distinguishes the following two cases:
If is regular, i.e., , for all , then the unit tangent (where the prime symbol refers to differentiation with respect to the parameter, , of the curve) of is defined everywhere in and the curvature is given by: Moreover, if the curvature numerator is not zero in , the binormal vector and the principal normal vector are defined everywhere in , as well as the torsion of the curve:
If is irregular at some point , i.e. such that , 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 , 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 to . 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 -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.
Section snippets
Generalized cubic curves
The notion of Chebyshev system is central to the introduction of generalized cubic curves. Adopting Definitions 2.20 and 2.35 of [37]:
Definition 2 Let be a set of two continuous functions, defined on . The system is called a (weak) Chebyshev system provided that the determinant: is (nonnegative) positive for any choice with .
Based on Definition 2.32 of [37], the following will also be found useful in the sequel:
Definition 3 Let be
Planar and non-planar generalized cubic curves
The first result is a direct consequence of the expression (2.46) of the torsion numerator.
Theorem 4 A generalized cubic curve , given by (2.26), is planar if and only if , , are linearly dependent.
Proof If , , are linearly dependent, then the torsion is zero and the curve is planar. Conversely, if is planar, then its torsion numerator is zero for all , i.e. the triple product is zero, thus , , are linearly dependent, i.e., the control polygon , ,
Generalized cubic curves in two dimensions
Corollary 2 implies that, for a generalized cubic curve, a singularity, i.e., a zero of the curvature numerator can only be found if the curve is planar. Talking about planar curves, it can be assumed, without any loss of generality, that they lie in the -plane. The cross product of any two vectors on the -plane is a vector along the -direction, thus it can be regarded just as a signed quantity. The same applies to the curvature (numerator), which becomes a signed function. In this
Generalized cubic functions
Consider the generalized cubic function as a 2D generalized cubic curve of (2.26) with control points given by (2.41). Root finding, monotonicity and convexity analysis are integral parts of any 1D algorithm for sign, monotonicity and convexity (shape) preserving interpolation; see e.g, [9], [15]. Then, for , the results of Section 4 regarding convexity hold, but regarding monotonicity and root-finding, one needs to study explicitly the -component of the 2D curve.
The variation
Generalized cubic curves via the Pruess spline representations [20]
The work done by Pruess [20], on alternatives to the exponential spline, provides two equivalent generic representations for spline segments, which satisfy either Hermite or Lidstone boundary conditions, and lead to -continuous splines. Although [20] was focused on functions , the proposed curve models directly generalize to the case of parametric curves as detailed below. With the aid of Pruess’ representations [20, eqn (1.1)] and [20, eqn (1.2)], one obtains generalized cubic basis
Examples & applications
The examples presented in this section illustrate the most important results of Sections 3–5. Moreover, Example 4 presents the construction of the Bernstein-like form for given basis functions and with Hermite and Lidstone boundary conditions, as in Sections 2.2 Control points solving a Hermite problem, 2.3 Control points solving a Lidstone problem. The images have all been produced using the CAD-CAE software package Genie [54].
Example 1 This example illustrates the similarities in theShape Similarities Between a Cubic Polynomial and an Exponential Spline-Segment in Tension [20], [55] with the Same Control Points
Conclusions & open issues
The present work is a study on the differential–geometric characteristics of the generalized cubic curves. This work primarily aims to further the comprehension of this class of curves, which are met in many occasions in CAGD, especially in shape-preserving interpolation algorithms.
The definition of the generalized cubic function (2.8) imposes condition (2.10), i.e., that is a Chebyshev system. Based on this requirement, the generalized cubic curve admits a Bernstein-like
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments, which improved this work significantly.
The authors would like also to express their thankfulness to Mr Jan Land, Head of Segment Fixed Structures (D-SS-X), DNV GL — Digital Solutions, for his continuous support to the research activities of the members of the Section.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
References (74)
comonotone Hermite interpolation via parametric cubics
J Comput Appl Math
(1996)- et al.
Cubic polynomial and cubic rational sign, monotonicity and convexity preserving Hermite interpolation
J Comput Appl Math
(2019) - et al.
Monotone linear rational spline interpolation
Comput Aided Geom Design
(1992) - et al.
Convexity-preserving interpolatory parametric splines of non-uniform polynomial degree
Comput Aided Geom Design
(1995) - et al.
sign, monotonicity and convexity preserving Hermite polynomial splines of variable degree
J Comput Appl Math
(2018) - et al.
An algorithm for constructing convexity and monotonicity-preserving splines in tension
Comput Aided Geom Design
(1988) C-curves: An extension of cubic curves
Comput Aided Geom Design
(1996)Shape classification of the parametric cubic curve and parametric B-spline cubic curve
Comput-Aided Des
(1981)- et al.
Shape determination of planar uniform cubic B-spline segments
Comput-Aided Des
(1990) Inflections on curves in two and three dimensions
Comput Aided Geom Design
(1991)
Managing the shape of planar splines by their control polygons
Comput-Aided Des
Inflection points and singularities on planar rational cubic curve segments
Comput Aided Geom Design
Computing singular points of plane rational curves
J Symbolic Comput
Computing real inflection points of cubic algebraic curves
Comput Aided Geom Design
On the singularity of a class of parametric curves
Comput Aided Geom Design
Detecting cusps and inflection points in curves
Comput Aided Geom Design
Identification of inflection points and cusps on rational curves
Comput Aided Geom Design
On cubics: A survey
Comput Graph Image Process
Strictly totally positive systems
J Approx Theory
Polynomial cubic splines with tension properties
Comput Aided Geom Design
Chebyshev–Bernstein bases
Comput Aided Geom Design
Blossoming beyond extended Chebyshev spaces
J Approx Theory
Shape preserving representations
Curvature combs and curvature plots
Comput-Aided Des
Calculating the self-intersections of Bézier curves
Comput Ind
A review on approaches for handling Bézier curves in CAD for manufacturing
Procedia Eng
Point-tangent/point-normal B-spline curve interpolation by geometric algorithms
Comput-Aided Des
Rational quartic interpolation spline with local shape preserving property
Appl Math Lett
Shape preserving approximation by spatial cubic splines
Comput Aided Geom Design
A new approach on curves of constant precession
Appl Math Comput
Analysis of inflection and singular points on a parametric curve with a shape factor
Math Comput Appl
A method for constructing local monotone piecewise cubic interpolants
SIAM J Sci Stat Comput
Monotone and convex cubic spline interpolation
Calcolo
Shape preserving cubic spline interpolation
IMA J Numer Anal
Piecewise -shape-preserving Hermite interpolation
Computing
Spatial shape-preserving interpolation using -splines
Numer Algorithms
Shape-preserving functional interpolation via parametric cubics
Numer Algorithms
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This paper has been recommended for acceptance by Takashi Maekawa.