Consider for a moment a length of uniform wire or chain which goes between two posts of equal height. If we assume the earth to be flat then we can predict the shape of the curve using
$$y = a \cosh \left(\frac{x}{a}\right)$$
But if we had a very long wire between two very tall posts which were angle theta apart on a planet which is a perfect sphere then what would the equation be which gives the height of the wire as a function of the angle between the two posts.
Note that in the "flat earth" wire example the gravitational field strength which the wire is in will be constant, while in the case of a wire which goes between two very tall towers the gravitational field strength is not constant. This is likely to make the problem more complex.