### Tobler - Mercator

**A New Companion for Mercator**

Waldo
Tobler

Professor
emeritus

Geography
Department

University
of California

Santa
Barbara, CA 93106-4060

http://www.geog.ucsb.edu/~tobler

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
purpose

^{1}.
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 equation We are left with dx/dl
* dy/df,
and dy/df for Mercator’s projection is sec Thus dx/dl
= cos are
therefore:

^{2}for an equal area projection of a spherical surface of radius 1, is dx/dl * dy/df - dx/df * dy/dl = cos^{2}f . Requiring that the latitude spacing be the same as that for Mercator’s map fixes dy/df^{2}f . The final equations for an equal area projection with latitudinal spacing as on the Mercator projection^{2}f

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.

Citations:

^{1}Suggested 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., 2^{nd}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

Figura 2: Tobler-Mercator and Mercator

© Waldo Tobler
2016

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