Building two communication towers over a terrain

We consider visibility problems on a terrain. Given two stations located at different points on the terrain, we would like to find the minimum heights of communication towers at the two locations so that they can communicate freely. Formally, given two points q₁ and q₂ on the terrain, we would to find the height h₁ and h₂ such that if one lift q_i up by h_i, for i = 1, 2, the lifted points see each other with the minimum cost that depends linearly on h₁ and h₂. We consider two versions of the problem. When given two points q₁ and q₂ on a terrain with n vertices, with the preprocessing time of O(n log n), we can answer the query in time O(log n + d) where d is the number of terrain edges between q₁ and q₂. When one point q₁ is fixed, given q₂ and ε > 0, we can give an approximate answer with additive error at most ε in time O(log² (h_max/ε) · log n) where h_max is the maximum height in terrain. In this case, the preprocessing time is Õ(n₂).