Abstract
Let P = p1; : : : ; pn and Q = q1; : : : ; qn be two point sets lying in the interior of rectangles in the plane. We show how to construct a piecewise linear homeomorphism of size O(n2 ) between the rectangles which maps pi to qi for each i. This bound is optimal in the worst case; i.e., there exist point sets for which any piecewise linear homeomorphism has size (n2 ).