Approximating Offset Curves by Rational Bézier Cubics and Quartics

Offset curves arise in a variety of industrial applications such as robot’s path planning and numerical control machining in the textile, shoe and automobile industries. Rational curves, in particular the rational cubics, are widely accepted as a standard representation for design problems and geometric modellers but their offset curves are in general not rational. Given a rational cubic or quartic spline, we present two local methods to approximate its offset curve using a rational Bézier spline of the same degree. This approximate offset curve interpolates the positions and unit tangents at both ends of the exact offset curve segments and its curvatures at these endpoints are consistent with the offset distance and the corresponding curvatures of the given curve. It has second order geometric continuity if the given curve is so. The accuracy of the approximation can be refined by a local iterative subdivision process.