Geometry of rational helices and its applications

TL;DR

This paper explores the geometric properties of rational helices, offering simpler construction methods, applications in interpolation and surface modeling, and investigating rational rotation minimizing frames with approximation techniques.

Contribution

It introduces a natural geometric approach to rational PH helices, simplifies their construction, and applies these concepts to interpolation and sweep surface modeling.

Findings

Simplified construction of rational helices using geometric features

Development of rational approximation for rotation minimizing frames

Application of rational helices in interpolation and surface modeling

Abstract

The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this to related subjects. One of these applications is Geometric C1 Hermite interpolation (i.e. interpolation of end points with associated unit tangents) by rational helices. Furthermore, we investigate the existence of rational rotation minimizing frames (RRMFs) on rational helices. A rational approximation procedure to rotation minimizing frames (RMFs) is suggested. Subsequently, we deploy the approximate frame for modeling a rational sweep surface. The resulting algorithms are illustrated by several examples.