Efficient circular arc interpolation based on active tolerance control

In this paper, we present an ef®cient sub-optimal algorithm for ®tting smooth planar parametric curves by G 1 arc splines. To ®t a parametric curve by an arc spline within a prescribed tolerance, we ®rst sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of ®tting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal ®tting arc spline in the end. Even more, the ®tting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide ®elds such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and ef®ciency of the method. q 2002 Elsevier Science Ltd. All rights reserved.