Formula to derive angle and radius from Bezier circular curve control points

Question

I know the x,y coordinates for the 2 endpoints and the 2 control points for a Bezier circular curve that is less than 180 degrees. I do not know the radius of the circle or the angle of the curve. Could someone give me a formula--or two--where I could plug in the 4 sets of coordinates and return the radius and angle?

Answer 1

One way ...

Compute the point on the curve at $t=\tfrac12$. If the four control points are $P_0$, $P_1$, $P_2$, $P_3$, then this point is $$ P_m = \tfrac18 P_0 + \tfrac38 P_1 + \tfrac38 P_2 + \tfrac18 P_3 $$ The three points $P_0$, $P_m$, $P_1$ define a circle whose center and radius you can compute fairly easily. These answers show you how.