We consider a geometric matching of two realistic terrains, each of which is modeled as a piecewise-linear bivariate function. For two realistic terrains f and g where the domain of g is relatively larger than that of f, we seek to find a translated copy f' of f such that the domain of f' is a sub-domain of g and the L_∞ or the L_1 distance of f' and g restricted to the domain of f' is minimized. In this paper, we show a tight bound on the number of different combinatorial structures that f and g can have under translation in their projections on the xy-plane. We give a deterministic algorithm and a randomized algorithm that compute an optimal translation of f with respect to g under L_∞ metric. We also give a deterministic algorithm that computes an optimal translation of f with respect to g under L_1 metric.
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