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We present a computationally simple and inexpensive method to induce coincidence between one map and another, regardless of their respective projections. We imagine the map to be transformed on a rubber sheet stretched to coincide with a stable base map at an average of 15 control points; for more precise coincidence, we add more control points. We compute triangles on the rubber-sheet map, using control points as vertices, and with a procedure called piecewise linear homeomorphism, map these triangles linearly onto corresponding triangles on the stable map.

Cartograms are proving increasingly popular, but relatively little is known about the problems related to their use. A test program measured the performance of relatively inexperienced map users in identifying locations on a topological cartogram and on the geographic map from which it was derived. The spatial pattern of errors and the variation in task difficulty showed the dominant influence of two map transformation characteristics: change in angular location with respect to other units and change in unit shape. Also, it took significantly longer to perform the task on the map than on the cartogram.

A computationally efficient piecewise-linear rubber-sheet map transformation is derived by analyzing geometrical relationships of areas, linear equations and inequalities, and simplicial coordinates. The use of simplicial coordinates permits a representation of the rubber-sheet transformation, which is completely independent of the image points. The geometric analysis used in deriving the simplicial coordinates leads to special point-in-polygon algorithms for the rubber-sheeting, which may be employed in more general situations. The geometric analysis also leads to convenient formulas for areas of triangulated regions.