Waldo Tobler

University of California, Santa Barbara | UCSB · Department of Geography

Distance cartograms can be viewed as a special type of azimuthal map projection. Area cartograms are a generalization of equal area maps in the sense that conventional equal area maps are a special case of area cartograms in which the physical earth surface is the propeerty that is preserved. This can be shown by examination of the map projection e...

The inappropriate use of the Mercator projection has declined but still occasionally occurs. One method of contrasting the Mercator projection is to present an alternative in the form of an equal area projection. The map projection derived here is thus not simply a pretty Christmas tree ornament: it is instead a complement to Mercator’s conformal n...

Some geographical phenomena are continuous and exist in whole number dimensions. Topography, for example. Other phenomena, such as population density, depend heavily on the area used in their computation. Some refer to this as existing in 2 12 dimensions. Is the difference just because it is a computed, rather than an observed quantity? I argue the...

Article by Kos and Pogany

Isard has argued for greater presentation of empirical data in regional studies by displaying tables and geographical maps. In this note his suggestion is followed by displaying two sets of computer maps using tables given in his classic 1960s book

Editors' overviewThe map as a data storage elementThe map as a computer inputThe map as data processing outputOther applications of data processing procedureThe possibilities for automationReferencesFurther readingSee also

The present contribution can be divided into three sections. In the first part we describe the Dirichlet problem for polygonal areas. Here we specify an abstract mathematical problem and its resolution and present a simple computed example. In the second part we speculate on possible scientific interpretation of variants of this problem; we then ev...

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 1...

A method of spatial interpolation from an arbitrary arrangement of numerical observations is derived, with examples shown as contour maps. The fit to the original data and the smoothness of the resulting lattice values are optimized simultaneously by an iterative numerical technique. The algorithm does not require specification of distance dependen...

A potential model, which allows one to estimate a net flow table from observed geographical changes, is derived. The relation of this model to Poisson's equation is established; numerical evaluation then permits the computation of a continuous spatial flow field. The model may have promise as a device for the forecasting of net geographical migrati...

The mathematics of a push-pull model are shown to incorporate many of Ravenstein's laws of migration, to be equivalent to a quadratic transportation problem, and to be related to the mathematics of classical continuous-flow models. These results yield an improved class of linear spatial interaction models. Empirical results are presented for one co...

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...

Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/60276/1/tobler.pdf

In this paper I look at azimuthal projections, equal area projections, and spherical and map distances, with emphasis on less known variants.

Computer manipulation of geographical data involves record matching, testing of theories, production of maps, and spatial pattern analysis. A brief review is presented of the operational status of computer programs in these areas.

Volume 219, Democratic Representation and Apportionment: Quantitative Methods, Measures, and Criteria pages 215–220, November 1973

This document was delivered over the internet. The purpose of Solstice is to promote interaction between geography and mathematics. Articles in which elements of one discipline are used to shed light on the other are particularly sought. Also welcome, are original contributions that are purely geographical or purely mathematical. These may be prefa...

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In this paper we present a methodology for comparing the adequacy of two or more models in terms of how well they represent a given data set. a set of interregional migration models is tested, including several variations of push-pull models. Wilson's entropy maximizing model, a quadratic programing solution, and an ANOVA model. Testing is undertak...

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The notion of a cartogram is reviewed. Then, based on a presentation from the 1960s, a direct and simple introduction is given to the design of a computer algorithm for the construction of contiguous value-by-area cartograms. As an example, a table of latitude/longitude to rectangular plane coordinates is included for a cartogram of the United Stat...

The qibla problem—determination of the direction to Mecca—has given rise to retro-azimuthal map projections, an interesting, albeit unusual and little known, class of map projections. Principal contributors to this subject were Craig and Hammer, both writing in 1910. A property of retro-azimuthal projections is that the parallels are bent downwards...

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I initiated a course with the title 'Analytical Cartography' in the late 1960's at the University of Michigan in Ann Arbor. At the request of Dr. H. Moellering of Ohio State University (who was at one time a student in the course) a short personal historical perspective of the development of the course was presented at the recent Hawaii meeting of...

program to demonstrate this, with five commodities, including strawberries, wheat, other foodstuffs, and Cadillacs. Given the other inputs I could demonstrate computer printouts of circularly arranged optimal land uses, and could even run the program with multiple market places to get mixtures of partially concentric patterns, and could include a c...

"We report on a project that converted subnational population data to a raster of cells on the earth. We note that studies using satellites as collection devices yield results indexed by latitude and longitude. Thus it makes sense to assemble the terrestrial arrangement of people in a compatible manner. This alternative is explored here, using lati...

This study examines the relationship between the size of U.S. cities and the size of street-level databases containing information on the urban street network. Models deduced from work in two research domains, allometric theory and spatial database theory, are calibrated using regression analysis based on data for 240 Census Bureau Dual Independent...

The traditional equidistant assignment of ocean area rights to coastal neighbors is not always equitable. A well-defined and easily computed alternative is provided by a line dividing the sea into areas of equal dominance relative to each of several countries. The Gulf of Maine is used to illustrate the concept. © 1997 John Wiley & Sons, Inc.

A transition is made from an initial discrete model of migration to a spatially continuous version by letting the number of geographic places increase without bound When this happens the parameters become continuous functions of location and the model is written in terms of partial djfferential equations. It is then shown how this formulation needs...

It is important that cartographers, and others who use geographic information, understand the procedures used to adjust empirical measurements made for the purpose of specifying locations. Two new graphical interpretations of reconciling distance or directional observations are presented in this paper. These graphics are consistent with standard pr...

Over one hundred years have passed since Ernst Georg Ravenstein published his “Laws of Migration.” This paper addresses the question of how these laws have withstood the test of time. Ravenstein’s 1885 paper also includes a map of “Currents of Migration,” not mentioned in the text, which this article examines for its impacts on the study of migrati...

There now exist several microcomputer processing systems which incorporate algorithms and display techniques appropriate for the class of objects known as images. Zooming, histogram equalization, contrast stretching, ratioing, edge enhancement, and filtering are common options used in these systems. In the present discussion a system is proposed fo...

Since its invention by Francis Galton in 1877 regression analysis has been found useful in almost all disciplines. Comparison of geographic phenomena requires a two-dimensional extension of this technique. In this manner geographic maps can be compared with each other. Possible applications include geometric comparison of ancient and modern maps, o...

Continuous transportation modelling was introduced by Martin Beckmann in 1952, and continues in some of his recent books with Tonu Puu. My own recent work has also dealt with this topic and with spatial interaction models in both continuous and discrete form, especially as related to migration. The general class of models here includes the PP/D gra...

The geographical arrangement of people on the surface of the earth is approximated by a mathematical equation of 361 terms. This is a convenient form for comparison with other distributions and for use in global change studies, and it has other advantages, but it must be considered preliminary because of data limitations.

A satellite picture of adjacent cities clearly illustrates the analytical dynamics of urban growth.

Mathematical procedures for the generalization of isarithmic maps are derived. It is shown that the level of generalization for different purposes may be specified in advance, and that under certain conditions the original map can be recreated from the generalized map. Les procédures mathématiques utilisées pour la généralisation des cartes d'isari...

The results of an analysis of geographical data should not depend on the spatial coordinates used - the results should be frame independent. This should also apply when areal units are used as the spatial data collection entity. Previous work has shown that some analysis procedures do not yield the same results under alternate areal aggregations, b...

The anchor-point hypothesis of spatial cognition, according to which primary nodes or reference points anchor distinct regions in cognitive space, brings together certain frequently reported apparent properties of mental maps: the regionalization and hierarchical organization of cognitive space, and the active role of salient cues in structuring sp...

Migration maps represent patterns of geographical movement by arrows or bands between places, using information arriving in "from-to" tables. In the most interesting cases the tables are of large size, suggesting that computer assistance would be useful in the preparation of the maps. A computer program prepared for this purpose shows that graphica...

A numerical model applied to the density and flow of wealth in the United States between 1970 and 1976 shows that wealth moves with people. The maps representing this flow resemble those of the net population movement in the same period. Wealth can be created, destroyed, and moved from place to place. Actual from-to tables of the geographical movem...

Kinetic perimetry is still a widely accepted and used means of assessing the visual field. Attempts at quantifying the area of vision for given isopters have been limited to area measurements on visual field charts, which are azimuthal equidistant polar map projections of the inside surface of the perimetry bowl. Such measurements distort the true...

The departure of one map projection from another can be calculated from the lengths of the vectors connecting corresponding points. This allows one to decide the relative similarity of two map projections and suggests a procedure for finding a projection close to a given one but with different properties. The departure of a map from the sphere can...

Specific polycylindric map projections are derived and illustrated, one of which is equal area. Equations are given for the entire class of polycylindric projections.

Pseudo-cartograms provide a convenient approximation to pycnomirastic map projections, and are easily computed from one-dimensional integrals of marginal density distributions. Examples are given using world population.

A geographically continuous movement model can be deduced from the quadratic transhipment problem by considering the transportation network as a discrete mesh. The resulting system of coupled Helmholz equations can be considered to solve simultaneously the spatially continuous versions of the traffic “distribution” and “assignment” problems. The as...

The globular map projection is a special case of the interrupted world map presented here.

Elementary geometric assumptions are used to derive a spatial-interaction model in which places are related to each other through a set of simultaneous linear equations. The system has simple properties with respect to aggregation and turnover, yet incorporates spatial competition, adjacency, and effects of geographic shadowing. The objective funct...

Approaches to improving spatial modelling for remote sensing are presented. Examples of variograms of various scenes are shown.

In this paper we present a methodology for comparing the adequacy of two or more models in terms of how well they represent a given data set. A set of interregional migration models is tested, including several variations of push-pull models, Wilson's entropy maximizing model, a quadratic programming solution, and and ANOVA model. Testing is undert...

A method is described for cartographically illustrating interstate flows of federal monies within the contiguous United States, using as data the differences between government collections and expenditures by state. It is assumed that movements between surplus and deficit areas are governed by a potential field, for which the driving potential can...

Cartographic transformations are applied to locative geographic data and to substantive geographic data. Conversion between locative aliases are between points, lines, and areas. Substantive transformations occur in map interpolation, filtering, and generalization, and in map reading. The theoretical importance of the inverses is in the study of er...

Census enumerations are usually packaged in irregularly shaped geographical regions. Interior values can be interpolated for such regions, without specification of "control points," by using an analogy to elliptical partial differential equations. A solution procedure is suggested, using finite difference methods with classical boundary conditions....

Net geographical interactions are assumed to be related to differences in attractivity, reduced by spatial separation. It is demonstrated that only the differences of the marginals of the interaction table are required to calculate the geographical attractivities thus defined. Variants of this potential model are confronted with a 48 x 48 table of...

A method of piecewise quadratic polynomial approximation of histograms or digital pictures is described. An example is given of the use of this method to reconstruct a smooth image from averaged and sampled data.

Captain Ahab, in the film version of Moby Dick, searches for the white whale with the aid of a geographical map on which are noted sighting-frequencies within 5° cells bounded by lines of latitude and longitude. The written version of the story, dating from circa 1830, does not contain this scene, but the technique of recording geographical data in...

A procedure is described to obtain an empirical transformation regressing an independent plane configuration against a similar dependent configuration. As implemented the procedure constructs the mapping by estimating two bivariate tables, with arguments on a square lattice interpolated from the irregularly arranged original observations. The table...

As Geobase Information Systems increase in number, size and complexity, the format compatability of satellite remote sensing data becomes increasingly more important. Because of the vast and continually increasing quantity of data available from remote sensing systems the utility of these data is increasingly dependent on the degree to which their...

The literature on migration is extremely large and has been reviewed repeatedly (Greenwood, 1974; Price and Sikes, 1975; Shaw, 1975). But this literature is also rather disappointing and frustrating. Most of the models are of the push-pu11, economic motivation type and are stated in the form of single equations in which migration is the dependent v...

Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/23038/1/0000609.pdf

An algebraic examination of spatial models leads to the conclusion that a convenient description of the pattern of flows implicit in a geographical interaction table is obtained by displaying a field of vectors computed from the relative net exchanges. The vector field approximates the gradient of a scalar potential, and this may be invoked to expl...

Editors' overviewReferencesFurther readingSee also

Working Papers are not intended for distribution outside of IIASA, and are solely for discussion and infor-mation purposes. The views expressed are those of the author, and do not necessarily reflect those of IIASA.

Working Papers are not intended for distribution outside of IIASA, and are solely for discussion and infor~ mation purposes. The views expressed are those of the author, and do not necessarily reflect those of IIASA.

A computationally simple map projection is derived and is shown to have a correspondingly direct inverse, a single empirical parameter, and modest distortional properties. This is especially appropriate for rapidly and inexpensively displaying geographical data on an electronic tube under computer control, assuming that any computation is to be don...