Matching Terrains under a Linear Transformation∗

We study the problem of matching two polyhedral terrains, where one can be changed vertically by a linear transformation of the third coordinate (scaling and translation). We give an algorithm that minimizes the maximum distance over all linear transformations in O(npolylog n) expected time. We also study matching two 1-dimensional terrains, and give a (1+ ε)-approximation algorithm for minimizing the area in between that runs in O(n/ √ ε) time, for any fixed ε > 0.