Construction and evaluation of PH curves in exponential-polynomial spaces

TL;DR

This paper explores exponential-polynomial PH curves, providing conditions for their control polygons, analyzing their properties, and introducing a new evaluation algorithm that outperforms existing methods.

Contribution

It introduces necessary and sufficient conditions for EPH curves to be PH, analyzes their properties, and proposes a novel, efficient evaluation algorithm.

Findings

Conditions for control polygons to produce PH EPH curves

Analysis of parametric speed and arc length properties

New evaluation algorithm with improved performance

Abstract

In the past few decades polynomial curves with Pythagorean Hodograph (for short PH curves) have received considerable attention due to their usefulness in various CAD/CAM areas, manufacturing, numerical control machining and robotics. This work deals with classes of PH curves built-upon exponential-polynomial spaces (for short EPH curves). In particular, for the two most frequently encountered exponential-polynomial spaces, we first provide necessary and sufficient conditions to be satisfied by the control polygon of the B\'{e}zier-like curve in order to fulfill the PH property. Then, for such EPH curves, fundamental characteristics like parametric speed or cumulative and total arc length are discussed to show the interesting analogies with their well-known polynomial counterparts. Differences and advantages with respect to ordinary PH curves become commendable when discussing the solutions to application problems like the interpolation of first-order Hermite data. Finally, a new evaluation algorithm for EPH curves is proposed and shown to compare favorably with the celebrated de Casteljau-like algorithm and two recently proposed methods: Wo\'zny and Chudy's algorithm and the dynamic evaluation procedure by Yang and Hong.