Determining circle-line intersection on sphere

a third idea is:
- take the plane of the great circle, and thus its normal vector $\bf n$ (or just determine it by the cross product of the position vectors of two points on the "line")
- take the vector of the center of the small circle $\bf c$
- determine the angle $\alpha$ between $\bf c$ and $\bf n$ ($\arccos$ of the normalized dot product)
- $R (\pi/2-\alpha)$ is the distance along the earth surface from the center of the small circle to the great circle considered.

First idea : (probably the simplest) Determine the plane of the great circle and the plane of the small circle and determine if their intersection line $L$ is exterior or not to the sphere (for example by testing if the shortest distance of the origin to line $L$ is larger than radius $R$ or not).

Second idea : Use stereographic projection with direct and inverse formulas you will find there.

In this document, it is also established that the image of circles, either great circles or others, are still circles in the projection plane ; your issue becomes a plane (analytical) geometry question : do two circles intersect, and if such is the case, in which points (using in a further step inverse formulas to obtain the spherical coordinates of these points).

There are exceptional cases (circles passing though the pole) that should (and can) be avoided, for example by doing a preliminary rotation that will be compensated later on.