## Class ObliqueMercator

• All Implemented Interfaces:
`Serializable`, `MathTransform`, `MathTransform2D`
Direct Known Subclasses:
`HotineObliqueMercator`

```public class ObliqueMercator
extends MapProjection```
Oblique Mercator Projection. A conformal, oblique, cylindrical projection with the cylinder touching the ellipsoid (or sphere) along a great circle path (the central line). The Mercator and Transverse Mercator projections can be thought of as special cases of the oblique mercator, where the central line is along the equator or a meridian, respectively. The Oblique Mercator projection has been used in Switzerland, Hungary, Madagascar, Malaysia, Borneo and the panhandle of Alaska.

The Oblique Mercator projection uses a (U,V) coordinate system, with the U axis along the central line. During the forward projection, coordinates from the ellipsoid are projected conformally to a sphere of constant total curvature, called the "aposphere", before being projected onto the plane. The projection coordinates are further convented to a (X,Y) coordinate system by rotating the calculated (u,v) coordinates to give output (x,y) coordinates. The rotation value is usually the same as the projection azimuth (the angle, east of north, of the central line), but some cases allow a separate rotation parameter.

There are two forms of the oblique mercator, differing in the origin of their grid coordinates. The Hotine Oblique Mercator (EPSG code 9812) has grid coordinates start at the intersection of the central line and the equator of the aposphere. The Oblique Mercator (EPSG code 9815) is the same, except the grid coordinates begin at the central point (where the latitude of center and central line intersect). ESRI separates these two case by appending `"Natural_Origin"` (for the ``` "Hotine_Oblique_Mercator"```) and `"Center"` (for the `"Oblique_Mercator"`) to the projection names.

Two different methods are used to specify the central line for the oblique mercator: 1) a central point and an azimuth, east of north, describing the central line and 2) two points on the central line. The EPSG does not use the two point method, while ESRI separates the two cases by putting `"Azimuth"` and `"Two_Point"` in their projection names. Both cases use the point where the `"latitude_of_center"` parameter crosses the central line as the projection's central point. The central meridian is not a projection parameter, and is instead calculated as the intersection between the central line and the equator of the aposphere.

For the azimuth method, the central latitude cannot be ±90.0 degrees and the central line cannot be at a maximum or minimum latitude at the central point. In the two point method, the latitude of the first and second points cannot be equal. Also, the latitude of the first point and central point cannot be ±90.0 degrees. Furthermore, the latitude of the first point cannot be 0.0 and the latitude of the second point cannot be -90.0 degrees. A change of 10-7 radians can allow calculation at these special cases. Snyder's restriction of the central latitude being 0.0 has been removed, since the equations appear to work correctly in this case.

Azimuth values of 0.0 and ±90.0 degrees are allowed (and used in Hungary and Switzerland), though these cases would usually use a Mercator or Transverse Mercator projection instead. Azimuth values > 90 degrees cause errors in the equations.

The oblique mercator is also called the "Rectified Skew Orthomorphic" (RSO). It appears is that the only difference from the oblique mercator is that the RSO allows the rotation from the (U,V) to (X,Y) coordinate system to be different from the azimuth. This separate parameter is called `"rectified_grid_angle"` (or ``` "XY_Plane_Rotation"``` by ESRI) and is also included in the EPSG's parameters for the Oblique Mercator and Hotine Oblique Mercator. The rotation parameter is optional in all the non-two point projections and will be set to the azimuth if not specified.

Projection cases and aliases implemented by the `ObliqueMercator` are:

• `Oblique_Mercator` (EPSG code 9815)
grid coordinates begin at the central point, has `"rectified_grid_angle"` parameter.
• `Hotine_Oblique_Mercator_Azimuth_Center` (ESRI)
grid coordinates begin at the central point.
• `Rectified_Skew_Orthomorphic_Center` (ESRI)
grid coordinates begin at the central point, has `"rectified_grid_angle"` parameter.
• `Hotine_Oblique_Mercator` (EPSG code 9812)
grid coordinates begin at the interseciton of the central line and aposphere equator, has `"rectified_grid_angle"` parameter.
• `Hotine_Oblique_Mercator_Azimuth_Natural_Origin` (ESRI)
grid coordinates begin at the interseciton of the central line and aposphere equator.
• `Rectified_Skew_Orthomorphic_Natural_Origin` (ESRI)
grid coordinates begin at the interseciton of the central line and aposphere equator, has `"rectified_grid_angle"` parameter.
• `Hotine_Oblique_Mercator_Two_Point_Center` (ESRI)
grid coordinates begin at the central point.
• `Hotine_Oblique_Mercator_Two_Point_Natural_Origin` (ESRI)
grid coordinates begin at the interseciton of the central line and aposphere equator.

References:

• `libproj4` is available at libproj4 Miscellanea
Relevent files are: `PJ_omerc.c`, `pj_tsfn.c`, `pj_fwd.c`, ``` pj_inv.c``` and `lib_proj.h`
• John P. Snyder (Map Projections - A Working Manual, U.S. Geological Survey Professional Paper 1395, 1987)
• "Coordinate Conversions and Transformations including Formulas", EPSG Guidence Note Number 7 part 2, Version 24.
• Gerald Evenden, 2004, Documentation of revised Oblique Mercator
Since:
2.1
Author:
Gerald I. Evenden (for original code in Proj4), Rueben Schulz
See Also:
Oblique Mercator projection on MathWorld, "hotine_oblique_mercator" on RemoteSensing.org, "oblique_mercator" on RemoteSensing.org, Serialized Form
• ### Nested Class Summary

Nested Classes
Modifier and Type Class Description
`static class ` `ObliqueMercator.Provider`
The math transform provider for an Oblique Mercator projection (EPSG code 9815).
`static class ` `ObliqueMercator.Provider_TwoPoint`
The math transform provider for a Oblique Mercator projection, specified with two points on the central line (instead of a central point and azimuth).
• ### Nested classes/interfaces inherited from class MapProjection

`MapProjection.AbstractProvider`
• ### Field Summary

Fields
Modifier and Type Field Description
`protected double` `azimuth`
The azimuth of the central line passing throught the centre of the projection, in radians.
`protected double` `latitudeOfCentre`
Latitude of the projection centre.
`protected double` `longitudeOfCentre`
Longitude of the projection centre.
`protected double` `rectifiedGridAngle`
The rectified bearing of the central line, in radians.
• ### Fields inherited from class MapProjection

`centralMeridian, en0, en1, en2, en3, en4, excentricity, excentricitySquared, falseEasting, falseNorthing, globalScale, invertible, isSpherical, latitudeOfOrigin, LOGGER, scaleFactor, semiMajor, semiMinor, SKIP_SANITY_CHECKS`
• ### Fields inherited from class Formattable

`SINGLE_LINE`
• ### Constructor Summary

Constructors
Modifier Constructor Description
`protected ` `ObliqueMercator​(ParameterValueGroup parameters)`
Constructs a new map projection from the supplied parameters.
• ### Method Summary

All Methods
Modifier and Type Method Description
`boolean` `equals​(Object object)`
Compares the specified object with this map projection for equality.
`ParameterDescriptorGroup` `getParameterDescriptors()`
Returns the parameter descriptors for this map projection.
`ParameterValueGroup` `getParameterValues()`
Returns the parameter values for this map projection.
`protected double` ```getToleranceForAssertions​(double longitude, double latitude)```
Maximal error (in metres) tolerated for assertion, if enabled.
`int` `hashCode()`
Returns a hash value for this projection.
`protected Point2D` ```inverseTransformNormalized​(double x, double y, Point2D ptDst)```
Transforms the specified coordinate and stores the result in `ptDst`.
`protected Point2D` ```transformNormalized​(double x, double y, Point2D ptDst)```
Transforms the specified coordinate and stores the result in `ptDst`.
• ### Methods inherited from class MapProjection

`checkReciprocal, getSourceDimensions, getTargetDimensions, inv_mlfn, inverse, mlfn, orthodromicDistance, resetWarnings, transform, transform, transform`
• ### Methods inherited from class AbstractMathTransform

`createTransformedShape, derivative, derivative, ensureNonNull, formatWKT, getName, isIdentity, needCopy, normalizeAngle, rollLongitude, transform, transform, transform`
• ### Methods inherited from class Formattable

`cleanupThreadLocals, toString, toWKT, toWKT, toWKT, toWKT`
• ### Methods inherited from class Object

`clone, finalize, getClass, notify, notifyAll, wait, wait, wait`
• ### Methods inherited from interface MathTransform

`derivative, isIdentity, toWKT, transform, transform, transform`
• ### Methods inherited from interface MathTransform2D

`createTransformedShape, derivative`
• ### Field Detail

• #### latitudeOfCentre

`protected final double latitudeOfCentre`
Latitude of the projection centre. This is similar to the `MapProjection.latitudeOfOrigin`, but the latitude of origin is the Earth equator on aposphere for the oblique mercator.
• #### longitudeOfCentre

`protected final double longitudeOfCentre`
Longitude of the projection centre. This is NOT equal to the `MapProjection.centralMeridian`, which is the meridian where the central line intersects the Earth equator on aposphere.

This parameter applies to the "azimuth" case only. It is set to `NaN` for the "two points" case.

• #### azimuth

`protected final double azimuth`
The azimuth of the central line passing throught the centre of the projection, in radians.
• #### rectifiedGridAngle

`protected final double rectifiedGridAngle`
The rectified bearing of the central line, in radians. This is equals to the azimuth if the `"rectified_grid_angle"` parameter value is not set.
• ### Constructor Detail

• #### ObliqueMercator

```protected ObliqueMercator​(ParameterValueGroup parameters)
throws ParameterNotFoundException```
Constructs a new map projection from the supplied parameters.
Parameters:
`parameters` - The parameter values in standard units.
Throws:
`ParameterNotFoundException` - if a mandatory parameter is missing.
Since:
2.4
• ### Method Detail

• #### getParameterDescriptors

`public ParameterDescriptorGroup getParameterDescriptors()`
Returns the parameter descriptors for this map projection. This is used for a providing a default implementation of `MapProjection.getParameterValues()`, as well as arguments checking.
Specified by:
`getParameterDescriptors` in class `MapProjection`
Returns:
The parameter descriptors for this math transform, or `null`.
See Also:
`OperationMethod.getParameters()`
• #### getParameterValues

`public ParameterValueGroup getParameterValues()`
Returns the parameter values for this map projection.
Overrides:
`getParameterValues` in class `MapProjection`
Returns:
A copy of the parameter values for this map projection.
See Also:
`Operation.getParameterValues()`
• #### transformNormalized

```protected Point2D transformNormalized​(double x,
double y,
Point2D ptDst)
throws ProjectionException```
Transforms the specified coordinate and stores the result in `ptDst`. This method is usually (but not guaranteed) to be invoked with values of x in the range `[-PI..PI]` and values of y in the range `[-PI/2..PI/2]`. Values outside those ranges are accepted (sometime with a warning logged) on the assumption that most implementations use those values only in trigonometric functions like sin and cos.

Coordinates have the `MapProjection.centralMeridian` removed from lambda before this method is invoked. After this method is invoked, the results in `ptDst` are multiplied by `MapProjection.globalScale`, and the `MapProjection.falseEasting` and `MapProjection.falseNorthing` are added. This means that projections that implement this method are performed on an ellipse (or sphere) with a semi-major axis of 1.

In PROJ.4, the same standardization, described above, is handled by `pj_fwd.c`. Therefore when porting projections from PROJ.4, the forward transform equations can be used directly here with minimal change. In the equations of Snyder, `MapProjection.falseEasting`, `MapProjection.falseNorthing` and `MapProjection.scaleFactor` are usually not given. When implementing these equations here, you will not need to remove the `MapProjection.centralMeridian` from lambda or apply the `MapProjection.semiMajor` (a or R).

Specified by:
`transformNormalized` in class `MapProjection`
Parameters:
`x` - The longitude of the coordinate, in radians.
`y` - The latitude of the coordinate, in radians.
`ptDst` - the specified coordinate point that stores the result of transforming ``` ptSrc```, or `null`. Ordinates will be in a dimensionless unit, as a linear distance on a unit sphere or ellipse.
Returns:
the coordinate point after transforming (`lambda`, `phi`) and storing the result in `ptDst`.
Throws:
`ProjectionException` - if the point can't be transformed.
• #### inverseTransformNormalized

```protected Point2D inverseTransformNormalized​(double x,
double y,
Point2D ptDst)
throws ProjectionException```
Transforms the specified coordinate and stores the result in `ptDst`. This method returns longitude as x values in the range `[-PI..PI]` and latitude as y values in the range `[-PI/2..PI/2]`. It will be checked by the caller, so this method doesn't need to performs this check.

Input coordinates have the `MapProjection.falseEasting` and `MapProjection.falseNorthing` removed and are divided by `MapProjection.globalScale` before this method is invoked. After this method is invoked, the `MapProjection.centralMeridian` is added to the `x` results in `ptDst`. This means that projections that implement this method are performed on an ellipse (or sphere) with a semi-major axis of 1.

In PROJ.4, the same standardization, described above, is handled by `pj_inv.c`. Therefore when porting projections from PROJ.4, the inverse transform equations can be used directly here with minimal change. In the equations of Snyder, `MapProjection.falseEasting`, `MapProjection.falseNorthing` and `MapProjection.scaleFactor` are usually not given. When implementing these equations here, you will not need to add the `MapProjection.centralMeridian` to the output longitude or remove the `MapProjection.semiMajor` (a or R).

Specified by:
`inverseTransformNormalized` in class `MapProjection`
Parameters:
`x` - The easting of the coordinate, linear distance on a unit sphere or ellipse.
`y` - The northing of the coordinate, linear distance on a unit sphere or ellipse.
`ptDst` - the specified coordinate point that stores the result of transforming ``` ptSrc```, or `null`. Ordinates will be in radians.
Returns:
the coordinate point after transforming `x`, `y` and storing the result in `ptDst`.
Throws:
`ProjectionException` - if the point can't be transformed.
• #### getToleranceForAssertions

```protected double getToleranceForAssertions​(double longitude,
double latitude)```
Maximal error (in metres) tolerated for assertion, if enabled.
Overrides:
`getToleranceForAssertions` in class `MapProjection`
Parameters:
`longitude` - The longitude in decimal degrees.
`latitude` - The latitude in decimal degrees.
Returns:
The tolerance level for assertions, in meters.
• #### hashCode

`public int hashCode()`
Returns a hash value for this projection.
Overrides:
`hashCode` in class `MapProjection`
• #### equals

`public boolean equals​(Object object)`
Compares the specified object with this map projection for equality.
Overrides:
`equals` in class `MapProjection`
Parameters:
`object` - The object to compare with this transform.
Returns:
`true` if the given object is a transform of the same class and if, given identical source position, the transformed position would be the equals.