 Solution

At the first endpoint, the Cartesian coordinates are $(r,0,0)$, so its great circle must have $l=0$. At the second endpoint, the plane condition gives $m\sin {\theta }_0\sin {\varphi }_0+n\cos {\theta }_0=0$. Since $\cos {\theta }_0>0$, an equivalent regular plane equation isWhere $\cos \theta \ne 0$, divide by $\cos {\theta }_0\cos \theta$ to obtainThe regular implicit equation includes the equatorial first endpoint; the tangent form is undefined there and must be read by a limit. The shorter connecting arc of this great circle is the shortest path. The endpoints are not antipodal, so this shorter arc is unique. For example, the central angle is $\delta =\arccos (\sin {\theta }_0\cos {\varphi }_0)$, giving spherical distance $r\delta$.