  We will develop simple relations between the arc-lengths of a pair of
geodesics that share common end-points. The two geodesics differ only by the
requirement that one is constrained to lie in a subspace of the parent
manifold. We will present two applications of our results. In the first example
we explore the convergence of Gaussian curvature estimates on a simple
triangular mesh. The second example demonstrates an improved error estimate for
the area of a Schwarz lantern.
