# Calculating originally arc approximated by cubic bezier curve

I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant:

$$\\k = \frac43*(\sqrt{2}-1)$$

This means, that the distance from the end points to the control points of the bezier curve is just r*k. The information is based on the riskus paper for bezier arc approximation.

Now I am looking for a solution (analytical or numerical) to revert the approximation and calculate the originally arc.

On this article, there is described the formula calculate the magnitude of the vector from each arc endpoint to its control point, where alpha is the arc angle.

$$\\length = k*r*tan(\frac\alpha2)$$

This formula only works if the arc is centered around the X axis. Is there an universal formula to calculate the radius or angle with the given control points using the kappa constant? Or are there some other numerical algorithms to revert the approximation?

Currently, I determine the normal vectors of the tangents (start-c1 and end-c2) and calculate the intersection point (arc center). The problem on this solution is the bad accuracy while processing big arcs caused by the radial drift of 0.027253% described here

Any suggestions?

The number $k$ that you cited is suitable only for circular arcs whose angular span is 90 degrees.