Superspirals

We present superspirals, a new and very general family of fair curves, whose radius of curvature is given in terms of a completely monotonic Gauss hypergeometric function. The superspirals are generalizations of log-aesthetic curves, as well as other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. High-accuracy computation of a superspiral segment is performed by the Gauss-Kronrod integration method. The proposed curves, despite their complexity, are the candidates for generating G2, and G3 non-linear superspiral splines.

The present work was motivated by an opportunity of finding a very general analytic way, in which so-called fair curves (Levien and Séquin, 2009; Wang et al., 2004) can be represented. The possibility to generate fair curves and surfaces that are visually pleasing is significant in computer graphics, computer-aided design, and other geometric modeling applications (Sapidis, 1994; Yamada et al., 1999).

A curve’s fairness is usually associated with its monotonically varying curvature, even though this concept still remains insufficiently defined (Levien and Séquin, 2009). The different mathematical definitions of fairness and aesthetic aspects of geometric modeling are briefly described by Sapidis (1994). The curves of monotone curvature were studied in recent works. Frey and Field (2000) analyzed the curvature distributions of segments of conic sections represented as rational quadratic Bézier curves in standard form. Farouki (1997) has used the Pythagorean-hodograph quintic curve as the monotone-curvature transition between a line and a circle. The monotone-curvature condition for rational quadratic B-spline curves is studied by Li et al. (2006). The use of Cornu spirals in drawing planar curves of controlled curvature was discussed by Meek and Walton (1989). The log-aesthetic curves (LACs), which are high-quality curves with linear logarithmic curvature graphs (Yoshida et al., 2010), have recently been developed to meet the requirements of industrial design for visually pleasing shapes (Harada et al., 1999; Miura et al., 2005; Miura, 2006; Yoshida and Saito, 2006; Yoshida et al., 2009; Ziatdinov et al., 2012b). LACs were reformulated based on variational principle, and their properties were analyzed by Miura et al. (2012). A planar spiral called generalized log-aesthetic curve segment (GLAC) (Gobithaasan and Miura, 2011) has been proposed using the curve synthesis process with two types of formulation: ρ-shift and κ-shift, and it was extended to three-dimensional case by Gobithaasan et al. (2012). According to the author of this work, a series of interesting works of Alexei Kurnosenko (2009, 2010a, 2010b) play an important role in the research on spirals.

Besides artificial objects, spirals, which are the curves with the monotone-curvature function, are important components of natural world objects: horns, seashells, bones, leaves, flowers, and tree trunks (Cook, 1903; Harary and Tal, 2011). In addition, they are used as a transition curves in rail-road and highway design (Walton and Meek, 1999, 2002; Habib and Sakai, 2003, 2004, 2005a, 2005b, 2005c, 2006, 2007, 2009; Dimulyo et al., 2009; Baykal et al., 1997; Walton et al., 2003; Ziatdinov et al., 2012a).

In this paper, we consider a radius of curvature function of a planar curve in terms of a very general Gauss hypergeometric function, which is completely monotonic under some constraints. It allows us to enclose many well-known spirals, the family of log-aesthetic curves, and other types of curves with monotone curvature, the properties of which can be still remain unexplored because of the curve’s complicated analytic expression in terms of special functions.

Our work has the following features:

• The proposed superspirals include a huge variety of fair curves with monotonic curvatures.

• The superspirals can be computed with high accuracy using the adaptive Gauss–Kronrod method.

• The superspirals might allow us to construct a two-point G2 Hermite interpolant, which seems to be impossible to do by means of log-aesthetic curves since insufficient degrees of freedom;

and several deficiencies:

• The proposed equations are integrals in terms of hypergeometric functions and cannot be represented in terms of analytic functions, despite its representation using infinite series.

• Since superspirals have no inflection points in non-polynomial cases, it cannot be considered as a G2 transition between a straight line and another curve.

• For highly accurate superspiraloid computation, significant time is necessary.

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