The best G1 cubic and G2 quartic Bézier approximations of circular arcs

We obtain cubic and quartic Bézier approximations of circular arcs that respectively satisfy G1 and G2 end-point interpolation conditions. We identify the necessary and sufficient conditions for such approximations to be the best, in the sense that they have the minimum Hausdorff distance to the circular arc. We then establish the existence and uniqueness of these best approximations and present practical methods to calculate them, which are verified by examples.