Four-arc approximation to ellipses: The best in general

摘要

So far, the four-arc approximations to an ellipse E are made under the condition that the major and minor axes keep strictly unchanged. In general, however, this condition is unnecessary. Then the fitting can be further improved. Considering a representative quadrant of £, we first draw two auxiliary circular arcs tangent to E at the axes but having a gap s at their boundary, such that the small arc S lies outside the large arc L. Meanwhile the extreme errors of S and L w.r.t. £ are e and -s respectively. Giving the radii of S and L a decrement — e/2 and an increment e/2 brings them to join smoothly. Thus, reducing the overall error to minimum, an analytic solution in implicit form is derived.