We call an embedding of a weighted graph G on a surface S isometric, if edges
of G are mapped to geodesic arcs on S so that arc lengths are equal to
corresponding edge weights. The metric conditions of an embedding correspond
to systems of nonlinear differential and algebraic equations, describing a net of
intersecting geodesic arcs on the surface.
As a real-world application (and as the actual incentive for this work), consider
fitting snow tire chains to a car tire. The chain segments under tension follow
geodesic curves along the convex hull of the tire surface. A tension chain will
pull out all possible slack, thus enforcing a minimum distance condition between
a set of vertices.
We present algorithms that either find a feasible embedding or, since there may
be no solution fulfilling all constraints, an embedding that at least approximately
fulfills all distance conditions.