Four-arc approximation to ellipses: The best in general

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 E, we first draw two auxiliary circular arcs tangent to E at the axes but having a gap ε 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. E are ε and −ε respectively. Giving the radii of S and L   a decrement −ε/2−ε/2 and an increment ε/2ε/2 brings them to join smoothly. Thus, reducing the overall error to minimum, an analytic solution in implicit form is derived.

► Allowing both diameters a little change, the best approximation in math is pursued. ► A novel approach (adding auxiliary gapped arcs) achieves it successfully. ► Being the crux of this approach, the desired error distribution proves to be true. ► The solution is simply an implicit (or even explicit) equation with one unknown. ► The new result cuts the fitting error down to about 0.7 of the existing best.