TITLE: Bézier curve approximation of a circular Arc
QUESTION [6 upvotes]: I would like to know how I can get the coordinates of four control points of a Bézier curve that represents the best approximation of a circular arc, knowing the coordinates of three points of the corresponding circle. I would like at least to know the solution to this problem in the case where two of the known circle points are the two ends of a diameter of the circle.

REPLY [2 votes]: You can use the following ways to find the control points of a cubic Bezier curve for approximating a circular arc with end points $P_0$, $P_1$, radius R and angular span A:
Denoting the control points as $Q_0$, $Q_1$, $Q_2$ and $Q_3$, then
$Q_0=P_0$, 
$Q_3=P_1$, 
$Q_1=P_0 + LT_0$ 
$Q_2=P_1 - LT_1$ 
where $T_0$ and $T_1$ are the unit tangent vector of the circular arc at $P_0$ and $P_1$ and $L = \frac{4R}{3}tan(\frac{A}{4})$. 
Please note that above formula will give you a pretty good approximation for the circular arc. But it is not "the best" approximation. We can achieve an even better approximation with more complicated formula for the $L$ value. But for practical purpose, above formula is typically good enough.