  We study distance functions from geodesics to points on Riemannian surfaces
with H\"older continuous Gauss curvature, and prove a finiteness principle in
the spirit of Whitney extension theory for such functions. Our result can be
viewed as a finiteness principle for isometric embedding of a certain type of
metric spaces into Riemannian surfaces, with control over the H\"older seminorm
of the Gauss curvature.
