Transforms.js

define([
        '../ThirdParty/when',
        './Cartesian2',
        './Cartesian3',
        './Cartesian4',
        './Cartographic',
        './Check',
        './defaultValue',
        './defined',
        './DeveloperError',
        './EarthOrientationParameters',
        './EarthOrientationParametersSample',
        './Ellipsoid',
        './Iau2006XysData',
        './Iau2006XysSample',
        './JulianDate',
        './Math',
        './Matrix3',
        './Matrix4',
        './Quaternion',
        './TimeConstants'
    ], function(
        when,
        Cartesian2,
        Cartesian3,
        Cartesian4,
        Cartographic,
        Check,
        defaultValue,
        defined,
        DeveloperError,
        EarthOrientationParameters,
        EarthOrientationParametersSample,
        Ellipsoid,
        Iau2006XysData,
        Iau2006XysSample,
        JulianDate,
        CesiumMath,
        Matrix3,
        Matrix4,
        Quaternion,
        TimeConstants) {
    'use strict';

    /**
     * Contains functions for transforming positions to various reference frames.
     *
     * @exports Transforms
     */
    var Transforms = {};

    var vectorProductLocalFrame = {
        up : {
            south : 'east',
            north : 'west',
            west : 'south',
            east : 'north'
        },
        down : {
            south : 'west',
            north : 'east',
            west : 'north',
            east : 'south'
        },
        south : {
            up : 'west',
            down : 'east',
            west : 'down',
            east : 'up'
        },
        north : {
            up : 'east',
            down : 'west',
            west : 'up',
            east : 'down'
        },
        west : {
            up : 'north',
            down : 'south',
            north : 'down',
            south : 'up'
        },
        east : {
            up : 'south',
            down : 'north',
            north : 'up',
            south : 'down'
        }
    };

    var degeneratePositionLocalFrame = {
        north : [-1, 0, 0],
        east : [0, 1, 0],
        up : [0, 0, 1],
        south : [1, 0, 0],
        west : [0, -1, 0],
        down : [0, 0, -1]
    };

    var localFrameToFixedFrameCache = {};

    var scratchCalculateCartesian = {
        east : new Cartesian3(),
        north : new Cartesian3(),
        up : new Cartesian3(),
        west : new Cartesian3(),
        south : new Cartesian3(),
        down : new Cartesian3()
    };
    var scratchFirstCartesian = new Cartesian3();
    var scratchSecondCartesian = new Cartesian3();
    var scratchThirdCartesian = new Cartesian3();
    /**
    * Generates a function that computes a 4x4 transformation matrix from a reference frame
    * centered at the provided origin to the provided ellipsoid's fixed reference frame.
    * @param  {String} firstAxis  name of the first axis of the local reference frame. Must be
    *  'east', 'north', 'up', 'west', 'south' or 'down'.
    * @param  {String} secondAxis  name of the second axis of the local reference frame. Must be
    *  'east', 'north', 'up', 'west', 'south' or 'down'.
    * @return {localFrameToFixedFrameGenerator~resultat} The function that will computes a
    * 4x4 transformation matrix from a reference frame, with first axis and second axis compliant with the parameters,
    */
    Transforms.localFrameToFixedFrameGenerator = function( firstAxis, secondAxis) {
      if (!vectorProductLocalFrame.hasOwnProperty(firstAxis) || !vectorProductLocalFrame[firstAxis].hasOwnProperty(secondAxis)) {
          throw new DeveloperError('firstAxis and secondAxis must be east, north, up, west, south or down.');
      }
      var thirdAxis = vectorProductLocalFrame[firstAxis][secondAxis];

      /**
       * Computes a 4x4 transformation matrix from a reference frame
       * centered at the provided origin to the provided ellipsoid's fixed reference frame.
       * @callback Transforms~LocalFrameToFixedFrame
       * @param {Cartesian3} origin The center point of the local reference frame.
       * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid whose fixed frame is used in the transformation.
       * @param {Matrix4} [result] The object onto which to store the result.
       * @returns {Matrix4} The modified result parameter or a new Matrix4 instance if none was provided.
       */
      var resultat;
      var hashAxis = firstAxis + secondAxis;
      if (defined(localFrameToFixedFrameCache[hashAxis])) {
          resultat = localFrameToFixedFrameCache[hashAxis];
      } else {
          resultat = function(origin, ellipsoid, result) {
              //>>includeStart('debug', pragmas.debug);
              if (!defined(origin)) {
                  throw new DeveloperError('origin is required.');
              }
              //>>includeEnd('debug');
              if (!defined(result)) {
                  result = new Matrix4();
              }
              // If x and y are zero, assume origin is at a pole, which is a special case.
              if (CesiumMath.equalsEpsilon(origin.x, 0.0, CesiumMath.EPSILON14) && CesiumMath.equalsEpsilon(origin.y, 0.0, CesiumMath.EPSILON14)) {
                  var sign = CesiumMath.sign(origin.z);

                  Cartesian3.unpack(degeneratePositionLocalFrame[firstAxis], 0, scratchFirstCartesian);
                  if (firstAxis !== 'east' && firstAxis !== 'west') {
                      Cartesian3.multiplyByScalar(scratchFirstCartesian, sign, scratchFirstCartesian);
                  }

                  Cartesian3.unpack(degeneratePositionLocalFrame[secondAxis], 0, scratchSecondCartesian);
                  if (secondAxis !== 'east' && secondAxis !== 'west') {
                      Cartesian3.multiplyByScalar(scratchSecondCartesian, sign, scratchSecondCartesian);
                  }

                  Cartesian3.unpack(degeneratePositionLocalFrame[thirdAxis], 0, scratchThirdCartesian);
                  if (thirdAxis !== 'east' && thirdAxis !== 'west') {
                      Cartesian3.multiplyByScalar(scratchThirdCartesian, sign, scratchThirdCartesian);
                  }
              } else {
                  ellipsoid = defaultValue(ellipsoid, Ellipsoid.WGS84);
                  ellipsoid.geodeticSurfaceNormal(origin, scratchCalculateCartesian.up);

                  var up = scratchCalculateCartesian.up;
                  var east = scratchCalculateCartesian.east;
                  east.x = -origin.y;
                  east.y = origin.x;
                  east.z = 0.0;
                  Cartesian3.normalize(east, scratchCalculateCartesian.east);
                  Cartesian3.cross(up, east, scratchCalculateCartesian.north);

                  Cartesian3.multiplyByScalar(scratchCalculateCartesian.up, -1, scratchCalculateCartesian.down);
                  Cartesian3.multiplyByScalar(scratchCalculateCartesian.east, -1, scratchCalculateCartesian.west);
                  Cartesian3.multiplyByScalar(scratchCalculateCartesian.north, -1, scratchCalculateCartesian.south);

                  scratchFirstCartesian = scratchCalculateCartesian[firstAxis];
                  scratchSecondCartesian = scratchCalculateCartesian[secondAxis];
                  scratchThirdCartesian = scratchCalculateCartesian[thirdAxis];
              }
              result[0] = scratchFirstCartesian.x;
              result[1] = scratchFirstCartesian.y;
              result[2] = scratchFirstCartesian.z;
              result[3] = 0.0;
              result[4] = scratchSecondCartesian.x;
              result[5] = scratchSecondCartesian.y;
              result[6] = scratchSecondCartesian.z;
              result[7] = 0.0;
              result[8] = scratchThirdCartesian.x;
              result[9] = scratchThirdCartesian.y;
              result[10] = scratchThirdCartesian.z;
              result[11] = 0.0;
              result[12] = origin.x;
              result[13] = origin.y;
              result[14] = origin.z;
              result[15] = 1.0;
              return result;
          };
          localFrameToFixedFrameCache[hashAxis] = resultat;
      }
      return resultat;
    };

    /**
     * Computes a 4x4 transformation matrix from a reference frame with an east-north-up axes
     * centered at the provided origin to the provided ellipsoid's fixed reference frame.
     * The local axes are defined as:
     * <ul>
     * <li>The <code>x</code> axis points in the local east direction.</li>
     * <li>The <code>y</code> axis points in the local north direction.</li>
     * <li>The <code>z</code> axis points in the direction of the ellipsoid surface normal which passes through the position.</li>
     * </ul>
     *
     * @function
     * @param {Cartesian3} origin The center point of the local reference frame.
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid whose fixed frame is used in the transformation.
     * @param {Matrix4} [result] The object onto which to store the result.
     * @returns {Matrix4} The modified result parameter or a new Matrix4 instance if none was provided.
     *
     * @example
     * // Get the transform from local east-north-up at cartographic (0.0, 0.0) to Earth's fixed frame.
     * var center = Cesium.Cartesian3.fromDegrees(0.0, 0.0);
     * var transform = Cesium.Transforms.eastNorthUpToFixedFrame(center);
     */
    Transforms.eastNorthUpToFixedFrame = Transforms.localFrameToFixedFrameGenerator('east','north');

    /**
     * Computes a 4x4 transformation matrix from a reference frame with an north-east-down axes
     * centered at the provided origin to the provided ellipsoid's fixed reference frame.
     * The local axes are defined as:
     * <ul>
     * <li>The <code>x</code> axis points in the local north direction.</li>
     * <li>The <code>y</code> axis points in the local east direction.</li>
     * <li>The <code>z</code> axis points in the opposite direction of the ellipsoid surface normal which passes through the position.</li>
     * </ul>
     *
     * @function
     * @param {Cartesian3} origin The center point of the local reference frame.
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid whose fixed frame is used in the transformation.
     * @param {Matrix4} [result] The object onto which to store the result.
     * @returns {Matrix4} The modified result parameter or a new Matrix4 instance if none was provided.
     *
     * @example
     * // Get the transform from local north-east-down at cartographic (0.0, 0.0) to Earth's fixed frame.
     * var center = Cesium.Cartesian3.fromDegrees(0.0, 0.0);
     * var transform = Cesium.Transforms.northEastDownToFixedFrame(center);
     */
    Transforms.northEastDownToFixedFrame = Transforms.localFrameToFixedFrameGenerator('north','east');

    /**
     * Computes a 4x4 transformation matrix from a reference frame with an north-up-east axes
     * centered at the provided origin to the provided ellipsoid's fixed reference frame.
     * The local axes are defined as:
     * <ul>
     * <li>The <code>x</code> axis points in the local north direction.</li>
     * <li>The <code>y</code> axis points in the direction of the ellipsoid surface normal which passes through the position.</li>
     * <li>The <code>z</code> axis points in the local east direction.</li>
     * </ul>
     *
     * @function
     * @param {Cartesian3} origin The center point of the local reference frame.
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid whose fixed frame is used in the transformation.
     * @param {Matrix4} [result] The object onto which to store the result.
     * @returns {Matrix4} The modified result parameter or a new Matrix4 instance if none was provided.
     *
     * @example
     * // Get the transform from local north-up-east at cartographic (0.0, 0.0) to Earth's fixed frame.
     * var center = Cesium.Cartesian3.fromDegrees(0.0, 0.0);
     * var transform = Cesium.Transforms.northUpEastToFixedFrame(center);
     */
    Transforms.northUpEastToFixedFrame = Transforms.localFrameToFixedFrameGenerator('north','up');

    /**
     * Computes a 4x4 transformation matrix from a reference frame with an north-west-up axes
     * centered at the provided origin to the provided ellipsoid's fixed reference frame.
     * The local axes are defined as:
     * <ul>
     * <li>The <code>x</code> axis points in the local north direction.</li>
     * <li>The <code>y</code> axis points in the local west direction.</li>
     * <li>The <code>z</code> axis points in the direction of the ellipsoid surface normal which passes through the position.</li>
     * </ul>
     *
     * @function
     * @param {Cartesian3} origin The center point of the local reference frame.
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid whose fixed frame is used in the transformation.
     * @param {Matrix4} [result] The object onto which to store the result.
     * @returns {Matrix4} The modified result parameter or a new Matrix4 instance if none was provided.
     *
      * @example
     * // Get the transform from local north-West-Up at cartographic (0.0, 0.0) to Earth's fixed frame.
     * var center = Cesium.Cartesian3.fromDegrees(0.0, 0.0);
     * var transform = Cesium.Transforms.northWestUpToFixedFrame(center);
     */
    Transforms.northWestUpToFixedFrame = Transforms.localFrameToFixedFrameGenerator('north','west');

    var scratchHPRQuaternion = new Quaternion();
    var scratchScale = new Cartesian3(1.0, 1.0, 1.0);
    var scratchHPRMatrix4 = new Matrix4();

    /**
     * Computes a 4x4 transformation matrix from a reference frame with axes computed from the heading-pitch-roll angles
     * centered at the provided origin to the provided ellipsoid's fixed reference frame. Heading is the rotation from the local north
     * direction where a positive angle is increasing eastward. Pitch is the rotation from the local east-north plane. Positive pitch angles
     * are above the plane. Negative pitch angles are below the plane. Roll is the first rotation applied about the local east axis.
     *
     * @param {Cartesian3} origin The center point of the local reference frame.
     * @param {HeadingPitchRoll} headingPitchRoll The heading, pitch, and roll.
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid whose fixed frame is used in the transformation.
     * @param {Transforms~LocalFrameToFixedFrame} [fixedFrameTransform=Transforms.eastNorthUpToFixedFrame] A 4x4 transformation
     *  matrix from a reference frame to the provided ellipsoid's fixed reference frame
     * @param {Matrix4} [result] The object onto which to store the result.
     * @returns {Matrix4} The modified result parameter or a new Matrix4 instance if none was provided.
     *
     * @example
     * // Get the transform from local heading-pitch-roll at cartographic (0.0, 0.0) to Earth's fixed frame.
     * var center = Cesium.Cartesian3.fromDegrees(0.0, 0.0);
     * var heading = -Cesium.Math.PI_OVER_TWO;
     * var pitch = Cesium.Math.PI_OVER_FOUR;
     * var roll = 0.0;
     * var hpr = new Cesium.HeadingPitchRoll(heading, pitch, roll);
     * var transform = Cesium.Transforms.headingPitchRollToFixedFrame(center, hpr);
     */
    Transforms.headingPitchRollToFixedFrame = function(origin, headingPitchRoll, ellipsoid, fixedFrameTransform, result) {
        //>>includeStart('debug', pragmas.debug);
        Check.typeOf.object( 'HeadingPitchRoll', headingPitchRoll);
        //>>includeEnd('debug');

        fixedFrameTransform = defaultValue(fixedFrameTransform, Transforms.eastNorthUpToFixedFrame);
        var hprQuaternion = Quaternion.fromHeadingPitchRoll(headingPitchRoll, scratchHPRQuaternion);
        var hprMatrix = Matrix4.fromTranslationQuaternionRotationScale(Cartesian3.ZERO, hprQuaternion, scratchScale, scratchHPRMatrix4);
        result = fixedFrameTransform(origin, ellipsoid, result);
        return Matrix4.multiply(result, hprMatrix, result);
    };

    var scratchENUMatrix4 = new Matrix4();
    var scratchHPRMatrix3 = new Matrix3();

    /**
     * Computes a quaternion from a reference frame with axes computed from the heading-pitch-roll angles
     * centered at the provided origin. Heading is the rotation from the local north
     * direction where a positive angle is increasing eastward. Pitch is the rotation from the local east-north plane. Positive pitch angles
     * are above the plane. Negative pitch angles are below the plane. Roll is the first rotation applied about the local east axis.
     *
     * @param {Cartesian3} origin The center point of the local reference frame.
     * @param {HeadingPitchRoll} headingPitchRoll The heading, pitch, and roll.
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid whose fixed frame is used in the transformation.
     * @param {Transforms~LocalFrameToFixedFrame} [fixedFrameTransform=Transforms.eastNorthUpToFixedFrame] A 4x4 transformation
     *  matrix from a reference frame to the provided ellipsoid's fixed reference frame
     * @param {Quaternion} [result] The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter or a new Quaternion instance if none was provided.
     *
     * @example
     * // Get the quaternion from local heading-pitch-roll at cartographic (0.0, 0.0) to Earth's fixed frame.
     * var center = Cesium.Cartesian3.fromDegrees(0.0, 0.0);
     * var heading = -Cesium.Math.PI_OVER_TWO;
     * var pitch = Cesium.Math.PI_OVER_FOUR;
     * var roll = 0.0;
     * var hpr = new HeadingPitchRoll(heading, pitch, roll);
     * var quaternion = Cesium.Transforms.headingPitchRollQuaternion(center, hpr);
     */
    Transforms.headingPitchRollQuaternion = function(origin, headingPitchRoll, ellipsoid, fixedFrameTransform, result) {
        //>>includeStart('debug', pragmas.debug);
        Check.typeOf.object( 'HeadingPitchRoll', headingPitchRoll);
        //>>includeEnd('debug');

        var transform = Transforms.headingPitchRollToFixedFrame(origin, headingPitchRoll, ellipsoid, fixedFrameTransform, scratchENUMatrix4);
        var rotation = Matrix4.getRotation(transform, scratchHPRMatrix3);
        return Quaternion.fromRotationMatrix(rotation, result);
    };

    var gmstConstant0 = 6 * 3600 + 41 * 60 + 50.54841;
    var gmstConstant1 = 8640184.812866;
    var gmstConstant2 = 0.093104;
    var gmstConstant3 = -6.2E-6;
    var rateCoef = 1.1772758384668e-19;
    var wgs84WRPrecessing = 7.2921158553E-5;
    var twoPiOverSecondsInDay = CesiumMath.TWO_PI / 86400.0;
    var dateInUtc = new JulianDate();

    /**
     * Computes a rotation matrix to transform a point or vector from True Equator Mean Equinox (TEME) axes to the
     * pseudo-fixed axes at a given time.  This method treats the UT1 time standard as equivalent to UTC.
     *
     * @param {JulianDate} date The time at which to compute the rotation matrix.
     * @param {Matrix3} [result] The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter or a new Matrix3 instance if none was provided.
     *
     * @example
     * //Set the view to in the inertial frame.
     * scene.postUpdate.addEventListener(function(scene, time) {
     *    var now = Cesium.JulianDate.now();
     *    var offset = Cesium.Matrix4.multiplyByPoint(camera.transform, camera.position, new Cesium.Cartesian3());
     *    var transform = Cesium.Matrix4.fromRotationTranslation(Cesium.Transforms.computeTemeToPseudoFixedMatrix(now));
     *    var inverseTransform = Cesium.Matrix4.inverseTransformation(transform, new Cesium.Matrix4());
     *    Cesium.Matrix4.multiplyByPoint(inverseTransform, offset, offset);
     *    camera.lookAtTransform(transform, offset);
     * });
     */
    Transforms.computeTemeToPseudoFixedMatrix = function (date, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(date)) {
            throw new DeveloperError('date is required.');
        }
        //>>includeEnd('debug');

        // GMST is actually computed using UT1.  We're using UTC as an approximation of UT1.
        // We do not want to use the function like convertTaiToUtc in JulianDate because
        // we explicitly do not want to fail when inside the leap second.

        dateInUtc = JulianDate.addSeconds(date, -JulianDate.computeTaiMinusUtc(date), dateInUtc);
        var utcDayNumber = dateInUtc.dayNumber;
        var utcSecondsIntoDay = dateInUtc.secondsOfDay;

        var t;
        var diffDays = utcDayNumber - 2451545;
        if (utcSecondsIntoDay >= 43200.0) {
            t = (diffDays + 0.5) / TimeConstants.DAYS_PER_JULIAN_CENTURY;
        } else {
            t = (diffDays - 0.5) / TimeConstants.DAYS_PER_JULIAN_CENTURY;
        }

        var gmst0 = gmstConstant0 + t * (gmstConstant1 + t * (gmstConstant2 + t * gmstConstant3));
        var angle = (gmst0 * twoPiOverSecondsInDay) % CesiumMath.TWO_PI;
        var ratio = wgs84WRPrecessing + rateCoef * (utcDayNumber - 2451545.5);
        var secondsSinceMidnight = (utcSecondsIntoDay + TimeConstants.SECONDS_PER_DAY * 0.5) % TimeConstants.SECONDS_PER_DAY;
        var gha = angle + (ratio * secondsSinceMidnight);
        var cosGha = Math.cos(gha);
        var sinGha = Math.sin(gha);

        if (!defined(result)) {
            return new Matrix3(cosGha, sinGha, 0.0,
                              -sinGha, cosGha, 0.0,
                                  0.0,    0.0, 1.0);
        }
        result[0] = cosGha;
        result[1] = -sinGha;
        result[2] = 0.0;
        result[3] = sinGha;
        result[4] = cosGha;
        result[5] = 0.0;
        result[6] = 0.0;
        result[7] = 0.0;
        result[8] = 1.0;
        return result;
    };

    /**
     * The source of IAU 2006 XYS data, used for computing the transformation between the
     * Fixed and ICRF axes.
     * @type {Iau2006XysData}
     *
     * @see Transforms.computeIcrfToFixedMatrix
     * @see Transforms.computeFixedToIcrfMatrix
     *
     * @private
     */
    Transforms.iau2006XysData = new Iau2006XysData();

    /**
     * The source of Earth Orientation Parameters (EOP) data, used for computing the transformation
     * between the Fixed and ICRF axes.  By default, zero values are used for all EOP values,
     * yielding a reasonable but not completely accurate representation of the ICRF axes.
     * @type {EarthOrientationParameters}
     *
     * @see Transforms.computeIcrfToFixedMatrix
     * @see Transforms.computeFixedToIcrfMatrix
     *
     * @private
     */
    Transforms.earthOrientationParameters = EarthOrientationParameters.NONE;

    var ttMinusTai = 32.184;
    var j2000ttDays = 2451545.0;

    /**
     * Preloads the data necessary to transform between the ICRF and Fixed axes, in either
     * direction, over a given interval.  This function returns a promise that, when resolved,
     * indicates that the preload has completed.
     *
     * @param {TimeInterval} timeInterval The interval to preload.
     * @returns {Promise.<undefined>} A promise that, when resolved, indicates that the preload has completed
     *          and evaluation of the transformation between the fixed and ICRF axes will
     *          no longer return undefined for a time inside the interval.
     *
     *
     * @example
     * var interval = new Cesium.TimeInterval(...);
     * when(Cesium.Transforms.preloadIcrfFixed(interval), function() {
     *     // the data is now loaded
     * });
     *
     * @see Transforms.computeIcrfToFixedMatrix
     * @see Transforms.computeFixedToIcrfMatrix
     * @see when
     */
    Transforms.preloadIcrfFixed = function(timeInterval) {
        var startDayTT = timeInterval.start.dayNumber;
        var startSecondTT = timeInterval.start.secondsOfDay + ttMinusTai;
        var stopDayTT = timeInterval.stop.dayNumber;
        var stopSecondTT = timeInterval.stop.secondsOfDay + ttMinusTai;

        var xysPromise = Transforms.iau2006XysData.preload(startDayTT, startSecondTT, stopDayTT, stopSecondTT);
        var eopPromise = Transforms.earthOrientationParameters.getPromiseToLoad();

        return when.all([xysPromise, eopPromise]);
    };

    /**
     * Computes a rotation matrix to transform a point or vector from the International Celestial
     * Reference Frame (GCRF/ICRF) inertial frame axes to the Earth-Fixed frame axes (ITRF)
     * at a given time.  This function may return undefined if the data necessary to
     * do the transformation is not yet loaded.
     *
     * @param {JulianDate} date The time at which to compute the rotation matrix.
     * @param {Matrix3} [result] The object onto which to store the result.  If this parameter is
     *                  not specified, a new instance is created and returned.
     * @returns {Matrix3} The rotation matrix, or undefined if the data necessary to do the
     *                   transformation is not yet loaded.
     *
     *
     * @example
     * scene.postUpdate.addEventListener(function(scene, time) {
     *   var icrfToFixed = Cesium.Transforms.computeIcrfToFixedMatrix(time);
     *   if (Cesium.defined(icrfToFixed)) {
     *     var offset = Cesium.Matrix4.multiplyByPoint(camera.transform, camera.position, new Cesium.Cartesian3());
     *     var transform = Cesium.Matrix4.fromRotationTranslation(icrfToFixed)
     *     var inverseTransform = Cesium.Matrix4.inverseTransformation(transform, new Cesium.Matrix4());
     *     Cesium.Matrix4.multiplyByPoint(inverseTransform, offset, offset);
     *     camera.lookAtTransform(transform, offset);
     *   }
     * });
     *
     * @see Transforms.preloadIcrfFixed
     */
    Transforms.computeIcrfToFixedMatrix = function(date, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(date)) {
            throw new DeveloperError('date is required.');
        }
        //>>includeEnd('debug');
        if (!defined(result)) {
            result = new Matrix3();
        }

        var fixedToIcrfMtx = Transforms.computeFixedToIcrfMatrix(date, result);
        if (!defined(fixedToIcrfMtx)) {
            return undefined;
        }

        return Matrix3.transpose(fixedToIcrfMtx, result);
    };

    var xysScratch = new Iau2006XysSample(0.0, 0.0, 0.0);
    var eopScratch = new EarthOrientationParametersSample(0.0, 0.0, 0.0, 0.0, 0.0, 0.0);
    var rotation1Scratch = new Matrix3();
    var rotation2Scratch = new Matrix3();

    /**
     * Computes a rotation matrix to transform a point or vector from the Earth-Fixed frame axes (ITRF)
     * to the International Celestial Reference Frame (GCRF/ICRF) inertial frame axes
     * at a given time.  This function may return undefined if the data necessary to
     * do the transformation is not yet loaded.
     *
     * @param {JulianDate} date The time at which to compute the rotation matrix.
     * @param {Matrix3} [result] The object onto which to store the result.  If this parameter is
     *                  not specified, a new instance is created and returned.
     * @returns {Matrix3} The rotation matrix, or undefined if the data necessary to do the
     *                   transformation is not yet loaded.
     *
     *
     * @example
     * // Transform a point from the ICRF axes to the Fixed axes.
     * var now = Cesium.JulianDate.now();
     * var pointInFixed = Cesium.Cartesian3.fromDegrees(0.0, 0.0);
     * var fixedToIcrf = Cesium.Transforms.computeIcrfToFixedMatrix(now);
     * var pointInInertial = new Cesium.Cartesian3();
     * if (Cesium.defined(fixedToIcrf)) {
     *     pointInInertial = Cesium.Matrix3.multiplyByVector(fixedToIcrf, pointInFixed, pointInInertial);
     * }
     *
     * @see Transforms.preloadIcrfFixed
     */
    Transforms.computeFixedToIcrfMatrix = function(date, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(date)) {
            throw new DeveloperError('date is required.');
        }
        //>>includeEnd('debug');

        if (!defined(result)) {
            result = new Matrix3();
        }

        // Compute pole wander
        var eop = Transforms.earthOrientationParameters.compute(date, eopScratch);
        if (!defined(eop)) {
            return undefined;
        }

        // There is no external conversion to Terrestrial Time (TT).
        // So use International Atomic Time (TAI) and convert using offsets.
        // Here we are assuming that dayTT and secondTT are positive
        var dayTT = date.dayNumber;
        // It's possible here that secondTT could roll over 86400
        // This does not seem to affect the precision (unit tests check for this)
        var secondTT = date.secondsOfDay + ttMinusTai;

        var xys = Transforms.iau2006XysData.computeXysRadians(dayTT, secondTT, xysScratch);
        if (!defined(xys)) {
            return undefined;
        }

        var x = xys.x + eop.xPoleOffset;
        var y = xys.y + eop.yPoleOffset;

        // Compute XYS rotation
        var a = 1.0 / (1.0 + Math.sqrt(1.0 - x * x - y * y));

        var rotation1 = rotation1Scratch;
        rotation1[0] = 1.0 - a * x * x;
        rotation1[3] = -a * x * y;
        rotation1[6] = x;
        rotation1[1] = -a * x * y;
        rotation1[4] = 1 - a * y * y;
        rotation1[7] = y;
        rotation1[2] = -x;
        rotation1[5] = -y;
        rotation1[8] = 1 - a * (x * x + y * y);

        var rotation2 = Matrix3.fromRotationZ(-xys.s, rotation2Scratch);
        var matrixQ = Matrix3.multiply(rotation1, rotation2, rotation1Scratch);

        // Similar to TT conversions above
        // It's possible here that secondTT could roll over 86400
        // This does not seem to affect the precision (unit tests check for this)
        var dateUt1day = date.dayNumber;
        var dateUt1sec = date.secondsOfDay - JulianDate.computeTaiMinusUtc(date) + eop.ut1MinusUtc;

        // Compute Earth rotation angle
        // The IERS standard for era is
        //    era = 0.7790572732640 + 1.00273781191135448 * Tu
        // where
        //    Tu = JulianDateInUt1 - 2451545.0
        // However, you get much more precision if you make the following simplification
        //    era = a + (1 + b) * (JulianDayNumber + FractionOfDay - 2451545)
        //    era = a + (JulianDayNumber - 2451545) + FractionOfDay + b (JulianDayNumber - 2451545 + FractionOfDay)
        //    era = a + FractionOfDay + b (JulianDayNumber - 2451545 + FractionOfDay)
        // since (JulianDayNumber - 2451545) represents an integer number of revolutions which will be discarded anyway.
        var daysSinceJ2000 = dateUt1day - 2451545;
        var fractionOfDay = dateUt1sec / TimeConstants.SECONDS_PER_DAY;
        var era = 0.7790572732640 + fractionOfDay + 0.00273781191135448 * (daysSinceJ2000 + fractionOfDay);
        era = (era % 1.0) * CesiumMath.TWO_PI;

        var earthRotation = Matrix3.fromRotationZ(era, rotation2Scratch);

        // pseudoFixed to ICRF
        var pfToIcrf = Matrix3.multiply(matrixQ, earthRotation, rotation1Scratch);

        // Compute pole wander matrix
        var cosxp = Math.cos(eop.xPoleWander);
        var cosyp = Math.cos(eop.yPoleWander);
        var sinxp = Math.sin(eop.xPoleWander);
        var sinyp = Math.sin(eop.yPoleWander);

        var ttt = (dayTT - j2000ttDays) + secondTT / TimeConstants.SECONDS_PER_DAY;
        ttt /= 36525.0;

        // approximate sp value in rad
        var sp = -47.0e-6 * ttt * CesiumMath.RADIANS_PER_DEGREE / 3600.0;
        var cossp = Math.cos(sp);
        var sinsp = Math.sin(sp);

        var fToPfMtx = rotation2Scratch;
        fToPfMtx[0] = cosxp * cossp;
        fToPfMtx[1] = cosxp * sinsp;
        fToPfMtx[2] = sinxp;
        fToPfMtx[3] = -cosyp * sinsp + sinyp * sinxp * cossp;
        fToPfMtx[4] = cosyp * cossp + sinyp * sinxp * sinsp;
        fToPfMtx[5] = -sinyp * cosxp;
        fToPfMtx[6] = -sinyp * sinsp - cosyp * sinxp * cossp;
        fToPfMtx[7] = sinyp * cossp - cosyp * sinxp * sinsp;
        fToPfMtx[8] = cosyp * cosxp;

        return Matrix3.multiply(pfToIcrf, fToPfMtx, result);
    };

    var pointToWindowCoordinatesTemp = new Cartesian4();

    /**
     * Transform a point from model coordinates to window coordinates.
     *
     * @param {Matrix4} modelViewProjectionMatrix The 4x4 model-view-projection matrix.
     * @param {Matrix4} viewportTransformation The 4x4 viewport transformation.
     * @param {Cartesian3} point The point to transform.
     * @param {Cartesian2} [result] The object onto which to store the result.
     * @returns {Cartesian2} The modified result parameter or a new Cartesian2 instance if none was provided.
     */
    Transforms.pointToWindowCoordinates = function (modelViewProjectionMatrix, viewportTransformation, point, result) {
        result = Transforms.pointToGLWindowCoordinates(modelViewProjectionMatrix, viewportTransformation, point, result);
        result.y = 2.0 * viewportTransformation[5] - result.y;
        return result;
    };

    /**
     * @private
     */
    Transforms.pointToGLWindowCoordinates = function(modelViewProjectionMatrix, viewportTransformation, point, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(modelViewProjectionMatrix)) {
            throw new DeveloperError('modelViewProjectionMatrix is required.');
        }

        if (!defined(viewportTransformation)) {
            throw new DeveloperError('viewportTransformation is required.');
        }

        if (!defined(point)) {
            throw new DeveloperError('point is required.');
        }
        //>>includeEnd('debug');

        if (!defined(result)) {
            result = new Cartesian2();
        }

        var tmp = pointToWindowCoordinatesTemp;

        Matrix4.multiplyByVector(modelViewProjectionMatrix, Cartesian4.fromElements(point.x, point.y, point.z, 1, tmp), tmp);
        Cartesian4.multiplyByScalar(tmp, 1.0 / tmp.w, tmp);
        Matrix4.multiplyByVector(viewportTransformation, tmp, tmp);
        return Cartesian2.fromCartesian4(tmp, result);
    };

    var normalScratch = new Cartesian3();
    var rightScratch = new Cartesian3();
    var upScratch = new Cartesian3();

    /**
     * @private
     */
    Transforms.rotationMatrixFromPositionVelocity = function(position, velocity, ellipsoid, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(position)) {
            throw new DeveloperError('position is required.');
        }

        if (!defined(velocity)) {
            throw new DeveloperError('velocity is required.');
        }
        //>>includeEnd('debug');

        var normal = defaultValue(ellipsoid, Ellipsoid.WGS84).geodeticSurfaceNormal(position, normalScratch);
        var right = Cartesian3.cross(velocity, normal, rightScratch);
        if (Cartesian3.equalsEpsilon(right, Cartesian3.ZERO, CesiumMath.EPSILON6)) {
            right = Cartesian3.clone(Cartesian3.UNIT_X, right);
        }

        var up = Cartesian3.cross(right, velocity, upScratch);
        Cartesian3.cross(velocity, up, right);
        Cartesian3.negate(right, right);

        if (!defined(result)) {
            result = new Matrix3();
        }

        result[0] = velocity.x;
        result[1] = velocity.y;
        result[2] = velocity.z;
        result[3] = right.x;
        result[4] = right.y;
        result[5] = right.z;
        result[6] = up.x;
        result[7] = up.y;
        result[8] = up.z;

        return result;
    };

    var swizzleMatrix = new Matrix4(
        0.0, 0.0, 1.0, 0.0,
        1.0, 0.0, 0.0, 0.0,
        0.0, 1.0, 0.0, 0.0,
        0.0, 0.0, 0.0, 1.0
    );

    var scratchCartographic = new Cartographic();
    var scratchCartesian3Projection = new Cartesian3();
    var scratchCenter = new Cartesian3();
    var scratchRotation = new Matrix3();
    var scratchFromENU = new Matrix4();
    var scratchToENU = new Matrix4();

    /**
     * @private
     */
    Transforms.basisTo2D = function(projection, matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(projection)) {
            throw new DeveloperError('projection is required.');
        }
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required.');
        }
        //>>includeEnd('debug');

        var rtcCenter = Matrix4.getTranslation(matrix, scratchCenter);
        var ellipsoid = projection.ellipsoid;

        // Get the 2D Center
        var cartographic = ellipsoid.cartesianToCartographic(rtcCenter, scratchCartographic);
        var projectedPosition = projection.project(cartographic, scratchCartesian3Projection);
        Cartesian3.fromElements(projectedPosition.z, projectedPosition.x, projectedPosition.y, projectedPosition);

        // Assuming the instance are positioned in WGS84, invert the WGS84 transform to get the local transform and then convert to 2D
        var fromENU = Transforms.eastNorthUpToFixedFrame(rtcCenter, ellipsoid, scratchFromENU);
        var toENU = Matrix4.inverseTransformation(fromENU, scratchToENU);
        var rotation = Matrix4.getRotation(matrix, scratchRotation);
        var local = Matrix4.multiplyByMatrix3(toENU, rotation, result);
        Matrix4.multiply(swizzleMatrix, local, result); // Swap x, y, z for 2D
        Matrix4.setTranslation(result, projectedPosition, result); // Use the projected center

        return result;
    };

    /**
     * @private
     */
    Transforms.wgs84To2DModelMatrix = function(projection, center, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(projection)) {
            throw new DeveloperError('projection is required.');
        }
        if (!defined(center)) {
            throw new DeveloperError('center is required.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required.');
        }
        //>>includeEnd('debug');

        var ellipsoid = projection.ellipsoid;

        var fromENU = Transforms.eastNorthUpToFixedFrame(center, ellipsoid, scratchFromENU);
        var toENU = Matrix4.inverseTransformation(fromENU, scratchToENU);

        var cartographic = ellipsoid.cartesianToCartographic(center, scratchCartographic);
        var projectedPosition = projection.project(cartographic, scratchCartesian3Projection);
        Cartesian3.fromElements(projectedPosition.z, projectedPosition.x, projectedPosition.y, projectedPosition);

        var translation = Matrix4.fromTranslation(projectedPosition, scratchFromENU);
        Matrix4.multiply(swizzleMatrix, toENU, result);
        Matrix4.multiply(translation, result, result);

        return result;
    };

    return Transforms;
});