Equations of the Mayr Projection

In this study, the pseudo-cylindrical projection of Franz Mayr is examined in detail. The computation of one of Mayr's projection equations depends on the solution of an elliptical integral. It is this characteristic of the projection that most likely contributes to it being the neglected one among the group of the pseudo-cylindrical projections available today. Franz Mayr used the Legendre tables for the elliptical functions E and F and gave the plane coordinates within one-degree latitude intervals on the 90° meridian. The research reported here derives analytical expressions instead of using the elliptical integral and the interpolation between the table values. Four different solutions have been introduced for mapping applications. The distortion quantities are also presented and discussed.     

Symbolization of Map Projection Distortion: A Review

One of the most fundamental steps in map creation is the transformation of information from the surface of a globe onto a flat map. Mapmakers have developed and used hundreds of different map projections over the past 2,000 years, yet there is no perfect choice because every map projection uniquely alters some aspect of space during the transformation process. Detailed information about the type, amount, and distribution of distortion is essential for choosing the best projection for a particular map or data set. The distortion inherent in projections can be measured and symbolized much like any other map variable. Methods for symbolizing map projection distortion are reviewed, with each method described and illustrated in graphical form. The symbolization methods are collected under ten separate headings organized from simple to more complex in terms of interpretation. Most of these methods are highly effective at communicating distortion, yet they are rarely used beyond textbooks and technical documentation. Map projections and the distortions they carry need to be better understood by spatial data developers, distributors, and users. Map distortion should be carried along with map data as confidence layers, and the easily accessible distortion displays should be available to help in the selection of map projections. There is a suitably wide array of symbolization methods to match any need from basic education to research.     

The Miller Oblated Stereographic Projection for Africa,Europe, Asia and Australasia

The Miller Oblated Stereographic Projection, implemented by both the American Geographical Society and the Defense Mapping Agency in small-scale mapping of Africa, Europe, Asia and Australasia, is conformal for most land masses, reducing overall scale and area distortion by using a double projection. “Fill-in” sections are not conformal. Developed in 1953–1955 by O. M. Miller, the projection is implemented via tables for map construction. With the advent of computer-assisted intelligence and operations systems that use this projection to provide electronic images, the computation of rectangular projection coordinates from geographic locations via table look-up is no longer adequate. This paper presents an algorithm, based upon the original Miller work, which performs this transformation in a form applicable to scientific programming languages.     

A Geodesic Map Projection for Quadrilaterals

Part of a cuboctahedron-based geodesic map projection by Buckminster Fuller is expanded to spherical quadrilaterals with opposite sides equal. Points on such a quadrilateral can be described by intersections of geodesics, which provide parameterized coordinate values in a Geodesic Coordinate System. Through a linear transform of the geodesic coordinates, a spherical quadrilateral maps to a parallelogram. The resulting map has the same constant scale on all four sides. Within the map, all lines parallel to the sides correspond to geodesics on the globe. Spherical rectangles, spherical squares, and diamonds (spherical rhombi) are specifically examined; various tiling sets of diamonds can be used to cover the globe. Non-projection applications in geometry, astronomy, and computer graphics are described.     

Variations of the Gringorten Square Equal-area Map Projection

Gringonen's square equal-area map projection has been forgotten since its appearance in 1972. I describe a modern implementation, including details of how to arrange, in different ways, the fundamental Gringonen projection of a sexadecant (one sixteenth of the surface of the sphere) onto a triangle. The Gringorten Mark I projection is an arrangement in which one hemisphere forms a square, with the other hemisphere disposed around it so that the whole sphere projects as a diamond, which may then be rotated to appear as a square. I introduce an alternative arrangement, the Gringorten Mark II, which is twice as high as it is wide, with one hemisphere on top of the other. These variants are compared with some other square map projections. Maps that fill a rectangular space completely can be very useful where, as on computer screens, space is limited and must be used efficiently.     

New Equal-Area Map Projections For Noncircular Regions

A series of new equal-area map projections has been devised. Called Oblated Equal-Area, its lines of constant distortion follow approximately oval or rectangular paths instead of the circles of the Lambert Azimuthal Equal-Area projection or the straight lines of the Cylindrical Equal-Area projection. The projection series permits design of equal-area maps of oblong regions with less overall distortion of shape and scale than equal-area maps on other projections.     

Designing Effective Bivariate Symbols: The Influence of Perceptual Grouping Processes

The purpose of this research was to empirically assess perceptual groupings of various combinations of symbol dimensions (e.g., graphic variables) used in designing bivariate map symbols. Perceptual grouping ability was assessed using the theory of selective attention, a construct first proposed in psychological research. Selective attention theory contends that one's ability to analyze a symbol's dimensions—such as color or size—is affected by other dimensions present in the same symbol. Symbol dimensions are described as either separable (capable of being attended to independently of other dimensions), integral (cannot be processed without interference from other dimensions), or configural (i.e., show characteristics of both integrality and separability, which may also form new, emergent properties). Without empirical evidence describing such interactions for various combinations of symbol dimensions, cartographers cannot truly evaluate the functionality of the symbols they use on maps. The symbol dimensions or graphic combinations chosen for this study were selected to incorporate a wide range of traditional cartographic symbolization, including line and lettering symbolization, areal shading, dot patterns, and point symbols. Combinations were examined in an abstract setting using a speeded classification task, which is the traditional means of studying selective attention. Subject reaction times provided an assessment of the levels of integrality, separability, and configurality. Results suggest that most symbol dimension combinations are either separable or exhibit evidence of asymmetrical dimensional interactions. Findings from this study will be integrated into subsequent experiments, the results of which will assist cartographers in the design of complex map symbols.     

Map-Projection Graphics from a Personal Computer

Some inexpensive personal computers may be programmed to produce, at a nominal incremental cost, map projection graphics useful as educational tools. Programs have been developed to produce outline maps based on anyone of dozens of projections, in almost any aspect, at a size of up to 13.5 by 27.1 cm or 20.3 by 20.3 cm [5?by 10? in. or 8 by 8 in.]. They are printed in a dot-matrix format normally containing up to 320 by 768 dots, with alternatives of 640 by 768 or 960 by 576 dots. Available options include features such as interrupted projection, miscellaneous great or small circles, and Tissot indicatrices. The maps often require many hours to prepare, but the programs can run unattended after initial parameters have been entered.     

Joint Leadership for Spatial Data Standards

The Defense Mapping Agency, the National Ocean Service, and the U.S. Geological Survey each has earned a reputation among respective user communities as a leader in producing quality data and map products that meet the highest standards. With the increased use of GIS, user communities are rapidly expanding and have now begun to overlap. With adoption of SDTS as a Federal Information Processing Standard, all federal agencies are required to use the SDTS domestically. There is additionally an emerging need to unify efforts in the development of international standards. Existing standards need to be harmonized and future standards activities need to be linked more closely. This paper reviews the roles of each agency in developing spatial data and standards for exchange and outlines leadership activities in standards development that are currently underway.     

Cartographic Consequences of a Planned Economy—50 Years of Soviet Cartography

For half a century all cartography in the Soviet Union has been under centralized control. It was recognized early that mapmaking was an activity of prime importance. All phases of the activity—from cartographic education and the preparation of topographic and thematic maps to the development of regional atlases and school maps—have been managed by and for the benefit of the state in accordance with its economic and social objectives. This has had a significant effect on the development of cartography, the degree of its standardization, and the relative emphasis placed on its various aspects.     

An atlas of Northamptonshire the medieval and early-modern landscape

National Geographic Society (NGS) has made several changes throughout the years in their choice of map projection for their world reference maps. The Van der Crinten I map projection was used from 1922 to 1988. Then, in 1988, it was replaced by the Robinson projection. Beginning in 1998, the Winkel Tripel became the map projection of choice for NCS' world maps. Given this change, cartographers and others who make maps may be interested in using the Winkel Tripel for custom applications. The goal of this paper is to show how Winkel Tripel's complex projection equations can be programmed using Visual Basic. Those who use other languages such as C++ can use this programming example to help them create a similar algorithm in their language of choice.     

Numerical Evaluation of the Robinson Projection

The Robinson projection is one of the most preferred projections for world reference maps in atlas cartography. The projection is constructed from Robinson's look-up table since there are no analytical formulas. This deficiency has led to a number of requests for the plotting formulas to which cartographers have responded by deriving analytical equations using different interpolation algorithms applied to Robinson's table values. The Robinson projection was examined with regard to its deformations calculated by four different algorithms, including the multiquadratic method. The numerical evaluations were then used to compare the algorithms. Solutions have been presented including some criticisms about this projection. The latitudes along which the scale is true and on which the maximum angular distortion equals zero have been determined.     

Combining Usability Techniques to Design Geovisualization Tools for Epidemiology

Designing usable geovisualization tools is an emerging problem in GIScience software development. We are often satisfied that a new method provides an innovative window on our data, but functionality alone is insufficient assurance that a tool is applicable to a problem in situ. As extensions of the static methods they evolved from, geovisualization tools are bound to enable new knowledge creation. We have yet to learn how to adapt techniques from interaction designers and usability experts toward our tools in order to maximize this ability. This is especially challenging because there is limited existing guidance for the design of usable geovisualization tools. Their design requires knowledge about the context of work within which they will be used, and should involve user input at all stages, as is the practice in any human-centered design effort. Toward that goal, we have employed a wide range of techniques in the design of ESTAT, an exploratory geovisualization toolkit for epidemiology. These techniques include; verbal protocol analysis, card-sorting, focus groups, and an in-depth case study. This paper reports the design process and evaluation results from our experience with the ESTAT toolkit.     

An Implementation of the Robinson Map Projection Based on Cubic Splines

The Robinson world map projection has been in existence since 1963. Mapping equations for the projection are based on table interpolation. Cubic splines with the choice of appropriate boundary conditions are shown to possess favorable characteristics as interpolative functions. The evaluation of cubic splines for given latitudes provides a basis for efficient forward mapping equations. Inverse mapping equations are based on an efficient inversion of the cubic spline functions utilizing Newton's method. Derivatives of the cubic spline functions lead to formulas for scale factors along projected meridians and parallels and for areal scale factors. Details for a computer implementation of the mapping equations and the computation of scale factors are given.     

Qibla,and Related,Map Projections

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 towards the equator. The resulting maps, when extended to the entire world, thus must overlap themselves. An unusual recent discovery from Iran suggests that Muslims might have been prior inventors of a similar projection, by at least several centuries. A later corollary by Schoy leads to a new "cylindrical" azimuthal map projection with parallels bending away from the equator, here illustrated for the first time.     

Complexity and Intractability: Limitations to Implementation in Analytical Cartography

The computational complexity of algorithms is an important consideration for all computer systems, including geographic information systems and mapping systems. Mathematical cartographers and GIS professionals need to understand and to take into account the limitations imposed on problem solving by the very nature of computation itself. We look at three active research sub-areas of analytical cartography to highlight the differences between traditional mathematical solutions and solutions with computationally tractable algorithms. The three sub-areas are map projections, map feature labeling, and map generalization.     

Recent literature in cartography and geographic information science

Tissot's Indicatrix and regular grids have been used for assessing map projection accuracies. Despite their broad applicability for accuracy assessment, they have limitations in quantifying resampling errors caused by map projections. This is due to the structural uncertainty with regard to the placement and pattern of grids. It is also difficult to calculate the absolute amount of resampling error in each projection. As an alternative to traditional testing methods, the use of random points was investigated. Specifically, random point generation, resampling with spherical block search algorithms, resampling accuracy with a perfect grid, and resampling accuracy with eight projections were investigated and are discussed here. Eight global referencing methods were tested: the equal-area cylindrical, sinusoidal, Mollweide, Eckert IV, Hammer-Aitoff, interrupted Goode homolosine, integerized sinusoidal projections, and the equal area global gridding with a fixed latitudinal metric distance. The resampling accuracy with a perfect grid is about 75 percent. Results showed the sinusoidal and the integerized sinusoidal projections and equal-area global gridding to achieve the highest accuracies.     

Political Ecology: An Integrative Approach to Geography and Environment-Development Studies (2003), edited by Karl S. Zimmerer and Thomas J. Bassett

Map reading is effective to the extent that the user has a basic understanding of the nature of geographical information, is familiar with the logic of various cartographic techniques, and grasps the map deciphering problems that may result through the matching of cartographic method to geographical data in specific mapping situations. The qualitative/quantitative dichotomy that was formerly so frequently resorted to when dealing with geographical information has been replaced by a more sensitive four-level measurement scheme or scaling system which is not only useful in sorting out the relations between environmental information and its cartographic portrayal, but which also provides an important key to understanding the special interpretative considerations that these relations create for the map reader. From a measurement perspective, the information yielded by a particular map symbol depends on the nature of the data that go into its construction, the nature of the data that survive the symbolic transformation, and the ability of the map user to decode the symbol effectively. The utility of measurement concepts in map reading is further demonstrated in the clarifying effect they can have in determining map information content and in the area of map generalization.     

An Analysis of a Modified van der Grinten Equatorial Arbitrary Oval Projection

The choice of a projection for world distributions depends on a number of considerations. Primarily this involves the properties of equivalence and conformality. But there are instances where requirements regarding the use of these mutually exclusive properties would allow a compromise projection. This article presents the mathematical basis, a construction procedure, and an analysis of area and angular deformation of one such projection, a modified van der Grinten.

Mathematical expressions are derived which are used to locate parallels and meridians. Parallels of the modified grid are straight lines located at an increasing distance from the equator. Meridians are arcs of circles intersecting the construction equator truly. Distribution of S and 2ω are shown.