Rational approximation of rotation minimizing frames using Pythagorean–hodograph cubics

This article is devoted to the rotation minimizing frames that are associated with spatial curves. Firstly we summarize some results concerning the differential geometry of the sweeping surfaces which are generated by these frames (the so–called profile or moulding surfaces). In the second part of the article we describe a rational approximation scheme. This scheme is based on the use of spatial Pythagorean hodograph (PH) cubics (also called cubic helices) as spine curves. We discuss the existence of solutions and the approximation order of G Hermite interpolation with PH cubics. It is shown that any spatial curve can approximately be converted into cubic PH spline form. By composing the rational Frenet–Serret frame of these curves with suitable rotations around the tangent we develop a highly accurate rational approximation of the rotation minimizing frame. This leads to an approximate rational representation of profile surfaces.