Short Communication
Note on planar Pythagorean hodograph curves of Tschirnhaus type☆

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Abstract

We discuss special Pythagorean hodograph curves which can be considered, from construction point of view, as degree n generalizations of the famous Tschirnhaus cubic. It will be proved that for each n there exists only one curve of Tschirnhaus type up to similarities.

Section snippets

Introduction and preliminaries

Ever since the Pythagorean hodograph (PH) curves first appeared in (Farouki and Sakkalis, 1990), they have been an undisputed phenomenon in geometric modeling and related disciplines to which a huge number of articles have been devoted. Very soon, a first survey on PH curves was summarized by Farouki in Section 17 of the Handbook of Computer Aided Geometric Design (2002), see Farin et al. (2002). A special issue of the journal CAGD devoted to “Pythagorean-hodograph curves and related topics”

Pythagorean hodograph curves of Tschirnhaus type

It is a well known fact that there exists a unique planar PH cubic, namely the so called Tschirnhaus cubic, see Farouki (2008). When considered as a Bézier curve then it admits simple constraints on the control polygon which enable us to identify it easily within the set of all cubics. In particular, the Bézier cubic is a PH curve if and only if it holdsd12=d0d2andθ1=θ2, where d0,d1,d2 are the lengths of the control polygon legs, and θ1,θ2 are the control polygon angles at the interior vertices

Curves of Tschirnhaus type and monotone curvature

Farin introduced the so called class A curves as special curves distinguished by the property that they have monotone curvature and (in space also) torsion. This prescribes some conditions on the matrix M in (3), see (Farin, 2006; Cao and Wang, 2008) for more details. Study of these curves was motivated by the notion “class A surfaces” from the automotive industry, where this term is used for those shapes which are essential for the aesthetic appearance of cars. Motivated by this we will

Unique Tschirnhaus degree n curve up to similarities

For n=2 (i.e., for the control polygon consisting of one triangle) one obtains the Bézier curve which is a parabola. And it is well known that all parabolas are similar, i.e., they can be transformed via suitable reparameterization and similarity to each other. The same holds for n=3 and for two suitable glued triangles as also all Tschirnhaus cubics are similar. This brings us to the idea to confirm this property for all degree n curves of Tschirnhaus type, in general.

We start with the

Conclusion and brief commentary on interpolations

In this note we briefly presented a class of Pythagorean hodograph curves which are, in some sense, a generalization of the famous Tschirnhaus PH cubic. We mainly discussed their PH property, their classification modulo similarities and their close connection to Bézier curves with monotone curvatures.

The research is mostly theoretical as applications of PH curves of Tschirnhaus type are as limited as in the case of the cubic. They do not admit inflections and moreover e.g. for C1 Hermite data

CRediT authorship contribution statement

Michal Bizzarri: Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. Miroslav Lávička: Conceptualization, Investigation, Methodology, Resources, Writing – original draft, Writing – review & editing. Jan Vršek: Investigation, Methodology, Validation, Writing – review & editing.

Declaration of Competing Interest

The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers' bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed

Acknowledgements

The work on this article was supported in part by the project 21-08009K of the Grant Agency of the Czech Republic. We thank to all referees for their valuable comments which helped us to improve the paper.

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