Geometric Hermite interpolation by a family of spatial algebraic–trigonometric PH curves

Abstract

This paper proposes to define a family of spatial algebraic–trigonometric Pythagorean Hodograph (ATPH) curves by integrals of scaled unit tangent vector fields which are originally defined as sphere curves. The obtained ATPH curves have only polynomial parametric speeds but the curves can be employed to represent several typical non-polynomial curves without rational form. A simple algorithm for geometric Hermite interpolation by the proposed spatial ATPH curves without or with arc length constraint has been given. Given two boundary points and two unit tangents at the points, possibly with a prescribed arc length, a unique interpolating ATPH curve can be obtained by solving a simple linear system.