Constructing Rotation-Minimizing Frame of Space Bézier Curve

The rotation-minimizing frame(RMF) is the most attractive frame for applications such as animation, sweep surface construction, and motion planning. However, it is difficult to construct rational RMF which is compatible with the current CAD system for Bézier and B-spline curves. Since Pythagorean hodograph (PH) curves admit rational Euler-Rodrigues frame(ERF) exactly and cubic PH curve shares optimal geometric properties, in this paper, a cubic PH-spline with G1 continuity is constructed according to the endpoint conditions of the subdivided segments after inserting knots into the Bézier curve. Correspondingly, the PH-spline is an approximation for the given space Bézier curve. The least number of subdivisions and the bound error can be efficiently controlled by our formula. We construct ERF for every cubic PH curve which belongs to the PH-spline, then the corresponding RMF for each cubic PH curve can be generated by rotating the last two vectors of ERF for the PH-spline on the normal planar. The total RMF in rational form for PH-spline can be regarded as approximate RMF for the Bézier curve. The method can also be used to solve same problem on B-spline curve.