Comparison of Plane Forms

Nonlinear two-dimensional grid transformations are used to compare plane figures. The method is based on the identification of homologous points, and thus can be considered as a bidimensional extension of ordinary curvilinear regression, but is here approached from the point of view of cartography. All of the techniques illustrated by Thompson in 1917 can be mimicked by the computer program. Extensions of the method, based on the derivatives of the transformations, permit the determination of the axes of principal change, and also suggest numerical measures for the degree of similarity of topologically equivalent figures. Applications are illustrated in biology and geography, and more general uses are suggested.     

REGIONAL INCOME EVOLUTION IN SOUTH AFRICA AFTER APARTHEID*

ABSTRACT South Africa is one of the wealthiest countries on the African continent. The high national level (and growth) of GDP per capita, however, masks significant differences in economic performance across South Africa's regions. This paper uses (spatial) Markov chain techniques to describe the evolution of the entire cross‐section regional income distribution in terms of its intra‐distributional characteristics during the post‐Apartheid period. The results indicate a heavily diverging regional income distribution. Relatively poor regions are likely to remain poor or become even poorer and the richest regions will maintain their lead in terms of income levels. Explicitly taking account of space furthermore shows that these high‐income regions are acting as local growth poles, absorbing economic activity from their immediate surroundings. Location, trade, education, and the variable fortune of the gold mining industry seem to be important determinants of the observed evolution.     

The Gender of Money: Byzantine Empresses on Coins (324–802)

In Byzantium, images on coins were a means to disseminate information about the state widely and quickly. Portraits of emperors were standard; empresses also sometimes appeared. Between 324 and 802, imperial women served as important ideological resources, and their portraits on coins always served as symbols of state well‐being and especially of legitimacy for the current ruling family. Changes in the coin imagery demonstrate how the mints used images of imperial women to respond to specific political situations, but also allow us to chart broader shifts in Byzantine gender strategies.     

Reconstructing Population Density Surfaces from Areal Data: A Comparison of Tobler's Pycnophylactic Interpolation Method and Area-to-Point Kriging

We compare Tobler's pycnophylactic interpolation method with the geostatistical approach of area-to-point kriging for distributing population data collected by areal unit in 18 census tracts in Ann Arbor for 1970 to reconstruct a population density surface. In both methods, (1) the areal data are reproduced when the predicted population density is upscaled; (2) physical boundary conditions are accounted for, if they exist; and (3) inequality constraints, such as the requirement of non-negative point predictions, are satisfied. The results show that when a certain variogram model, that is, the de Wijsian model corresponding to the free-space Green's function of Laplace's equation, is used in the geostatistical approach under the same boundary condition and constraints with Tobler's approach, the predicted population density surfaces are almost identical (up to numerical errors and discretization discrepancies). The implications of these findings are twofold: (1) multiple attribute surfaces can be constructed from areal data using the geostatistical approach, depending on the particular point variogram model adopted—that variogram model need not be the one associated with Tobler's solution and (2) it is the analyst's responsibility to justify whether the smoothness criterion employed in Tobler's approach is relevant to the particular application at hand. A notable advantage of the geostatistical approach over Tobler's is that it allows reporting the uncertainty or reliability of the interpolated values, with critical implications for uncertainty propagation in spatial analysis operations.