# Line-circle intersection in spherical geometry?

How does one calculate the intersections between a "line" (a Great Circle) and a circle in spherical geometry?

i.e. given start point (as lat,lon), heading, circle centre (as lat, lon) and circle radius (as a % of the sphere's radius), there will be between zero and two* locations where they meet.

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This is just a special case of circle-circle intersection, because a great circle is just a circle with radius $\frac{\pi R}{2}$. If you embed your sphere in Euclidean $\mathbb{R}^3$ then a circle on the sphere is the intersection of a plane with the sphere (where the plane passes through the sphere's centre iff it's a great circle).