If the relationship between any two coordinate reference systems is known, coordinates can be transformed or converted to another coordinate reference system. The UML model therefore specifies a source and a target coordinate reference system for such coordinate operations. For that reason, a coordinate operation is often popularly said to "transform coordinate reference system A into coordinate reference system B". Although this wording may be good enough for conversation, it should be realised that coordinate operations do not operate on coordinate reference systems, but on coordinates.
Coordinate operations are divided into two subtypes:
Coordinate conversion - mathematical operation on coordinates that does not include any change of Datum. The best-known example of a coordinate conversion is a map projection. The parameters describing coordinate conversions are defined rather than empirically derived. Note that some conversions have no parameters.
Coordinate transformation - mathematical operation on coordinates that usually includes a change of Datum. The parameters of a coordinate transformation are empirically derived from data containing the coordinates of a series of points in both coordinate reference systems. This computational process is usually 'over-determined', allowing derivation of error (or accuracy) estimates for the transformation. Also, the stochastic nature of the parameters may result in multiple (different) versions of the same coordinate transformation. Because of this several transformations may exist for a given pair of coordinate reference systems, differing in their transformation method, parameter values and accuracy characteristics.
A coordinate operation (transformation or conversion) can be time-varying, and must be time-varying if either the source or target CRS is time varying. When the coordinate operation is time-varying, the operation method used will also be time-varying, and some of the parameters used by that operation method will involve time. For example, some of the parameters may have time, velocity, and/or acceleration values and units.
Coordinate conversions are coordinate operations that make use of exact, defined (rather than measured or computed), and therefore error-free parameter values. Corresponding pairs of coordinate tuples in each of the two coordinate reference systems connected through a coordinate conversion have a fixed arithmetic relationship. Additionally one of the two tuples cannot exist without specification of the coordinate conversion and the 'source' coordinate reference system. Coordinate conversions are therefore intimately related to the concept of Derived CRS.
The best-known example of this source-derived relationship is a projected coordinate reference system, which is always based on a source geographic coordinate reference system. The associated map projection effectively defines the projected coordinate reference system from the geographic coordinate system.
Concatenated coordinate operation
A concatenated coordinate operation is an ordered sequence of coordinate operations. The sequence of operations is constrained by the requirement that the source coordinate reference system of step (n+1) must be the same as the target coordinate reference system of step (n). The source coordinate reference system of the first step and the target coordinate reference system of the last step are the source and target coordinate reference system associated with the concatenated operation.
The above constraint should not be interpreted as implying
that only those coordinate operations can be used in a concatenated operation
that have their source and a target coordinate reference system specified through
the association pair between
CoordinateReferenceSystem. This would exclude coordinate
conversions. Concatenated coordinate operations may contain coordinate transformations
and/or coordinate conversions.
The source and target coordinate reference system of a
coordinate conversion are defined in the
by specifying the base (i.e., source) CRS and the defining conversion. The derived
coordinate reference system itself is the target CRS in this situation. When used
in a concatenated operation, the conversion's source and target coordinate reference
system are equally subject to the above constraint as the source and target of a
transformation although they are specified in a different manner.
The concatenated coordinate operation class is primarily intended to provide a mechanism that forces application software to use a preferred path to go from source to target coordinate reference system, if a direct transformation between the two is not available.
Pass-through coordinate operation
Coordinate operations require input coordinate tuples of certain dimensions and produce output tuples of certain dimensions. The dimensions of these coordinate tuples and the dimensions of the coordinate reference system they are defined in must be the same.
The ability to define compound coordinate reference systems combining two or more other coordinate reference systems, not themselves compound, introduces a difficulty. For example, it may be required to transform only the horizontal or only the vertical component of a compound coordinate reference system, which will put them at odds with coordinate operations specified for either horizontal or vertical coordinates only. To the human mind this is a trivial problem, but not so for coordinate transformation software that ought to be capable of automatic operation, without human intervention; the software logic would be confronted with the problem of having to apply a 2-dimensional coordinate operation to 3-dimensional coordinate tuples.
This problem has been solved by defining a pass-through operation. This operation specifies what subset of a coordinate tuple is subject to a requested transformation. It takes the form of referencing another coordinate operation and specifying a sequence of numbers defining the positions in the coordinate tuple of the coordinates affected by that transformation. The order of the coordinates in a coordinate tuple shall agree with the order of the coordinate system axes as defined for the associated coordinate system.
Operation method and parameters
The algorithm used to execute the coordinate operation is defined in the operation method. Concatenated operations and pass-through operations do not require a coordinate operation to be specified. Each operation method uses a number of parameters (although some coordinate conversions use none), and each coordinate operation assigns value to these parameters.
As this class comes close to the heart of any coordinate transformation software, it is recommended to make extensive use of identifiers, referencing well-known datasets wherever possible. There is as yet no standard way of spelling or even naming the various coordinate operation methods. Client software requesting a coordinate operation to be executed by a coordinate transformation server implementation may therefore ask for an operation method this server doesn't recognise, although a perfectly valid method may be available. The same holds for coordinate operation parameters used by any coordinate operation method. To facilitate recognition and validation it is recommended that the operation formulae be included or referenced in the relevant object, and if possible a worked example.
This explanation is not complete without giving some thought to implementations. Coordinate transformation services should be able to automatically derive coordinate operations that are not stored explicitly in any permanent data store, in other words determine their own concatenated or inverse operations. The reason is that is practically impossible to store all possible pairs of coordinate reference systems in explicitly defined coordinate operations. The key to a successful software implementation is the ability to apply meaningful constraints and validations to this process. For example: it may be mathematically possible to derive a concatenated coordinate operation that will transform North American Datum 1983 coordinates to Australian National Datum; but in a practical sense that operation would be meaningless. The key validation that would flag such an operation as invalid would be a comparison of the two areas of validity and the conclusion that there is no overlap between these.
Coordinate transformation services should also be able to derive or infer the inverse of any coordinate operation (from B to A) from its complementary forward operation (A to B). Most permanent data stores for coordinate reference parameter data will record only one of the two operations that may exist between any two coordinate reference systems. The inverse operation is then inferred by the application software logic from the stored operation.
In some cases, the algorithm for the inverse operation is the same as the forward algorithm, and only the signs of the parameter values need to be reversed for the inverse operation to be fully defined. An example is the 7-parameter Helmert transformation (both position vector and coordinate frame rotation convention).
Some polynomial coordinate operations require the signs of only most, but not all, parameter values to be reversed. Other coordinate operation methods imply two algorithms, one for the forward and one for the inverse operation. The parameters are generally the same in that case. The latter situation generally applies to map projections.
Finally the same algorithm may be used for the inverse operation, with entirely different parameter values. This is the case with some polynomial and affine operations. In those cases the inverse operation cannot be inferred from the forward operation but must be explicitly defined.
The logic to derive the inverse transformation should be built into the application software, be it server or client, that performs the coordinate operation.
ClassDescriptionAn ordered sequence of two or more single coordinate operations.Base interface for conical map projections.An operation on coordinates that does not include any change of Datum.A mathematical operation on coordinates that transforms or converts coordinates to another coordinate reference system.Creates coordinate transformation objects from codes.Base interface for cylindrical map projections.Specification of the coordinate operation method formula.Transforms multi-dimensional coordinate points.Transforms one-dimensional coordinate points.Transforms two-dimensional coordinate points.Low level factory for creating
MathTransforminstances.A two dimensional array of numbers.Thrown when
MathTransform.inverse()is invoked but the transform can not be inverted.Definition of an algorithm used to perform a coordinate operation.Thrown when a coordinate operation is not found.A pass-through operation specifies that a subset of a coordinate tuple is subject to a specific coordinate operation.Base interface for for azimuthal (or planar) map projections.A conversion transforming (longitude,latitude) coordinates to Cartesian coordinates (x,y).A parameterized mathematical operation on coordinates that transforms or converts coordinates to another coordinate reference system.An operation on coordinates that usually includes a change of Datum.Common superclass for a number of transformation-related exceptions.