A Note on Superspirals of Confluent Type

Jun-Ichi Inoguchi,Rushan Ziatdinov,Kenjiro T Miura

doi:10.3390/math8050762

Jun-Ichi Inoguchi, Rushan Ziatdinov + Show 1 more

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https://doi.org/10.3390/math8050762

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Abstract

Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. In this work, we study superspirals of confluent type via similarity geometry. Through a detailed investigation of the similarity curvatures of superspirals of confluent type, we find a new class of planar curves with monotone curvature in terms of Tricomi confluent hypergeometric function. Moreover, the proposed ideas will be our guide to expanding superspirals.

Highlights

Log-aesthetic curves (LAC) were proposed a decade ago to meet the requirements of industrial design to produce visually pleasing shapes
Followed by the discovery of superspirals, curves with monotonic curvature are considered as an excellent tool for generating highly-smooth shapes which are useful in computer aided design and styling
As a generalization of logarithmic spiral, the second named author introduced a new class of fair curves—the superspiral of confluent type [3]

Summary

Introduction

Log-aesthetic curves (LAC) were proposed a decade ago to meet the requirements of industrial design to produce visually pleasing shapes. Followed by the discovery of superspirals, curves with monotonic curvature are considered as an excellent tool for generating highly-smooth shapes which are useful in computer aided design and styling. Such non-polynomial curves are determined in terms of complex special functions and can be precisely computed in modern computer algebra systems and programming languages. The location of the logarithmic spiral in the whole family is not clear To overcome these difficulties, we use a new framework “similarity geometry” for the study of aesthetic curves developed in our previous works [4,5]. The present study implies that planar curves whose radii of curvature are Tricomi’s hypergeometric function of confluent type are new candidates of fair curves in industrial shape design. We close this paper by exhibiting some pictures of those curves

Preliminaries

Conclusions