geodesia y cartografía

Thursday, April 06, 2017

Tobler - Mercator

A New Companion for Mercator
Waldo Tobler
Professor emeritus
Geography Department
University of California
Santa Barbara, CA 93106-4060
ABSRACT: 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 compliment to Mercator’s conformal navigation anamorphose and can be displayed as an alternative. The equations for the new map projection preserve the latitudinal stretching of the Mercator while adjusting the longitudinal spacing. This allows placement of the new map adjacent to that of Mercator. The surface area, while drastically warped, maintains the correct magnitude.
Keywords: Map Projection, Equal Area, Mercator
In 1569, nearly 450 years ago, Mercator introduced his famous map projection “Ad Usam Navigatorum”. It still is in use for that purpose today. In the equatorial case the rendering of rhumb lines was, and is, especially useful for ocean navigation. And, as a conformal projection, the preservation of local directional gradients makes a Mercator map useful in several contexts. In particular, Halley’s use of this projection in 1701 to represent magnetic deviations by isogons (Monmonier, Figure 1.6, page 10) is thus justified. Weather maps showing wind directions also take advantage of this property.
 The normal Mercator map with the horizontal equator at the center, with latitudes stretched towards the poles, was also frequently used in the past for purposes other than navigation – in classrooms to teach world geography, for example - and this use has been criticized as being inappropriate and misleading. The equal area map proposed here might be set next to the conformal version in order to contrast or complement the normal Mercator. This is because the latitudinal spacing is the same for both maps. The Lambert cylindrical equal area projection (Lambert 1772), for which the longitudinal spacing is the same as the Mercator projection, might also be set next to (south of) either of these maps for a similar purpose1.
The equations for Mercator’s projection were developed by mathematicians somewhat after its introduction (Monmonier, page 61 et sec.) and are now known, in the case of a map for the surface of a unit sphere, centered on the equator, as
                        X = l
                        Y = Ln tan (p/4 + f/2)
where gitude, the latitude. In order to obtain an equal area projection, while retaining the same latitudinal spacing, it is necessary to change the equation for the spacing of the longitudes in the first equation. The general equation2 for an equal area projection of a spherical surface of radius 1, is  dx/dl * dy/df - dx/df * dy/dl = cos2f . Requiring that the latitude spacing be the same as that for Mercator’s map fixes  dy/df  We are left with dx/dl * dy/df, and dy/df  for Mercator’s projection is sec  Thus dx/dl = cos2f . The final equations for an equal area projection with latitudinal spacing as on the Mercator projection are therefore:

   X =  l cos2f


    Y = Ln tan (p/4 + f/2)
A version of the new projection, centered on the Greenwich meridian, appears in Figure 1. The poles, as on the Mercator map of the world, are displaced to infinity, and thus cannot be shown. By convention, the projection is truncated at about 80 degrees north and south. By taking the vertical meridian of the map at 180 degrees west longitude, and extending the map to 180 degrees east longitude, the map can be positioned adjacent to Mercator’s world map centered on Greenwich.  
The original Mercator map is conformal, meaning that local angles are represented faithfully, while geographic areas are not correctly depicted. The new version, Figure 1, reverses this, and the maximum angular modification is now extreme. The map is not a bonnie representation of the equal area map projection class, but reinforces the notion that Mercator’s map, stretching towards the poles, distorts areas. As such the map provides an equal area companion to Mercator’s conformal projection.
1Suggested by a reviewer.
2 (McBryde & Thomas, Equation 1, page 12).
Lambert, J.H., (1772), Notes and Comments on the Composition of Terrestrial and Celestial Maps, W. R. Tobler, Trans., 2nd ed., 2011, Redlands, CA: ESRI Press.
        McBryde, F. W., Thomas, P. D.,1949, Equal-Area Projections for World Statistical Maps, Coast and Geodetic Survey Special Publication No. 245, Washington, Government Printing Office.
Monmonier, M., 2004, Rhumb Lines and Map Wars, Chicago, University Press.
Figure 1:  The Tobler-Mercator equal area map projection with Mercator spacing of the latitude lines
Figura 2: Tobler-Mercator and Mercator

© Waldo Tobler 2016