Abstract
In this paper, we study algebraic and geometric characteristics of a class of Algebraic-Hyperbolic curves that possess Pythagorean-Hodograph (PH) properties, which are called AHPH curves. These curves are defined on the space Γn=span{1,t,…,tn−3,sinht,cosht}, n∈N﹨{1,2}. We prove that all non-degenerate AHPH curves belong to Γ4. For this particular case, we give an explicit expression in terms of canonical basis functions. To reveal the geometric properties of AHPH curves, we study the geometric constraints on their Bézier like control polygons. The main result shows that an AH curve is an AHPH curve if and only if the interior angles of its control polygon are equal, and the second leg-length of the control polygon is the geometric mean of the other two leg-lengths. Our main idea is to represent a planar parametric curve in complex form. As an application, we give some examples of G1 Hermite interpolation using AHPH curves. We point out that there are no more than two AHPH curves for any given G1 Hermite conditions.
Published Version
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