Class TransverseMercator

All Implemented Interfaces:
Serializable, Parameterized, LenientComparable, org.opengis.referencing.operation.MathTransform, org.opengis.referencing.operation.MathTransform2D
Direct Known Subclasses:
TransverseMercator.Spherical

public class TransverseMercator extends NormalizedProjection
Transverse Mercator projection (EPSG codes 9807). This class implements the "JHS formulas" reproduced in IOGP Publication 373-7-2 – Geomatics Guidance Note number 7, part 2 – April 2015.

Description

This is a cylindrical projection, in which the cylinder has been rotated 90°. Instead of being tangent to the equator (or to another standard latitude), it is tangent to a central meridian. Deformation are more important as we are going further from the central meridian. The Transverse Mercator projection is appropriate for region which have a greater extent north-south than east-west.

There are a number of versions of the Transverse Mercator projection including the Universal (UTM) and Modified (MTM) Transverses Mercator projections. In these cases the earth is divided into zones. For the UTM the zones are 6 degrees wide, numbered from 1 to 60 proceeding east from 180 degrees longitude, and between latitude 84 degrees North and 80 degrees South. The central meridian is taken as the center of the zone and the latitude of origin is the equator. A scale factor of 0.9996 and false easting of 500000 metres is used for all zones and a false northing of 10000000 metres is used for zones in the southern hemisphere.

Domain of validity

The difference between longitude values λ and the central meridian λ₀ should be less than 60°. Differences larger than 90° of longitude cause a ProjectionException to be thrown. Differences between 60° and 90° are not rejected by Apache SIS but should be avoided. See the projection method for more information.
Since:
0.6
Version:
1.3
See Also:
  • Field Details

    • serialVersionUID

      private static final long serialVersionUID
      For cross-version compatibility.
      See Also:
    • ALLOW_TRIGONOMETRIC_IDENTITIES

      private static final boolean ALLOW_TRIGONOMETRIC_IDENTITIES
      false for using the original formulas as published by EPSG, or true for using formulas modified using trigonometric identities. The use of trigonometric identities is for reducing the amount of calls to the Math.sin(double) and similar methods. Some identities used are:
      • sin(2θ) = 2⋅sinθ⋅cosθ
      • cos(2θ) = cos²θ - sin²θ
      • sin(3θ) = (3 - 4⋅sin²θ)⋅sinθ
      • cos(3θ) = (4⋅cos³θ) - 3⋅cosθ
      • sin(4θ) = (4 - 8⋅sin²θ)⋅sinθ⋅cosθ
      • cos(4θ) = (8⋅cos⁴θ) - (8⋅cos²θ) + 1
      Hyperbolic formulas:
      • sinh(2θ) = 2⋅sinhθ⋅coshθ
      • cosh(2θ) = cosh²θ + sinh²θ = 2⋅cosh²θ - 1 = 1 + 2⋅sinh²θ
      • sinh(3θ) = (3 + 4⋅sinh²θ)⋅sinhθ
      • cosh(3θ) = ((4⋅cosh²θ) - 3)⋅coshθ
      • sinh(4θ) = (1 + 2⋅sinh²θ)⋅4.sinhθ⋅coshθ = 4.cosh(2θ).sinhθ⋅coshθ
      • cosh(4θ) = (8⋅cosh⁴θ) - (8⋅cosh²θ) + 1 = 8⋅cosh²(θ) ⋅ (cosh²θ - 1) + 1 = 8⋅cosh²(θ) ⋅ sinh²(θ) + 1 = 2⋅sinh²(2θ) + 1
      Note that since this boolean is static final, the compiler should exclude the code in the branch that is never executed (no need to comment-out that code).
      See Also:
    • cf2

      private final double cf2
      Coefficients in the series expansion of the forward projection, depending only on eccentricity value. The series expansion is of the following form:
      cf₂⋅f(2θ) + cf₄⋅f(4θ) + cf₆⋅f(6θ) + cf₈⋅f(8θ)
      Those coefficients are named h₁, h₂, h₃ and h₄ in §1.3.5.1 of IOGP Publication 373-7-2 – Geomatics Guidance Note number 7, part 2 – April 2015.
      Serialization note: we do not strictly need to serialize those fields since they could be computed after deserialization. Bu we serialize them anyway in order to simplify a little bit this class (it allows us to keep those fields final) and because values computed after deserialization could be slightly different than the ones computed after construction since the constructor uses the double-double values provided by Initializer.
    • cf4

      private final double cf4
      Coefficients in the series expansion of the forward projection, depending only on eccentricity value. The series expansion is of the following form:
      cf₂⋅f(2θ) + cf₄⋅f(4θ) + cf₆⋅f(6θ) + cf₈⋅f(8θ)
      Those coefficients are named h₁, h₂, h₃ and h₄ in §1.3.5.1 of IOGP Publication 373-7-2 – Geomatics Guidance Note number 7, part 2 – April 2015.
      Serialization note: we do not strictly need to serialize those fields since they could be computed after deserialization. Bu we serialize them anyway in order to simplify a little bit this class (it allows us to keep those fields final) and because values computed after deserialization could be slightly different than the ones computed after construction since the constructor uses the double-double values provided by Initializer.
    • cf6

      private final double cf6
      Coefficients in the series expansion of the forward projection, depending only on eccentricity value. The series expansion is of the following form:
      cf₂⋅f(2θ) + cf₄⋅f(4θ) + cf₆⋅f(6θ) + cf₈⋅f(8θ)
      Those coefficients are named h₁, h₂, h₃ and h₄ in §1.3.5.1 of IOGP Publication 373-7-2 – Geomatics Guidance Note number 7, part 2 – April 2015.
      Serialization note: we do not strictly need to serialize those fields since they could be computed after deserialization. Bu we serialize them anyway in order to simplify a little bit this class (it allows us to keep those fields final) and because values computed after deserialization could be slightly different than the ones computed after construction since the constructor uses the double-double values provided by Initializer.
    • cf8

      private final double cf8
      Coefficients in the series expansion of the forward projection, depending only on eccentricity value. The series expansion is of the following form:
      cf₂⋅f(2θ) + cf₄⋅f(4θ) + cf₆⋅f(6θ) + cf₈⋅f(8θ)
      Those coefficients are named h₁, h₂, h₃ and h₄ in §1.3.5.1 of IOGP Publication 373-7-2 – Geomatics Guidance Note number 7, part 2 – April 2015.
      Serialization note: we do not strictly need to serialize those fields since they could be computed after deserialization. Bu we serialize them anyway in order to simplify a little bit this class (it allows us to keep those fields final) and because values computed after deserialization could be slightly different than the ones computed after construction since the constructor uses the double-double values provided by Initializer.
    • ci2

      private final double ci2
      Coefficients in the series expansion of the reverse projection, depending only on eccentricity value.
    • ci4

      private final double ci4
      Coefficients in the series expansion of the reverse projection, depending only on eccentricity value.
    • ci6

      private final double ci6
      Coefficients in the series expansion of the reverse projection, depending only on eccentricity value.
    • ci8

      private final double ci8
      Coefficients in the series expansion of the reverse projection, depending only on eccentricity value.
  • Constructor Details

    • TransverseMercator

      public TransverseMercator(org.opengis.referencing.operation.OperationMethod method, Parameters parameters)
      Creates a Transverse Mercator projection from the given parameters. The method argument can be the description of one of the following:
      • "Transverse Mercator".
      • "Transverse Mercator (South Orientated)".
      Parameters:
      method - description of the projection parameters.
      parameters - the parameter values of the projection to create.
    • TransverseMercator

      TransverseMercator(Initializer initializer)
      Creates a new Transverse Mercator projection from the given initializer. This constructor is used also by ZonedGridSystem.
    • TransverseMercator

      TransverseMercator(TransverseMercator other)
      Creates a new projection initialized to the same parameters than the given one.
  • Method Details

    • identityEquals

      private static boolean identityEquals(double actual, double expected)
      Verifies if a trigonometric identity produced the expected value. This method is used in assertions only, for values close to the [-1 … +1] range. The tolerance threshold is approximately 1.5E-12 (note that it still about 7000 time greater than Math.ulp(1.0)).
      See Also:
    • initializer

      @Workaround(library="JDK", version="1.7") private static Initializer initializer(org.opengis.referencing.operation.OperationMethod method, Parameters parameters)
      Work around for RFE #4093999 in Sun's bug database ("Relax constraint on placement of this()/super() call in constructors").
    • createMapProjection

      public org.opengis.referencing.operation.MathTransform createMapProjection(org.opengis.referencing.operation.MathTransformFactory factory) throws org.opengis.util.FactoryException
      Returns the sequence of normalizationthisdenormalization transforms as a whole. The transform returned by this method expects (longitude, latitude) coordinates in degrees and returns (x,y) coordinates in metres.

      The non-linear part of the returned transform will be this transform, except if the ellipsoid is spherical. In the latter case, this transform may be replaced by a simplified implementation.

      Overrides:
      createMapProjection in class NormalizedProjection
      Parameters:
      factory - the factory to use for creating the transform.
      Returns:
      the map projection from (λ,φ) to (x,y) coordinates.
      Throws:
      org.opengis.util.FactoryException - if an error occurred while creating a transform.
      See Also:
    • getDomain

      public Optional<org.opengis.geometry.Envelope> getDomain(DomainDefinition criteria)
      Returns the domain of input coordinates. The limits defined by this method are arbitrary and may change in any future implementation. Current implementation sets a limit at 40° of longitude on each side of the central meridian (this limit is mentioned in EPSG guidance notes) and a limit at 84° of latitude (same as Mercator projection).
      Overrides:
      getDomain in class AbstractMathTransform
      Parameters:
      criteria - controls the definition of transform domain.
      Returns:
      estimation of a domain where this transform is considered numerically applicable.
      Since:
      1.3
      See Also:
    • outsideDomainOfValidity

      private static org.opengis.referencing.operation.Matrix outsideDomainOfValidity(double[] dstPts, int dstOff, boolean derivate)
      Implementation of transform(double[], int, double[], int, boolean) for points outside domain of validity. Should be invoked only when the longitude is at more than 90° from central meridian, in which case result does not exist. This method should not be invoked for points at Δλ ≤ 90° that we fail to compute, because in such cases a ProjectionException should be thrown instead.
    • transform

      public org.opengis.referencing.operation.Matrix transform(double[] srcPts, int srcOff, double[] dstPts, int dstOff, boolean derivate) throws ProjectionException
      Projects the specified (λ,φ) coordinates (units in radians) and stores the result in dstPts. In addition, opportunistically computes the projection derivative if derivate is true. The results must be multiplied by the denormalization matrix before to get linear distances.

      Accuracy and domain of validity

      Projection errors depend on the difference ∆λ between longitude λ and the central meridian λ₀. All Universal Transverse Mercator (UTM) projections aim for ∆λ ≤ 3°, but this implementation can nevertheless handle larger values. Results have been compared with values provided by Karney, C.F.F. (2009). Test data for the transverse Mercator projection [Data set]. Zenodo. On the WGS84 ellipsoid we observed the following errors compared to Karney's data:
      • Errors less than 1 centimetre for ∆λ < 60° at all latitudes.
      • At latitudes far enough from equator (|φ| ≥ 20°), the domain can be extended up to ∆λ < (1 − ℯ)⋅90° (≈ 82.63627282416406551° on WGS84) with errors less than 70 centimetres.
      Case of 82.6…° < ∆λ ≤ 90°
      Karney (2009) uses an “extended” domain of transverse Mercator projection for ∆λ ≥ (1 − ℯ)⋅90°, but Apache SIS does not support such extension. Consequently, ∆λ values between (1 − ℯ)⋅90° and 90° should be considered invalid but are not rejected by Apache SIS. Note that those invalid values are consistent with the reverse projection (i.e. applying a projection followed by a reverse projection gives approximately the original values).
      Rational: those coordinates are accepted despite the low accuracy of projection results because they are sometimes needed for expressing bounding boxes. A bounding box may have corners located in invalid projection area even if all features inside the box have valid coordinates. For "contains" and "intersects" tests between envelopes, we do not need accurate coordinates; a monotonic behavior of x = f(λ) can be sufficient.
      Case of ∆λ > 90°
      Longitude values at a distance greater than 90° from the central meridian are rejected. A ProjectionException is thrown in that case. This limit exists because the Transverse Mercator projection is conceptually a Mercator projection rotated by 90°. Consequently, x values tend toward infinity for ∆λ close to ±90°
      Specified by:
      transform in class NormalizedProjection
      Parameters:
      srcPts - the array containing the source point coordinates, as (longitude, latitude) angles in radians.
      srcOff - the offset of the single coordinate tuple to be converted in the source array.
      dstPts - the array into which the converted coordinates 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 coordinates 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 coordinates cannot be converted.
      See Also:
    • inverseTransform

      protected void inverseTransform(double[] srcPts, int srcOff, double[] dstPts, int dstOff) throws ProjectionException
      Transforms the specified (η, ξ) coordinates and stores the result in dstPts (angles in radians).
      Specified by:
      inverseTransform in class NormalizedProjection
      Parameters:
      srcPts - the array containing the source point coordinates, 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 coordinates 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 cannot be converted.