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Oblique Mercator

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The Oblique Mercator projection is a cylindrical map projection that closes the +gap between the Mercator and the Transverse Mercator projections.

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Classification

Conformal cylindrical

Available forms

Forward and inverse, spherical and ellipsoidal

Defined area

Global, but reasonably accurate only within 15 degrees +of the oblique central line

Alias

omerc

Domain

2D

Input type

Geodetic coordinates

Output type

Projected coordinates

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+Oblique Mercator +
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proj-string: +proj=omerc +lat_1=45 +lat_2=55

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Figuratively, the cylinder used for developing the Mercator projection touches +the planet along the Equator, while that of the Transverse Mercator touches the +planet along a meridian, i.e. along a line perpendicular to the Equator.

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The cylinder for the Oblique Mercator, however, touches the planet along a line +at an arbitrary angle with the Equator. Hence, the Oblique Mercator projection +is useful for mapping areas having their greatest extent along a direction that +is neither north-south, nor east-west.

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The Mercator and the Transverse Mercator projections are both limiting forms of +the Oblique Mercator: The Mercator projection is equivalent to an Oblique Mercator +with central line along the Equator, while the Transverse Mercator is equivalent +to an Oblique Mercator with central line along a meridian.

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For the sphere, the construction of the Oblique Mercator projection can be +imagined as “tilting the cylinder of a plain Mercator projection”, +so the cylinder, instead of touching the equator, touches an arbitrary great circle +on the sphere. The great circle is defined by the tilt angle of the central line, +hence putting land masses along that great circle near the centre of the map, +where the Equator would go in the plain Mercator case.

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The ellipsoidal case, developed by Hotine, and refined by Snyder [Snyder1987] +is more complex, involving initial steps projecting from the ellipsoid to another +curved surface, the “aposphere”, then projection from the aposphere to the skew +uv-plane, before finally rectifying the skew uv-plane onto the map XY plane.

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Usage

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The tilt angle (azimuth) of the central line can be given in two different ways. +In the first case, the azimuth is given directly, using the option +alpha +and defining the centre of projection using the options +lonc and ++lat_0. +In the second case, the azimuth is given indirectly by specifying two points on +the central line, using the options ++lat_1, +lon_1, +lat_2, and +lon_2.

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Example: Verify that the Mercator projection is a limiting form of the Oblique +Mercator

+
$ echo 12 55 | proj +proj=merc +ellps=GRS80
+1335833.89   7326837.71
+
+$ echo 12 55 | proj +proj=omerc +lonc=0 +alpha=90 +ellps=GRS80
+1335833.89   7326837.71
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+
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Example: Second case - indirectly given azimuth

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$ echo 12 55 | proj +proj=omerc +lon_1=-1 +lat_1=1 +lon_2=0 +lat_2=0 +ellps=GRS80
+  349567.57   6839490.50
+
+
+

Example: An approximation of the Danish “System 34” from [Rittri2012]

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$ echo 10.536498003 56.229892362 | proj +proj=omerc +axis=wnu +lonc=9.46 +lat_0=56.13333333 +x_0=-266906.229 +y_0=189617.957 +k=0.9999537 +alpha=-0.76324 +gamma=0 +ellps=GRS80
+200000.13   199999.89
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+
+

The input coordinate represents the System 34 datum point “Agri Bavnehoj”, with coordinates +(200000, 200000) by definition. So at the datum point, the approximation is off by about 17 cm. +This use case represents a datum shift from a cylinder projection on an old, slightly +misaligned datum, to a similar projection on a modern datum.

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Parameters

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+

Central point and azimuth method

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++alpha=<value>
+

Azimuth of centerline clockwise from north at the center point of the line. +If +gamma is not given then +alpha determines the value of ++gamma.

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++gamma=<value>
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Azimuth of centerline clockwise from north of the rectified +bearing of centre line. If +alpha is not given, then ++gamma is used to determine +alpha.

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++lonc=<value>
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Longitude of the central point.

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++lat_0=<value>
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Latitude of the central point.

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Two point method

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++lon_1=<value>
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Longitude of first point.

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++lat_1=<value>
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Latitude of first point.

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++lon_2=<value>
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Longitude of second point.

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++lat_2=<value>
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Latitude of second point.

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Optional

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++no_rot
+

No rectification (not “no rotation” as one may well assume). +Do not take the last step from the skew uv-plane to the map +XY plane.

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Note

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This option is probably only marginally useful, but remains for (mostly) historical reasons.

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++no_off
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Do not offset origin to center of projection.

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++k_0=<value>
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Scale factor. Determines scale factor used in the projection.

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Defaults to 1.0.

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++lon_0=<value>
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Longitude of projection center.

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Defaults to 0.0.

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++x_0=<value>
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False easting.

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Defaults to 0.0.

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++y_0=<value>
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False northing.

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Defaults to 0.0.

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