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Area Deformation on the Robinson Projection
Abstract
The principal result of the authors analysis is to reduce considerably the amount of area deformation at higher latitudes. Robinson observed that more than 75 percent of the Earth's surface was shown on the project with less than 20 percent departure from its ture size. Now it appears more than 87 percent of the Earth's surface is shown with less than 20 percent departure from true size. The author's analysis also shows that the Robinson projection, which grew from a trial-and-error approach to yield certain characteristics, has no point on it that is free of deformation. The 38th parallels are true in length, but scale is only true along them.
... Interpolating methods use a function that passes exactly through the reference points. Ipbüker (1991), Bretterbauer (1994), and Evenden (2008) use cubic spline interpolation (which is also used in Flex Projector); and Richardson (1989) reports that Robinson applied the Aitken interpolation scheme. ...
The Natural Earth projection is a new projection for representing the entire Earth on small-scale maps. It was designed in Flex Projector, a specialized software application that offers a graphical approach for the creation of new projections. The original Natural Earth projection defines the length and spacing of parallels in a tabular form for every five degrees of increased latitude. It is a true pseudocylindrical projection, and is neither conformal nor equal-area. In the original definition, piece-wise cubic spline interpolation is used to project intermediate values that do not align with the five-degree grid.
This graduation thesis introduces alternative polynomial equations that are considerably simpler to compute. The polynomial expression also improves the smoothness of the rounded corners where the meridians meet the horizontal pole lines, a distinguished mark of the Natural Earth projection which suggests to readers that the Earth is spherical in shape. An inverse projection is presented. The formulas are simple to implement in cartographic software and libraries. Distortion values of this new graticule are not significantly different from the original piece-wise projection.
The development of the polynomial equations was inspired by a similar study of the Robinson projection. The polynomial coefficients were determined with least square adjustment of indirect observations with additional constraints to preserve the height and width of the graticule. The inverse procedure uses the Newton-Raphson method and converges in a few iterations.
... Indeed, Robinson (1974) did not stipulate which interpolation to use, which leads to incompatible implementations of his projection, as various methods have been used, each resulting in a slightly different projection. For example, Snyder (1990) applies the central-difference formula by Stirling, which the United States Geological Survey (USGS) adopted for GCTP. 1 Cubic spline interpolation is used by Ratner (1991) for Intergraph software and by Evenden (2005) for the PROJ.4 library, which is widely used by various open-source Geographic Information Systems (GIS). 2 It was not until 1989 that Richardson (1989) reported that Robinson used the Aitken interpolation procedure. ...
The design of new map projections has up until now required mathematical and cartographic expertise that has limited this activity to a small group of specialists. This article introduces the background mathematics for a software-based method that enables cartographers to easily design new small-scale world map projections. The software is usable even by those without mathematical expertise. A new projection is designed interactively in an iterative process that allows the designer to graphically and numerically assess the graticule, the representation of the continents, and the distortion properties of the new projection. The method has been implemented in Flex Projector, a free and open-source application enabling users to quickly create new map projections and modify existing projections. We also introduce new tools that help evaluate the distortion properties of projections, namely a configurable acceptance index to assess areal and angular distortion, a derived acceptance visualization, and interactive profiles through the distortion space of a projection. To illustrate the proposed method, a new projection, the Cropped Ginzburg VIII projection, is presented.
Die grundsätzliche Pflicht zur Aufstellung eines Konzernabschlusses ist eng an die Frage nach der Abgrenzung des Konsolidierungskreises geknüpft. Grundsätzlich müssen nach dem neuen Konzernbilanzrecht alle Unternehmen in den Konzernabschluß einbezogen werden, bei denen ein Mutter-Tochter-Verhältnis i.S.v. § 290 HGB vorliegt, ohne Rücksicht auf deren Rechtsform oder Sitz.1 Diese gem. § 290 HGB nach den Grundsätzen der Vollkonsolidierung in den Konzernabschluß einzubeziehenden Tochterunternehmen bilden gemeinsam mit dem Mutterunternehmen den Konsolidierungskreis im engeren Sinne. Bezieht man auch die Gemeinschaftsunternehmen nach §310 HGB und die assoziierten Unternehmen nach §§311, 312 HGB ein, so erhält man den Konsolidierungskreis im weiteren Sinne.2 Nicht mehr zur wirtschaftlichen Einheit und damit nicht mehr zum Konsolidierungskreis im weiteren Sinne zählen die Beteiligungsunternehmen nach § 271 Abs. 1 HGB.3 Die handelsrechtlichen Konzernrechnungslegungsvorschriften fuhren somit zu einem Konzernabschluß, der den stufenweisen Übergang vom eigentlichen Kern des Konzerns zu seiner Umwelt durch verschiedene Grade der Einflußnahme des Mutterunternehmens auf andere Unternehmen abbildet (sog. Stufenkonzeption).4
The Natural Earth projection is a new projection for representing the entire Earth on small-scale maps. It was designed in Flex Projector, a specialized software application that offers a graphical approach for the creation of new projections. The original Natural Earth projection defines the length and spacing of parallels in tabular form for every five degrees of increasing latitude. It is a pseudocylindrical projection, and is neither conformal nor equal-area. In the original definition, piece-wise cubic spline interpolation is used to project intermediate values that do not align with the five-degree grid. This paper introduces alternative polynomial equations that closely approximate the original projection. The polynomial equations are considerably simpler to compute and program, and require fewer parameters, which should facilitate the implementation of the Natural Earth projection in geospatial software. The polynomial expression also improves the smoothness of the rounded corners where the meridians meet the horizontal pole lines, a distinguishing trait of the Natural Earth projection that suggests to readers that the Earth is spherical in shape. Details on the least squares adjustment for obtaining the polynomial formulas are provided, including constraints for preserving the geometry of the graticule. This technique is applicable to similar projections that are defined by tabular parameters. For inverting the polynomial projection the Newton-Raphson root finding algorithm is suggested.
The 1988 adoption by the National Geographic Society of the Robinson projection for its world maps (Garver 1988), and the resulting widespread publicity and expanded use of the projection by Rand McNally and Company, have led to a number of requests for the plotting formulas. Since the projection itself is in the public domain, no legal problem is presented.
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.
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