# Hotine oblique Mercator

## Description

The Hotine projection, also known as oblique cylindrical orthomorphic or rectified skew orthomorphic, is one version of the oblique Mercator projection derivations. It is used for conformal mapping of areas that are obliquely oriented and do not follow a north-south or east-west trend.

The formulas for the projection were presented by Martin Hotine in 1946. It is available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0 and later.

## Projection properties

The subsections below describe the Hotine oblique Mercator projection properties.

### Graticule

Hotine oblique Mercator is an oblique cylindric projection. In general, the meridians and parallels are projected as complex curves. Only two meridians, exactly 180 degrees apart, can be projected as straight lines, crossing the poles. Both poles are presented as points inside the projection outline.

### Distortion

Hotine oblique Mercator is a conformal map projection. It does not maintain true directions, but angles and shapes are maintained at infinitesimal scale. Distances are accurate along the central line if the scale factor is 1.0. If it is less than 1.0, there are two approximately straight lines parallel to the central line with accurate scale. Area, distance, and scale distortions increase away from the central line or two straight lines parallel to the central line.

## Usage

Hotine oblique Mercator projection is appropriate for mapping large-scale areas or smaller areas with an oblique orientation that do not follow a north-south or east-west predominant extent.

## Variants

There are five variants of projection available in ArcGIS:

• Hotine oblique Mercator azimuth center, available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0.1 and later
• Hotine oblique Mercator azimuth center old, available in ArcGIS Pro 3.0 and later and in ArcGIS Desktop 10.8.3 and later
• Hotine oblique Mercator azimuth natural origin, available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0.1 and later
• Hotine oblique Mercator two-point center, available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0.1 and later
• Hotine oblique Mercator two-point natural origin, available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0 and later

The azimuth variants define the central line with a point on the line and an angle measuring east of north (an azimuth). The two-point variants specify the line with two points. The natural origin variants have the coordinate origin of the projected coordinate system where the central line of the projection crosses the equator. The center variants have the origin at the latitude of the center along the central line.

The Hotine oblique Mercator azimuth center old variant is based on the mathematics used for the projection in releases prior to the ArcGIS Pro 3.0 and ArcGIS Desktop 10.8.3.

## Limitations

The projection implementation in ArcGIS is limited to not show approximately one degree of latitude and longitude around the antipodal point. When using ellipsoids, constant scale along the central line or along straight lines parallel to the central line is not maintained.

## Parameters

Hotine oblique Mercator azimuth center parameters are as follows:

• False Easting
• False Northing
• Scale Factor
• Azimuth
• Longitude Of Center
• Latitude Of Center

Hotine oblique Mercator azimuth center old parameters are as follows:

• False Easting
• False Northing
• Scale Factor
• Azimuth
• Longitude Of Center
• Latitude Of Center

Hotine oblique Mercator azimuth natural origin parameters are as follows:

• False Easting
• False Northing
• Scale Factor
• Azimuth
• Longitude Of Center
• Latitude Of Center

Hotine oblique Mercator two-point center parameters are as follows:

• False Easting
• False Northing
• Latitude Of 1st Point
• Latitude Of 2nd Point
• Scale Factor
• Longitude Of 1st Point
• Longitude Of 2nd Point
• Latitude Of Center

Hotine oblique Mercator two-point natural origin parameters are as follows:

• False Easting
• False Northing
• Latitude Of 1st Point
• Latitude Of 2nd Point
• Scale Factor
• Longitude Of 1st Point
• Longitude Of 2nd Point
• Latitude Of Center

### Particular parameter cases

If both central line points are set on the same meridian, or if the azimuth is 0 degrees or 180 degrees, the resulting projection appears as the transverse Mercator projection. If both central line points are set on the equator, or the center is set on the equator and the azimuth parameter is either ±90 degrees, the projection appears as the Mercator projection.

## Sources

Snyder, J. P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395. Washington, DC: United States Government Printing Office.

Snyder, J. P. (1993). Flattening the Earth. Two Thousand Years of Map Projections. Chicago and London: University of Chicago Press.

Snyder, J. P. and Voxland, P. M. (1989). An Album of Map Projections. U.S. Geological Survey Professional Paper 1453.Washington, DC: United States Government Printing Office.