[coherentpony]

It depends on how you define "distance". My initial thought when reading your comment was that you couldn't possibly use the Euclidean distance because you'd be drawing a path between two points on the circle that involves points _not_ on the circle.

[taneq]

Using the Euclidean distance (Cartesian distance? I dunno) works because it's a monotonic underestimate of the distance along the circle. So you know if cartesian_dist(A, B) > cartesian_dist(A, C) then circle_dist(A, B) > circle_dist(A, C).

[Sniffnoy]

The original problem, though, is about computing distance, not about comparing distances. Comparing distances is just used as an intermediate step in the initial method given.