Quote by teodron
Thank you all for your replies!
As for cutting the surface with a perpendicular knife, it's dubbed the "greedy algorithm" to approximate geodesics on manifolds (probably not producing genuine geodesics..)  how can I prove or disprove this?
Again, thank you..

What do you mean by the greedy algorithm?
The perpendicular knife determines a plane that contains the normal to the surface. So the acceleration of a unit speed curve that lies on the surface must be normal to the surface.