Say I have an arc from $P_0$ to $P_3$, that has unit tangents $t_0$ and $t_3$, respectively. Let $L$ denote the distance between $A$ and $B$.
If $P_0 = (0, 1)$, $P_3 = (1, 0)$, $t_0 = (1, 0)$ and $t_3 = (0, 1)$, then the other two points of the cubic Bézier approximating the arc are $P_1 = (k, 1)$ and $P_2 = (1, k)$ where $k = \frac 4 3 (\sqrt 2 - 1)$. More generally, if the angle between $t_0$ and $t_3$ is 90°, then $P_1 = P_0 + k_n L t_0$ where $k_n = \frac{4 - 2 \sqrt 2}{3}$ (and similar for $P_2$).
What is a general form approximation for an arbitrary convex arc, expressed in terms of $P_0$, $P_3$, $t_0$ and $t_3$? More specifically, what is $k$, generally, expressed in terms of the angle $\theta$ between $t_0$ and $t_3$?
For what it's worth:
| $\theta$ | $k_n$ |
|---|---|
| 180° | $k_n \approx 0.6575$¹ |
| 90° | $k_n = \frac {4 - 2 \sqrt 2}{3} \approx 0.39$ |
| 0° | $k_n = \frac 1 3$ |
(¹ This was eye-balled in Inkscape and may not be as accurate as implied by the number of digits.)
This answer is relevant, but only gives a partial solution, in that a) I don't have the arc length, and b) I need to be able to calculate a value for the $\theta = 0$ case wherein the arc radius is infinite. Thus, I am still looking for an answer expressed in terms of the distance between $P_0$ and $P_3$, and particularly, one that gives a finite answer over $0° \le \theta \le 180°$.
