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 G 1 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.

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