I have read that the intersection of a surface (or any higher dimensional manifold) with a plane gives the geodesics of that surface. For example, the intersection of a sphere with a plane through the sphere's centre and the points A and B gives the geodesic between A and B (a segment of the great circle through A and B).
I have also read that we can do this on a hyperboloid. We take a plane through the origin (or centre if it is displaced) and points A and B and again this gives a geodesic.
However, this appears to be defining geodesics without knowing the metric which seems wrong. For example, if I choose a metric that makes it twice as expensive to move in the z direction as it does in the x and y directions then these geodesics will clearly need to change.
I think however that the example above is using a metric with respect to some ambient space (x,y,z) and so any change in metric there would cause the surface itself to change, since the sphere and hyperboloid are quadratic surfaces with respect to this metric. Am I correct in thinking that if I change the ambient metric then the surface geometry will adjust such that geodesics are still correctly found by taking intersections with planes?
Secondly, and more importantly, why do plane intersections give geodesics? Can we formalise this?