How to find midpoint of an arc in MATLAB

Question

It is necessary for me to find midpoint of some arc in MATLAB. For simplicity let this arc is a part of circle. Also, we know the radius, center and starting point and ending point of the arc. I used $atan2(y,x)$ to get the angle for each line constructed by start and end points. Then find $t_{mid}=\frac{t_1+t_2}{2}$ and use $x_{mid}=r*cos(t_{mid})+xc,~y_{mid}=r*sin(t_{mid})+yc.$ But the problem is that $atan2$ gives the angle between $-\pi$ and $\pi$. Then for example for third quarter if $t_1=\pi$ and $t_2=-\pi/2$, $t_{mid}=\pi/4$ and it is not true. If I add $2\pi$ to the angles lower than zero, then I have the same problem for fourth quarter, for example $t_1=\frac{3\pi}{2},t_2=0$, the mid angle is $\frac{3\pi}{4}$!!. How can I solve this problem?

Answer 1

Compute the coordinates of the midpoint $I$ of the endpoints (which is outside of the arc, of course): its polar angle is the same as the polar angle of the point you are looking for. The rest is easy.