Construction and Evaluation of Pythagorean Hodograph Curves in Exponential-Polynomial Spaces

Abstract

In the past few decades polynomial curves with Pythagorean hodograph (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 (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 Bézier-like curve in order to fulfill the PH property. Then, for such EPH curves, fundamental characteristics like parametric speed or 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: Woźny and Chudy's algorithm and the dynamic evaluation procedure by Yang and Hong.

On G1 and G2 Hermite interpolation by spatial Algebraic-Trigonometric Pythagorean Hodograph curves with polynomial parametric speed

“…Proof. First of all we observe that conditions (20) are two more than the ones previously considered in (15), thus the minimum degree of ρ is now increased to 6. There follows that the conditions to obtain a C 0 -continuous curvature now read as…”

“…and then impose the last two conditions in (20). Unfortunately, κ ′ (ξ) yields a quadratic expression in the unknowns, and thus it cannot be used to generate the two missing linear equations needed to set up the linear system in the unknowns c i , i = 0, .…”

“…The most known examples of non-polynomial Pythagorean Hodograph (PH for short) curves are certainly rational Pythagorean Hodograph curves [9]. However, in the last decade, other instances of non-polynomial PH curves have appeared, such as, the so-called Algebraic-Trigonometric PH curves (ATPH for short) [19,12,18,4] and their hyperbolic counterpart [17,5,20]. Recently, a special subfamily of ATPH curves has been proposed by Wu and Yang [22,23].…”