Package org.apache.sis.referencing.operation.projection

## Class NormalizedProjection

- Object
- - FormattableObject
  - - AbstractMathTransform
    - - AbstractMathTransform2D
      - - NormalizedProjection

- All Implemented Interfaces:
    `Serializable`, `Parameterized`, `LenientComparable`, `MathTransform`, `MathTransform2D`
  Direct Known Subclasses:
    `AlbersEqualArea`, `CylindricalEqualArea`, `LambertConicConformal`, `Mercator`, `Mollweide`, `ObliqueMercator`, `ObliqueStereographic`, `PolarStereographic`, `Polyconic`, `Sinusoidal`, `TransverseMercator`

      public abstract class NormalizedProjection
      extends AbstractMathTransform2D
      implements Serializable

  Base class for conversion services between ellipsoidal and cartographic projections. This conversion works on a normalized spaces, where angles are expressed in radians and computations are performed for a sphere having a semi-major axis of 1. More specifically:
  - On input, the `transform(…)` method expects (longitude, latitude) angles in radians, sometime pre-multiplied by other projection-specific factors (see point #3 below). Longitudes have the central meridian (λ₀) removed before the transform method is invoked. The conversion from degrees to radians and the longitude rotation are applied by the normalization affine transform.
  - On output, the `transform(…)` method returns (x, y) values on a sphere or ellipse having a semi-major axis length (a) of 1, sometime divided by other projection-specific factors (see point #3 below). The multiplication by the scale factor (k₀) and the translation by false easting (FE) and false northing (FN) are applied by the denormalization affine transform.
  - In addition to above-cited conversions, subclasses may opportunistically concatenate other linear operations (scales and translations). They do that by changing the normalization and denormalization matrices shown below. When such changes are applied, the `transform(…)` inputs are no longer angles in radians but some other derived values.
  The normalization and denormalization steps are represented below by the matrices immediately on the left and right sides of `NormalizedProjection` respectively. Those matrices show only the basic parameters common to most projections. Some projections will put more elements in those matrices.
  0 1 0 1 0 0 0 0 1
  →
  π/180 0 -λ 0 ⋅ (π/180) 0 π/180 0 0 0 1
  →
  `NormalizedProjection`
  →
  a ⋅ k 0 0 FE 0 a ⋅ k 0 FN 0 0 1
  Note: The first matrix on the left side is for swapping axes from (latitude, longitude) to (longitude, latitude) order. This matrix is shown here for completeness, but is not managed by this projection package. Axes swapping is managed at a higher level.
  `NormalizedProjection` does not store the above cited parameters (central meridian, scale factor, etc.) on intent (except indirectly), in order to make clear that those parameters are not used by subclasses. The ability to recognize two `NormalizedProjection`s as equivalent without consideration for the scale factor (among other) allow more efficient concatenation in some cases (typically some combinations of inverse projection followed by a direct projection).

  All angles (either fields, method parameters or return values) in this class and subclasses are in radians. This is the opposite of `Parameters` where all angles are in CRS-dependent units, typically decimal degrees.

  Serialization
  Serialization of this class is appropriate for short-term storage or RMI use, but may not be compatible with future versions. For long term storage, WKT (Well Know Text) or XML are more appropriate.
  Since:
    0.6
  See Also:
    `ContextualParameters`, Map projections on MathWorld, Serialized Form

  Defined in the `sis-referencing` module

- - ### Nested Class Summary

    Nested Classes
    Modifier and Type | Class | Description
    --- | --- | ---
    `protected static class` | `NormalizedProjection.ParameterRole` | Maps the parameters to be used for initializing `NormalizedProjection` and its normalization / denormalization matrices.

  - ### Field Summary

    Fields
    Modifier and Type | Field | Description
    --- | --- | ---
    `protected double` | `eccentricity` | Ellipsoid eccentricity, equals to `sqrt(eccentricitySquared)`.
    `protected double` | `eccentricitySquared` | The square of eccentricity: ℯ² = (a²-b²)/a² where ℯ is the eccentricity, a is the semi-major axis length and b is the semi-minor axis length.

  - ### Constructor Summary

    Constructors
    Modifier | Constructor | Description
    --- | --- | ---
    `protected` | `NormalizedProjection​(OperationMethod method, Parameters parameters, Map<NormalizedProjection.ParameterRole,​? extends ParameterDescriptor<? extends Number>> roles)` | Constructs a new map projection from the supplied parameters.

  - ### Method Summary

    All Methods Instance Methods Abstract Methods Concrete Methods
    Modifier and Type | Method | Description
    --- | --- | ---
    `protected int` | `computeHashCode()` | Computes a hash code value for this `NormalizedProjection`.
    `MathTransform` | `createMapProjection​(MathTransformFactory factory)` | Returns the sequence of normalization → `this` → denormalization transforms as a whole.
    `boolean` | `equals​(Object object, ComparisonMode mode)` | Compares the given object with this transform for equivalence.
    `protected ContextualParameters` | `getContextualParameters()` | Returns the parameters used for creating the complete map projection.
    `ParameterDescriptorGroup` | `getParameterDescriptors()` | Returns a description of the non-linear internal parameters of this `NormalizedProjection`.
    `ParameterValueGroup` | `getParameterValues()` | Returns a copy of non-linear internal parameter values of this `NormalizedProjection`.
    `MathTransform2D` | `inverse()` | Returns the inverse of this map projection.
    `protected abstract void` | `inverseTransform​(double[] srcPts, int srcOff, double[] dstPts, int dstOff)` | Inverse converts the single coordinate in `srcPts` at the given offset and stores the result in `ptDst` at the given offset.
    `abstract Matrix` | `transform​(double[] srcPts, int srcOff, double[] dstPts, int dstOff, boolean derivate)` | Converts a single coordinate in `srcPts` at the given offset and stores the result in `dstPts` at the given offset.
    `protected MathTransform` | `tryConcatenate​(boolean applyOtherFirst, MathTransform other, MathTransformFactory factory)` | Concatenates or pre-concatenates in an optimized way this projection with the given transform, if possible.

    - ### Methods inherited from class AbstractMathTransform2D

      `createTransformedShape, derivative, getSourceDimensions, getTargetDimensions, transform`

    - ### Methods inherited from class AbstractMathTransform

      `derivative, equals, formatTo, hashCode, isIdentity, transform, transform, transform, transform, transform`

    - ### Methods inherited from class FormattableObject

      `print, toString, toString, 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, transform, transform`

- - ### Field Detail

    - #### eccentricity

          protected final double eccentricity

      Ellipsoid eccentricity, equals to `sqrt(eccentricitySquared)`. Value 0 means that the ellipsoid is spherical.

    - #### eccentricitySquared

          protected final double eccentricitySquared

      The square of eccentricity: ℯ² = (a²-b²)/a² where ℯ is the eccentricity, a is the semi-major axis length and b is the semi-minor axis length.

  - ### Constructor Detail

    - #### NormalizedProjection

          protected NormalizedProjection​(OperationMethod method,
                                         Parameters parameters,
                                         Map<NormalizedProjection.ParameterRole,​? extends ParameterDescriptor<? extends Number>> roles)

      Constructs a new map projection from the supplied parameters. This constructor applies the following operations on the contextual parameters:
      - On the normalization matrix (to be applied before `this` transform):
        - Subtract the central meridian value.
        - Convert from degrees to radians.
      - On the denormalization matrix (to be applied after `this` transform):
        - Scale by the semi-major axis length.
        - If a scale factor is present (not all map projections have a scale factor), apply that scale.
        - Translate by the false easting and false northing (after the scale).
      - On the contextual parameters (not the parameters of `this` transform):
        - Store the values for semi-major axis length, semi-minor axis length, scale factor (if present), central meridian, false easting and false northing values.
      In matrix form, this constructor creates the following matrices (subclasses are free to modify):

      Initial matrix coefficients after construction
      Normalization | Denormalization
      --- | ---
      π/180 0 -λ 0 ⋅ (π/180) 0 π/180 0 0 0 1 | a ⋅ k 0 0 FE 0 a ⋅ k 0 FN 0 0 1

      Which parameters are considered
      The `roles` map specifies which parameters to look for central meridian, scale factor, false easting, false northing and other values. All entries in the `roles` map are optional. All descriptors in the map shall comply to the following constraints:
      - Descriptors associated to `NormalizedProjection​.ParameterRole​.SEMI_MAJOR`, `SEMI_MINOR`, `FALSE_EASTING` and `FALSE_NORTHING` shall have the same linear unit of measurement (usually metre).
      - Descriptors associated to angular measures (`NormalizedProjection​.ParameterRole​.CENTRAL_MERIDIAN` and `LATITUDE_OF_CONFORMAL_SPHERE_RADIUS`) shall use degrees.
      Note that users can still use units of their choice in the `Parameters` object given in argument to this constructor. But those values will be converted to the units of measurement specified by the parameter descriptors in the `roles` map, which must be the above-cited units.
      Parameters:
        `method` - description of the map projection parameters.
        `parameters` - the parameters of the projection to be created.
        `roles` - parameters to look for central meridian, scale factor, false easting, false northing and other values.

  - ### Method Detail

    - #### createMapProjection

          public MathTransform createMapProjection​(MathTransformFactory factory)
                                            throws FactoryException

      Returns the sequence of normalization → `this` → denormalization transforms as a whole. The transform returned by this method expects (longitude, latitude) coordinates in degrees and returns (x,y) coordinates in metres. Conversion to other units and changes in axis order are not managed by the returned transform.

      The default implementation is as below:

      >     return getContextualParameters().completeTransform(factory, this);

      Subclasses can override this method if they wish to use alternative implementations under some circumstances. For example many subclasses will replace `this` by a specialized implementation if they detect that the ellipsoid is actually spherical.
      Parameters:
        `factory` - the factory to use for creating the transform.
      Returns:
        the map projection from (λ,φ) to (x,y) coordinates.
      Throws:
        `FactoryException` - if an error occurred while creating a transform.
      See Also:
        `ContextualParameters​.completeTransform(MathTransformFactory, MathTransform)`

    - #### getContextualParameters

          protected final ContextualParameters getContextualParameters()

      Returns the parameters used for creating the complete map projection. Those parameters describe a sequence of normalize → `this` → denormalize transforms, not including axis swapping. Those parameters are used for formatting Well Known Text (WKT) and error messages. Subclasses shall not use the values defined in the returned object for computation purpose, except at construction time.
      Overrides:
        `getContextualParameters` in class `AbstractMathTransform`
      Returns:
        the parameters values for the sequence of normalize → `this` → denormalize transforms, or `null` if unspecified.

    - #### getParameterValues

          @Debug
          public ParameterValueGroup getParameterValues()

      Returns a copy of non-linear internal parameter values of this `NormalizedProjection`. The returned group contains at least the `eccentricity` parameter value. Some subclasses add more non-linear parameters, but most of them do not because many parameters like the scale factor or the false easting/northing are handled by the (de)normalization affine transforms instead.
      Note: This method is mostly for debugging purposes since the isolation of non-linear parameters in this class is highly implementation dependent. Most GIS applications will instead be interested in the contextual parameters.
      Specified by:
        `getParameterValues` in interface `Parameterized`
      Overrides:
        `getParameterValues` in class `AbstractMathTransform`
      Returns:
        a copy of the internal parameter values for this normalized projection.
      See Also:
        `AbstractMathTransform​.getContextualParameters()`, `SingleOperation​.getParameterValues()`

    - #### getParameterDescriptors

          @Debug
          public ParameterDescriptorGroup getParameterDescriptors()

      Returns a description of the non-linear internal parameters of this `NormalizedProjection`. The returned group contains at least a descriptor for the `eccentricity` parameter. Subclasses may add more parameters.

      This method is for inspecting the parameter values of this non-linear kernel only, not for inspecting the contextual parameters. Inspecting the kernel parameter values is usually for debugging purpose only.

      Specified by:
        `getParameterDescriptors` in interface `Parameterized`
      Overrides:
        `getParameterDescriptors` in class `AbstractMathTransform`
      Returns:
        a description of the internal parameters.
      See Also:
        `DefaultOperationMethod​.getParameters()`

    - #### transform

          public abstract Matrix transform​(double[] srcPts,
                                           int srcOff,
                                           double[] dstPts,
                                           int dstOff,
                                           boolean derivate)
                                    throws ProjectionException

      Converts a single coordinate in `srcPts` at the given offset and stores the result in `dstPts` at the given offset. In addition, opportunistically computes the transform derivative if requested.
      Normalization
      The input coordinates are (λ,φ) (the variable names for longitude and latitude respectively) angles in radians, eventually pre-multiplied by projection-specific factors. Input coordinate shall have the central meridian removed from the longitude by the caller before this method is invoked. After this method is invoked, the caller will need to multiply the output coordinate by the global scale factor, apply the (false easting, false northing) offset and eventually other projection-specific factors. This means that projections that implement this method are performed on a sphere or ellipse having a semi-major axis length of 1.
      Note 1: it is generally not necessary to know the projection-specific additional factors applied by subclasses on the input and output values, because `NormalizedProjection` should never be used directly. `NormalizedProjection` instances are used only indirectly as a step in a concatenated transform that include the normalization and denormalization matrices documented in this class javadoc.
      Note 2: in Proj.4, the same standardization, described above, is handled by `pj_fwd​.c`, except for the projection-specific additional factors.
      Argument checks
      The input longitude and latitude are usually (but not always) in the range [-π … π] and [-π/2 … π/2] respectively. However values outside those ranges are accepted on the assumption that most implementations use those values only in trigonometric functions like sine and cosine. If this assumption is not applicable to a particular subclass, then it is implementer responsibility to check the range.
      Specified by:
        `transform` in class `AbstractMathTransform`
      Parameters:
        `srcPts` - the array containing the source point coordinate, as (longitude, latitude) angles in radians.
        `srcOff` - the offset of the single coordinate to be converted in the source array.
        `dstPts` - the array into which the converted coordinate is returned (may be the same than `srcPts`). Coordinates will be expressed in a dimensionless unit, as a linear distance on a unit sphere or ellipse.
        `dstOff` - the offset of the location of the converted coordinate that is stored in the destination array.
        `derivate` - `true` for computing the derivative, or `false` if not needed.
      Returns:
        the matrix of the projection derivative at the given source position, or `null` if the `derivate` argument is `false`.
      Throws:
        `ProjectionException` - if the coordinate can not be converted.
      See Also:
        `AbstractMathTransform​.derivative(DirectPosition)`, `AbstractMathTransform​.transform(DirectPosition, DirectPosition)`, `MathTransforms​.derivativeAndTransform(MathTransform, double[], int, double[], int)`

    - #### inverseTransform

          protected abstract void inverseTransform​(double[] srcPts,
                                                   int srcOff,
                                                   double[] dstPts,
                                                   int dstOff)
                                            throws ProjectionException

      Inverse converts the single coordinate in `srcPts` at the given offset and stores the result in `ptDst` at the given offset. The output coordinates are (longitude, latitude) angles in radians, usually (but not necessarily) in the range [-π … π] and [-π/2 … π/2] respectively.
      Normalization
      Input coordinate shall have the (false easting, false northing) removed by the caller and the result divided by the global scale factor before this method is invoked. After this method is invoked, the caller will need to add the central meridian to the longitude in the output coordinate. This means that projections that implement this method are performed on a sphere or ellipse having a semi-major axis of 1. Additional projection-specific factors may also need to be applied (see class javadoc).
      Note: in Proj.4, the same standardization, described above, is handled by `pj_inv​.c`, except for the projection-specific additional factors.
      Parameters:
        `srcPts` - the array containing the source point coordinate, as linear distance on a unit sphere or ellipse.
        `srcOff` - the offset of the point to be converted in the source array.
        `dstPts` - the array into which the converted point coordinate is returned (may be the same than `srcPts`). Coordinates will be (longitude, latitude) angles in radians.
        `dstOff` - the offset of the location of the converted point that is stored in the destination array.
      Throws:
        `ProjectionException` - if the point can not be converted.

    - #### inverse

          public MathTransform2D inverse()

      Returns the inverse of this map projection. Subclasses do not need to override this method, as they should override `inverseTransform(…)` instead.
      Specified by:
        `inverse` in interface `MathTransform`
      Specified by:
        `inverse` in interface `MathTransform2D`
      Overrides:
        `inverse` in class `AbstractMathTransform2D`
      Returns:
        the inverse of this map projection.

    - #### tryConcatenate

          protected MathTransform tryConcatenate​(boolean applyOtherFirst,
                                                 MathTransform other,
                                                 MathTransformFactory factory)
                                          throws FactoryException

      Concatenates or pre-concatenates in an optimized way this projection with the given transform, if possible. If transforms are concatenated in an (inverse projection) → (affine) → (projection) sequence where the (projection) and (inverse projection) steps are the inverse of each other, then in some particular case the sequence can be replaced by a single affine transform. If no such simplification is possible, this method returns `null`.
      Overrides:
        `tryConcatenate` in class `AbstractMathTransform`
      Parameters:
        `applyOtherFirst` - `true` if the transformation order is `other` followed by `this`, or `false` if the transformation order is `this` followed by `other`.
        `other` - the other math transform to (pre-)concatenate with this transform.
        `factory` - the factory which is (indirectly) invoking this method, or `null` if none.
      Returns:
        the simplified (usually affine) transform, or `null` if no such optimization is available.
      Throws:
        `FactoryException` - if an error occurred while combining the transforms.
      Since:
        0.8
      See Also:
        `DefaultMathTransformFactory​.createConcatenatedTransform(MathTransform, MathTransform)`

    - #### computeHashCode

          protected int computeHashCode()

      Computes a hash code value for this `NormalizedProjection`.
      Overrides:
        `computeHashCode` in class `AbstractMathTransform`
      Returns:
        the hash code value.

    - #### equals

          public boolean equals​(Object object,
                                ComparisonMode mode)

      Compares the given object with this transform for equivalence. The default implementation checks if `object` is an instance of the same class than `this`, then compares the eccentricity.

      If this method returns `true`, then for any given identical source position, the two compared map projections shall compute the same target position. Many of the contextual parameters used for creating the map projections are irrelevant and do not need to be known. Those projection parameters will be compared only if the comparison mode is `ComparisonMode​.STRICT` or `BY_CONTRACT`.

      Example: a Mercator projection can be created in the 2SP case with a standard parallel value of 60°. The same projection can also be created in the 1SP case with a scale factor of 0.5. Nevertheless those two map projections applied on a sphere gives identical results. Considering them as equivalent allows the referencing module to transform coordinates between those two projections more efficiently.
      Specified by:
        `equals` in interface `LenientComparable`
      Overrides:
        `equals` in class `AbstractMathTransform`
      Parameters:
        `object` - the object to compare with this map projection for equivalence.
        `mode` - the strictness level of the comparison. Default to `ComparisonMode​.STRICT`.
      Returns:
        `true` if the given object is equivalent to this map projection.
      See Also:
        `Utilities​.deepEquals(Object, Object, ComparisonMode)`