I just started reviewing Differential Geometry extrinsically. I came across the following information:
“... .Now suppose our user is content to have a $map^1$ which makes it easy to navigate along the shortest path connecting the two paths. Ideally the user would use a straight edge, magnetic compass, and protractor to do this. S/he would draw a straight line on the map connecting p and q and steer a course which maintains a constant angle (on the map) between the course and meridians. This can be done by the method of stereographic projection. This chart is conformal (which means that it preserves angles). According to Wikipedia stereographic projection was known to the ancient Greeks and a map using stereographic projection was constructed in the early 16th century.”
$^1$ By a map here is meant a chart on the sphere $S^2$ as in the theory of topological manifolds.
I did not understand if the straight line would be drawn in the image of the chart or in the ambient space where the sphere lies. Also, where would this line end up useful in the construction of the minimal geodesic?