Robust computation of the rotation minimizing frame for sweep surface modeling

Abstract

The rotation minimizing frame is superior to the Frenet frame for modeling sweep surfaces [F. Klok, Computer Aided Geometric Design 3, 217-229 (1986)]. However, the existing techniques for computing the rotation minimizing frame either have low approximation degree or are unrobust numerically. We present a method to compute an approximate rotation minimizing frame in a robust and efficient manner. The following problem is studied. Given an axial curve A(u) in space and a 2D cross-section curve C(v), generate a sweep surface S(u, v) = A(u) + F(u)C(v), where F(u) is a rotation minimizing frame defined on A(u). Our method works by approximating A(u) with a Gl circular-arc spline curve and then sweeping C(v) with a rotation minimizing frame along the approximating circular-arc spline curve; the sweep surface thus generated is an approximation of S(u, v). The advantages of this method are: (1) the approximate rotation minimizing frame is computed robustly, with its error being much smaller than would be obtained by Klok's linear method with the same number of segmentations; (2) the sweep surface generated is a NURBS surface if the cross-section curve is a NURBS curve; (3) the method is easily adapted to generating a smooth and closed sweep surface when A(u) is a closed smooth curve. © 1997 Elsevier Science Ltd. All rights reserved.