Tobler-Mercator

New in version 6.0.0.

Equal area cylindrical projection with the same latitudinal spacing as Mercator projection.

Classification

Cylindrical equal area

Available forms

Forward and inverse, spherical only

Defined area

Global, conventionally truncated at about 80 degrees north and south

Alias

tobmerc

Domain

2D

Input type

Geodetic coordinates

Output type

Projected coordinates

Tobler-Mercator

proj-string: +proj=tobmerc

Usage

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 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.

Parameters

Note

All parameters for the projection are optional.

+k_0=<value>

Scale factor. Determines scale factor used in the projection.

Defaults to 1.0.

+lon_0=<value>

Longitude of projection center.

Defaults to 0.0.

+x_0=<value>

False easting.

Defaults to 0.0.

+y_0=<value>

False northing.

Defaults to 0.0.

+R=<value>

Radius of the sphere given in meters. If used in conjunction with +ellps, +R takes precedence.

Mathematical definition

The formulas describing the Tobler-Mercator are taken from Waldo Tobler’s article [Tobler2018]

Spherical form

For the spherical form of the projection we introduce the scaling factor:

\[k_0 = \cos^2 \phi_{ts}\]

Forward projection

\[x = k_0 \lambda\]
\[y = k_0 \ln \left[ \tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right]\]

Inverse projection

\[\lambda = \frac{x}{k_0}\]
\[\phi = \frac{\pi}{2} - 2 \arctan \left[ e^{-y/k_0} \right]\]