  We explicitly construct simple, piecewise minimizing geodesic, arbitrarily
fine interpolation of simple and Jordan curves on a Riemannian manifold. In
particular, a finite sequence of partition points can be specified in advance
to be included in our construction. Then we present two applications of our
main results: the generalized Green's theorem and the uniqueness of signature
for planar Jordan curves with finite p-variation for 1<=p<2.
