Computer-Aided Design 125 (2020) 102849 Contents lists available at ScienceDirect Computer-Aided Design journal homepage: www.elsevier.com/locate/cad Shape Analysis of Generalized Cubic Curves ✩ Nikolaos C. Gabrielides a , Nickolas S. Sapidis b,∗ a Principal Specialist in Geometric Modelling, DNV GL - Digital Solutions, Veritasveien 1, 1363, Høvik, Norway b Department of Mechanical Engineering, University of Western Macedonia, Bakola & Sialvera Str., GR-50132, Kozani, Greece article info Article history: Received 2 October 2019 Received in revised form 24 February 2020 Accepted 10 April 2020 Keywords: Generalized cubic curve Shape analysis Monotonicity Convexity Torsion Inflection 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 R n , n = 1, 2, 3, proving that well-known theorems, which hold for cubic polynomials, hold also for this class of curves. © 2020 Elsevier Ltd. All rights reserved. 1. 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 seg- ments alternative to the cubic polynomial. The majority of these methods employ cubic polynomials [1–8] and [9], however, there are indeed successful alternative methods employing rational polynomials [9–11] or variable degree polynomials [8,12–14] and [15], or exponential spline segments [16–19], 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), ana- lyzing 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 ✩ This paper has been recommended for acceptance by Takashi Maekawa. ∗ Corresponding author. E-mail addresses: nikolaos.gavriilidis@dnvgl.com (N.C. Gabrielides), nsapidis@uowm.gr (N.S. Sapidis). planar curves (Section 4), concluding with the generalized cubic functions (Section 5). To begin this investigation, considering the curvature of a parametric curve C(s) :[a, b] −→ R 3 [22, Ch. 2], one distin- guishes the following two cases: Case (i). If C(s) is regular, i.e., C ′ (s) = 0, for all s ∈[a, b], then the unit tangent t(s) = C ′ (s) ∥C ′ (s)∥ , (1.1) (where the prime symbol refers to differentiation with respect to the parameter, s, of the curve) of C(s) is defined everywhere in [a, b] and the curvature is given by: κ (s) = ∥C ′ (s) × C ′′ (s)∥ ∥C ′ (s)∥ . (1.2) Moreover, if the curvature numerator is not zero in [a, b], the binormal vector b(s) = C ′ (s) × C ′′ (s) ∥C ′ (s) × C ′′ (s)∥ (1.3) and the principal normal vector n(s) = b(s) × t(s) (1.4) are defined everywhere in [a, b], as well as the torsion of the curve: τ (s) = ( C ′ (s) × C ′′ (s) ) · C ′′′ (s) ∥C ′ (s) × C ′′ (s)∥ 2 . (1.5) https://doi.org/10.1016/j.cad.2020.102849 0010-4485/© 2020 Elsevier Ltd. All rights reserved.