of area cartograms. Anamorphic Projection This is a type of contiguous cartogram that uses a single parametric mathematical formula (such as a polynomial curved surface) to distort space itself to equalize the spatial distribution of the chosen variable, rather than distorting the individual features. Because of this distinction, some have preferred to call the result a pseudo-cartogram. Tobler's first computer cartogram algorithm was based on this strategy, for which he developed the general mathematical construct on which his and subsequent algorithms are based. This approach first models the distribution of the chosen variable as a continuous density function (usually using a least squares fitting), then uses the inverse of that function to adjust the space such that the density is equalized. The Gastner-Newman algorithm, one of the most popular tools used today, is a more advanced version of this approach. Because they do not directly scale the districts, there is no guarantee that the area of each district is exactly equal to its value. Shape-warping contiguous cartograms Also called irregular cartograms or deformation cartograms, This is a family of very different algorithms that scale and deform the shape of each district while maintaining adjacent edges. This approach has its roots in the early 20th Century cartograms of Haack and Weichel and others, although these were rarely as mathematically precise as current computerized versions. The variety of approaches that have been proposed include cellular automata, quadtree partitions, cartographic generalization, medial axes, spring-like forces, and simulations of inflation and deflation. Some attempt to preserve some semblance of the original shape (and may thus be termed homomorphic), but these are often more complex and slower algorithms than those that severely distort shape. Non-contiguous isomorphic cartograms This is perhaps the simplest method for constructing a cartogram, in which each district is simply reduced or enlarged in size according to the variable without altering its shape at all. In most cases, a second step adjusts the location of each shape to reduce gaps and overlaps between the shapes, but their boundaries are not actually adjacent. While the preservation of shape is a prime advantage of this approach, the results often have a haphazard appearance because the individual districts do not fit together well. Diagrammatic (Dorling) cartograms In this approach, each district is replaced with a simple geometric shape of proportional size. Thus, the original shape is completely eliminated, and contiguity may be retained in a