Mercator¶
The Mercator projection is a cylindrical map projection that origins from the 16th century. It is widely recognized as the first regularly used map projection. It is a conformal projection in which the equator projects to a straight line at constant scale. The projection has the property that a rhumb line, a course of constant heading, projects to a straight line. This makes it suitable for navigational purposes.
Classification |
Conformal cylindrical |
Available forms |
Forward and inverse, spherical and ellipsoidal |
Defined area |
Global, but best used near the equator |
Alias |
merc |
Domain |
2D |
Input type |
Geodetic coordinates |
Output type |
Projected coordinates |
Usage¶
Applications should be limited to equatorial regions, but is frequently
used for navigational charts with latitude of true scale (+lat_ts
) specified within
or near chart’s boundaries.
It is considered to be inappropriate for world maps because of the gross
distortions in area; for example the projected area of Greenland is
larger than that of South America, despite the fact that Greenland’s
area is \(\frac18\) that of South America [Snyder1987].
Example using latitude of true scale:
$ echo 56.35 12.32 | proj +proj=merc +lat_ts=56.5
3470306.37 759599.90
Example using scaling factor:
echo 56.35 12.32 | proj +proj=merc +k_0=2
12545706.61 2746073.80
Note that +lat_ts
and +k_0
are mutually exclusive.
If used together, +lat_ts
takes precedence over +k_0
.
Parameters¶
Note
All parameters for the projection are optional.
- +lat_ts=<value>¶
Latitude of true scale. Defines the latitude where scale is not distorted. Takes precedence over
+k_0
if both options are used together.Defaults to 0.0.
- +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¶
Spherical form¶
For the spherical form of the projection we introduce the scaling factor:
Forward projection¶
The quantity \(\psi\) is the isometric latitude.
Inverse projection¶
Ellipsoidal form¶
For the ellipsoidal form of the projection we introduce the scaling factor:
where
\(a\,m(\phi)\) is the radius of the circle of latitude \(\phi\).
Forward projection¶
Inverse projection¶
The latitude \(\phi\) is found by inverting the equation for \(\psi\) iteratively.