Optimising the four-arc approximation to ellipses

The four-arc approximation to ellipses is important for such applications as image processing, curve fitting, NC machining, cam design, and collision avoidance. As approximation quality is evaluated in engineering usually by the maximum error, this paper aims to minimize the maximum error, starting from an analysis of the traditional method, by which, in a quarter of an ellipse, the maximum errors of the four arcs appear at two points: one at the small circular arc (positive error), the other at the large arc (negative error), but the latter's absolute value is quite less than the former. Since on the condition for tangent continuity one of the four-arc parameters (e.g., p) is free, we can adjust the parameter to reduce the former error with a trade-off of increasing the latter's absolute value, so as to improve the overall accuracy of the approximation. An analytical function of the optimal p/a versus b/a in implicit form, and furthermore in explicit form with negligible error, are obtained.