Suppose I have two circles that do not intersect.
$(x-x_1)^2+(y-y_1)^2=r_1$
$(x-x_2)^2+(y-y_2)^2=r_2$
If I know the two outer common tangent line segments:
$a_1y+b_1x=c_1$ for $x_{t_{11}}<x<x_{t_{21}}$ and $y_{t_{11}}<y<y_{t_{21}}$
$a_2y+b_2x=c_2$ for $x_{t_{12}}<x<x_{t_{22}}$ and $y_{t_{12}}<y<y_{t_{22}}$
where the $x_ts$ and $y_ts$ are the tangent points. What would be the parametric formula for the shape of the two circles connected with their outer tangent lines? An example is shown in below figure.
