Quaternion.js

/*global define*/
define([
        './Cartesian3',
        './defaultValue',
        './defined',
        './DeveloperError',
        './FeatureDetection',
        './freezeObject',
        './Math',
        './Matrix3'
    ], function(
        Cartesian3,
        defaultValue,
        defined,
        DeveloperError,
        FeatureDetection,
        freezeObject,
        CesiumMath,
        Matrix3) {
    'use strict';

    /**
     * A set of 4-dimensional coordinates used to represent rotation in 3-dimensional space.
     * @alias Quaternion
     * @constructor
     *
     * @param {Number} [x=0.0] The X component.
     * @param {Number} [y=0.0] The Y component.
     * @param {Number} [z=0.0] The Z component.
     * @param {Number} [w=0.0] The W component.
     *
     * @see PackableForInterpolation
     */
    function Quaternion(x, y, z, w) {
        /**
         * The X component.
         * @type {Number}
         * @default 0.0
         */
        this.x = defaultValue(x, 0.0);

        /**
         * The Y component.
         * @type {Number}
         * @default 0.0
         */
        this.y = defaultValue(y, 0.0);

        /**
         * The Z component.
         * @type {Number}
         * @default 0.0
         */
        this.z = defaultValue(z, 0.0);

        /**
         * The W component.
         * @type {Number}
         * @default 0.0
         */
        this.w = defaultValue(w, 0.0);
    }

    var fromAxisAngleScratch = new Cartesian3();

    /**
     * Computes a quaternion representing a rotation around an axis.
     *
     * @param {Cartesian3} axis The axis of rotation.
     * @param {Number} angle The angle in radians to rotate around the axis.
     * @param {Quaternion} [result] The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter or a new Quaternion instance if one was not provided.
     */
    Quaternion.fromAxisAngle = function(axis, angle, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(axis)) {
            throw new DeveloperError('axis is required.');
        }
        if (typeof angle !== 'number') {
            throw new DeveloperError('angle is required and must be a number.');
        }
        //>>includeEnd('debug');

        var halfAngle = angle / 2.0;
        var s = Math.sin(halfAngle);
        fromAxisAngleScratch = Cartesian3.normalize(axis, fromAxisAngleScratch);

        var x = fromAxisAngleScratch.x * s;
        var y = fromAxisAngleScratch.y * s;
        var z = fromAxisAngleScratch.z * s;
        var w = Math.cos(halfAngle);
        if (!defined(result)) {
            return new Quaternion(x, y, z, w);
        }
        result.x = x;
        result.y = y;
        result.z = z;
        result.w = w;
        return result;
    };

    var fromRotationMatrixNext = [1, 2, 0];
    var fromRotationMatrixQuat = new Array(3);
    /**
     * Computes a Quaternion from the provided Matrix3 instance.
     *
     * @param {Matrix3} matrix The rotation matrix.
     * @param {Quaternion} [result] The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter or a new Quaternion instance if one was not provided.
     *
     * @see Matrix3.fromQuaternion
     */
    Quaternion.fromRotationMatrix = function(matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required.');
        }
        //>>includeEnd('debug');

        var root;
        var x;
        var y;
        var z;
        var w;

        var m00 = matrix[Matrix3.COLUMN0ROW0];
        var m11 = matrix[Matrix3.COLUMN1ROW1];
        var m22 = matrix[Matrix3.COLUMN2ROW2];
        var trace = m00 + m11 + m22;

        if (trace > 0.0) {
            // |w| > 1/2, may as well choose w > 1/2
            root = Math.sqrt(trace + 1.0); // 2w
            w = 0.5 * root;
            root = 0.5 / root; // 1/(4w)

            x = (matrix[Matrix3.COLUMN1ROW2] - matrix[Matrix3.COLUMN2ROW1]) * root;
            y = (matrix[Matrix3.COLUMN2ROW0] - matrix[Matrix3.COLUMN0ROW2]) * root;
            z = (matrix[Matrix3.COLUMN0ROW1] - matrix[Matrix3.COLUMN1ROW0]) * root;
        } else {
            // |w| <= 1/2
            var next = fromRotationMatrixNext;

            var i = 0;
            if (m11 > m00) {
                i = 1;
            }
            if (m22 > m00 && m22 > m11) {
                i = 2;
            }
            var j = next[i];
            var k = next[j];

            root = Math.sqrt(matrix[Matrix3.getElementIndex(i, i)] - matrix[Matrix3.getElementIndex(j, j)] - matrix[Matrix3.getElementIndex(k, k)] + 1.0);

            var quat = fromRotationMatrixQuat;
            quat[i] = 0.5 * root;
            root = 0.5 / root;
            w = (matrix[Matrix3.getElementIndex(k, j)] - matrix[Matrix3.getElementIndex(j, k)]) * root;
            quat[j] = (matrix[Matrix3.getElementIndex(j, i)] + matrix[Matrix3.getElementIndex(i, j)]) * root;
            quat[k] = (matrix[Matrix3.getElementIndex(k, i)] + matrix[Matrix3.getElementIndex(i, k)]) * root;

            x = -quat[0];
            y = -quat[1];
            z = -quat[2];
        }

        if (!defined(result)) {
            return new Quaternion(x, y, z, w);
        }
        result.x = x;
        result.y = y;
        result.z = z;
        result.w = w;
        return result;
    };

    var scratchHPRQuaternion = new Quaternion();

    /**
     * Computes a rotation from the given heading, pitch and roll angles. Heading is the rotation about the
     * negative z axis. Pitch is the rotation about the negative y axis. Roll is the rotation about
     * the positive x axis.
     *
     * @param {Number} heading The heading angle in radians.
     * @param {Number} pitch The pitch angle in radians.
     * @param {Number} roll The roll angle in radians.
     * @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.
     */
    Quaternion.fromHeadingPitchRoll = function(heading, pitch, roll, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(heading)) {
            throw new DeveloperError('heading is required.');
        }
        if (!defined(pitch)) {
            throw new DeveloperError('pitch is required.');
        }
        if (!defined(roll)) {
            throw new DeveloperError('roll is required.');
        }
        //>>includeEnd('debug');

        var rollQuaternion = Quaternion.fromAxisAngle(Cartesian3.UNIT_X, roll, scratchHPRQuaternion);
        var pitchQuaternion = Quaternion.fromAxisAngle(Cartesian3.UNIT_Y, -pitch, result);
        result = Quaternion.multiply(pitchQuaternion, rollQuaternion, pitchQuaternion);
        var headingQuaternion = Quaternion.fromAxisAngle(Cartesian3.UNIT_Z, -heading, scratchHPRQuaternion);
        return Quaternion.multiply(headingQuaternion, result, result);
    };

    var sampledQuaternionAxis = new Cartesian3();
    var sampledQuaternionRotation = new Cartesian3();
    var sampledQuaternionTempQuaternion = new Quaternion();
    var sampledQuaternionQuaternion0 = new Quaternion();
    var sampledQuaternionQuaternion0Conjugate = new Quaternion();

    /**
     * The number of elements used to pack the object into an array.
     * @type {Number}
     */
    Quaternion.packedLength = 4;

    /**
     * Stores the provided instance into the provided array.
     *
     * @param {Quaternion} value The value to pack.
     * @param {Number[]} array The array to pack into.
     * @param {Number} [startingIndex=0] The index into the array at which to start packing the elements.
     *
     * @returns {Number[]} The array that was packed into
     */
    Quaternion.pack = function(value, array, startingIndex) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(value)) {
            throw new DeveloperError('value is required');
        }

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

        startingIndex = defaultValue(startingIndex, 0);

        array[startingIndex++] = value.x;
        array[startingIndex++] = value.y;
        array[startingIndex++] = value.z;
        array[startingIndex] = value.w;

        return array;
    };

    /**
     * Retrieves an instance from a packed array.
     *
     * @param {Number[]} array The packed array.
     * @param {Number} [startingIndex=0] The starting index of the element to be unpacked.
     * @param {Quaternion} [result] The object into which to store the result.
     * @returns {Quaternion} The modified result parameter or a new Quaternion instance if one was not provided.
     */
    Quaternion.unpack = function(array, startingIndex, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(array)) {
            throw new DeveloperError('array is required');
        }
        //>>includeEnd('debug');

        startingIndex = defaultValue(startingIndex, 0);

        if (!defined(result)) {
            result = new Quaternion();
        }
        result.x = array[startingIndex];
        result.y = array[startingIndex + 1];
        result.z = array[startingIndex + 2];
        result.w = array[startingIndex + 3];
        return result;
    };

    /**
     * The number of elements used to store the object into an array in its interpolatable form.
     * @type {Number}
     */
    Quaternion.packedInterpolationLength = 3;

    /**
     * Converts a packed array into a form suitable for interpolation.
     *
     * @param {Number[]} packedArray The packed array.
     * @param {Number} [startingIndex=0] The index of the first element to be converted.
     * @param {Number} [lastIndex=packedArray.length] The index of the last element to be converted.
     * @param {Number[]} result The object into which to store the result.
     */
    Quaternion.convertPackedArrayForInterpolation = function(packedArray, startingIndex, lastIndex, result) {
        Quaternion.unpack(packedArray, lastIndex * 4, sampledQuaternionQuaternion0Conjugate);
        Quaternion.conjugate(sampledQuaternionQuaternion0Conjugate, sampledQuaternionQuaternion0Conjugate);

        for (var i = 0, len = lastIndex - startingIndex + 1; i < len; i++) {
            var offset = i * 3;
            Quaternion.unpack(packedArray, (startingIndex + i) * 4, sampledQuaternionTempQuaternion);

            Quaternion.multiply(sampledQuaternionTempQuaternion, sampledQuaternionQuaternion0Conjugate, sampledQuaternionTempQuaternion);

            if (sampledQuaternionTempQuaternion.w < 0) {
                Quaternion.negate(sampledQuaternionTempQuaternion, sampledQuaternionTempQuaternion);
            }

            Quaternion.computeAxis(sampledQuaternionTempQuaternion, sampledQuaternionAxis);
            var angle = Quaternion.computeAngle(sampledQuaternionTempQuaternion);
            result[offset] = sampledQuaternionAxis.x * angle;
            result[offset + 1] = sampledQuaternionAxis.y * angle;
            result[offset + 2] = sampledQuaternionAxis.z * angle;
        }
    };

    /**
     * Retrieves an instance from a packed array converted with {@link convertPackedArrayForInterpolation}.
     *
     * @param {Number[]} array The array previously packed for interpolation.
     * @param {Number[]} sourceArray The original packed array.
     * @param {Number} [startingIndex=0] The startingIndex used to convert the array.
     * @param {Number} [lastIndex=packedArray.length] The lastIndex used to convert the array.
     * @param {Quaternion} [result] The object into which to store the result.
     * @returns {Quaternion} The modified result parameter or a new Quaternion instance if one was not provided.
     */
    Quaternion.unpackInterpolationResult = function(array, sourceArray, firstIndex, lastIndex, result) {
        if (!defined(result)) {
            result = new Quaternion();
        }
        Cartesian3.fromArray(array, 0, sampledQuaternionRotation);
        var magnitude = Cartesian3.magnitude(sampledQuaternionRotation);

        Quaternion.unpack(sourceArray, lastIndex * 4, sampledQuaternionQuaternion0);

        if (magnitude === 0) {
            Quaternion.clone(Quaternion.IDENTITY, sampledQuaternionTempQuaternion);
        } else {
            Quaternion.fromAxisAngle(sampledQuaternionRotation, magnitude, sampledQuaternionTempQuaternion);
        }

        return Quaternion.multiply(sampledQuaternionTempQuaternion, sampledQuaternionQuaternion0, result);
    };

    /**
     * Duplicates a Quaternion instance.
     *
     * @param {Quaternion} quaternion The quaternion to duplicate.
     * @param {Quaternion} [result] The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter or a new Quaternion instance if one was not provided. (Returns undefined if quaternion is undefined)
     */
    Quaternion.clone = function(quaternion, result) {
        if (!defined(quaternion)) {
            return undefined;
        }

        if (!defined(result)) {
            return new Quaternion(quaternion.x, quaternion.y, quaternion.z, quaternion.w);
        }

        result.x = quaternion.x;
        result.y = quaternion.y;
        result.z = quaternion.z;
        result.w = quaternion.w;
        return result;
    };

    /**
     * Computes the conjugate of the provided quaternion.
     *
     * @param {Quaternion} quaternion The quaternion to conjugate.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.conjugate = function(quaternion, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        result.x = -quaternion.x;
        result.y = -quaternion.y;
        result.z = -quaternion.z;
        result.w = quaternion.w;
        return result;
    };

    /**
     * Computes magnitude squared for the provided quaternion.
     *
     * @param {Quaternion} quaternion The quaternion to conjugate.
     * @returns {Number} The magnitude squared.
     */
    Quaternion.magnitudeSquared = function(quaternion) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required');
        }
        //>>includeEnd('debug');

        return quaternion.x * quaternion.x + quaternion.y * quaternion.y + quaternion.z * quaternion.z + quaternion.w * quaternion.w;
    };

    /**
     * Computes magnitude for the provided quaternion.
     *
     * @param {Quaternion} quaternion The quaternion to conjugate.
     * @returns {Number} The magnitude.
     */
    Quaternion.magnitude = function(quaternion) {
        return Math.sqrt(Quaternion.magnitudeSquared(quaternion));
    };

    /**
     * Computes the normalized form of the provided quaternion.
     *
     * @param {Quaternion} quaternion The quaternion to normalize.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.normalize = function(quaternion, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var inverseMagnitude = 1.0 / Quaternion.magnitude(quaternion);
        var x = quaternion.x * inverseMagnitude;
        var y = quaternion.y * inverseMagnitude;
        var z = quaternion.z * inverseMagnitude;
        var w = quaternion.w * inverseMagnitude;

        result.x = x;
        result.y = y;
        result.z = z;
        result.w = w;
        return result;
    };

    /**
     * Computes the inverse of the provided quaternion.
     *
     * @param {Quaternion} quaternion The quaternion to normalize.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.inverse = function(quaternion, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var magnitudeSquared = Quaternion.magnitudeSquared(quaternion);
        result = Quaternion.conjugate(quaternion, result);
        return Quaternion.multiplyByScalar(result, 1.0 / magnitudeSquared, result);
    };

    /**
     * Computes the componentwise sum of two quaternions.
     *
     * @param {Quaternion} left The first quaternion.
     * @param {Quaternion} right The second quaternion.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.add = function(left, right, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(left)) {
            throw new DeveloperError('left is required');
        }
        if (!defined(right)) {
            throw new DeveloperError('right is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        result.x = left.x + right.x;
        result.y = left.y + right.y;
        result.z = left.z + right.z;
        result.w = left.w + right.w;
        return result;
    };

    /**
     * Computes the componentwise difference of two quaternions.
     *
     * @param {Quaternion} left The first quaternion.
     * @param {Quaternion} right The second quaternion.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.subtract = function(left, right, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(left)) {
            throw new DeveloperError('left is required');
        }
        if (!defined(right)) {
            throw new DeveloperError('right is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        result.x = left.x - right.x;
        result.y = left.y - right.y;
        result.z = left.z - right.z;
        result.w = left.w - right.w;
        return result;
    };

    /**
     * Negates the provided quaternion.
     *
     * @param {Quaternion} quaternion The quaternion to be negated.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.negate = function(quaternion, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        result.x = -quaternion.x;
        result.y = -quaternion.y;
        result.z = -quaternion.z;
        result.w = -quaternion.w;
        return result;
    };

    /**
     * Computes the dot (scalar) product of two quaternions.
     *
     * @param {Quaternion} left The first quaternion.
     * @param {Quaternion} right The second quaternion.
     * @returns {Number} The dot product.
     */
    Quaternion.dot = function(left, right) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(left)) {
            throw new DeveloperError('left is required');
        }
        if (!defined(right)) {
            throw new DeveloperError('right is required');
        }
        //>>includeEnd('debug');

        return left.x * right.x + left.y * right.y + left.z * right.z + left.w * right.w;
    };

    /**
     * Computes the product of two quaternions.
     *
     * @param {Quaternion} left The first quaternion.
     * @param {Quaternion} right The second quaternion.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.multiply = function(left, right, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(left)) {
            throw new DeveloperError('left is required');
        }
        if (!defined(right)) {
            throw new DeveloperError('right is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var leftX = left.x;
        var leftY = left.y;
        var leftZ = left.z;
        var leftW = left.w;

        var rightX = right.x;
        var rightY = right.y;
        var rightZ = right.z;
        var rightW = right.w;

        var x = leftW * rightX + leftX * rightW + leftY * rightZ - leftZ * rightY;
        var y = leftW * rightY - leftX * rightZ + leftY * rightW + leftZ * rightX;
        var z = leftW * rightZ + leftX * rightY - leftY * rightX + leftZ * rightW;
        var w = leftW * rightW - leftX * rightX - leftY * rightY - leftZ * rightZ;

        result.x = x;
        result.y = y;
        result.z = z;
        result.w = w;
        return result;
    };

    /**
     * Multiplies the provided quaternion componentwise by the provided scalar.
     *
     * @param {Quaternion} quaternion The quaternion to be scaled.
     * @param {Number} scalar The scalar to multiply with.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.multiplyByScalar = function(quaternion, scalar, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required');
        }
        if (typeof scalar !== 'number') {
            throw new DeveloperError('scalar is required and must be a number.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        result.x = quaternion.x * scalar;
        result.y = quaternion.y * scalar;
        result.z = quaternion.z * scalar;
        result.w = quaternion.w * scalar;
        return result;
    };

    /**
     * Divides the provided quaternion componentwise by the provided scalar.
     *
     * @param {Quaternion} quaternion The quaternion to be divided.
     * @param {Number} scalar The scalar to divide by.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.divideByScalar = function(quaternion, scalar, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required');
        }
        if (typeof scalar !== 'number') {
            throw new DeveloperError('scalar is required and must be a number.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        result.x = quaternion.x / scalar;
        result.y = quaternion.y / scalar;
        result.z = quaternion.z / scalar;
        result.w = quaternion.w / scalar;
        return result;
    };

    /**
     * Computes the axis of rotation of the provided quaternion.
     *
     * @param {Quaternion} quaternion The quaternion to use.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Quaternion.computeAxis = function(quaternion, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var w = quaternion.w;
        if (Math.abs(w - 1.0) < CesiumMath.EPSILON6) {
            result.x = result.y = result.z = 0;
            return result;
        }

        var scalar = 1.0 / Math.sqrt(1.0 - (w * w));

        result.x = quaternion.x * scalar;
        result.y = quaternion.y * scalar;
        result.z = quaternion.z * scalar;
        return result;
    };

    /**
     * Computes the angle of rotation of the provided quaternion.
     *
     * @param {Quaternion} quaternion The quaternion to use.
     * @returns {Number} The angle of rotation.
     */
    Quaternion.computeAngle = function(quaternion) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required');
        }
        //>>includeEnd('debug');

        if (Math.abs(quaternion.w - 1.0) < CesiumMath.EPSILON6) {
            return 0.0;
        }
        return 2.0 * Math.acos(quaternion.w);
    };

    var lerpScratch = new Quaternion();
    /**
     * Computes the linear interpolation or extrapolation at t using the provided quaternions.
     *
     * @param {Quaternion} start The value corresponding to t at 0.0.
     * @param {Quaternion} end The value corresponding to t at 1.0.
     * @param {Number} t The point along t at which to interpolate.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.lerp = function(start, end, t, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(start)) {
            throw new DeveloperError('start is required.');
        }
        if (!defined(end)) {
            throw new DeveloperError('end is required.');
        }
        if (typeof t !== 'number') {
            throw new DeveloperError('t is required and must be a number.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        lerpScratch = Quaternion.multiplyByScalar(end, t, lerpScratch);
        result = Quaternion.multiplyByScalar(start, 1.0 - t, result);
        return Quaternion.add(lerpScratch, result, result);
    };

    var slerpEndNegated = new Quaternion();
    var slerpScaledP = new Quaternion();
    var slerpScaledR = new Quaternion();
    /**
     * Computes the spherical linear interpolation or extrapolation at t using the provided quaternions.
     *
     * @param {Quaternion} start The value corresponding to t at 0.0.
     * @param {Quaternion} end The value corresponding to t at 1.0.
     * @param {Number} t The point along t at which to interpolate.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     *
     * @see Quaternion#fastSlerp
     */
    Quaternion.slerp = function(start, end, t, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(start)) {
            throw new DeveloperError('start is required.');
        }
        if (!defined(end)) {
            throw new DeveloperError('end is required.');
        }
        if (typeof t !== 'number') {
            throw new DeveloperError('t is required and must be a number.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var dot = Quaternion.dot(start, end);

        // The angle between start must be acute. Since q and -q represent
        // the same rotation, negate q to get the acute angle.
        var r = end;
        if (dot < 0.0) {
            dot = -dot;
            r = slerpEndNegated = Quaternion.negate(end, slerpEndNegated);
        }

        // dot > 0, as the dot product approaches 1, the angle between the
        // quaternions vanishes. use linear interpolation.
        if (1.0 - dot < CesiumMath.EPSILON6) {
            return Quaternion.lerp(start, r, t, result);
        }

        var theta = Math.acos(dot);
        slerpScaledP = Quaternion.multiplyByScalar(start, Math.sin((1 - t) * theta), slerpScaledP);
        slerpScaledR = Quaternion.multiplyByScalar(r, Math.sin(t * theta), slerpScaledR);
        result = Quaternion.add(slerpScaledP, slerpScaledR, result);
        return Quaternion.multiplyByScalar(result, 1.0 / Math.sin(theta), result);
    };

    /**
     * The logarithmic quaternion function.
     *
     * @param {Quaternion} quaternion The unit quaternion.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Quaternion.log = function(quaternion, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var theta = CesiumMath.acosClamped(quaternion.w);
        var thetaOverSinTheta = 0.0;

        if (theta !== 0.0) {
            thetaOverSinTheta = theta / Math.sin(theta);
        }

        return Cartesian3.multiplyByScalar(quaternion, thetaOverSinTheta, result);
    };

    /**
     * The exponential quaternion function.
     *
     * @param {Cartesian3} cartesian The cartesian.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     */
    Quaternion.exp = function(cartesian, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian is required.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var theta = Cartesian3.magnitude(cartesian);
        var sinThetaOverTheta = 0.0;

        if (theta !== 0.0) {
            sinThetaOverTheta = Math.sin(theta) / theta;
        }

        result.x = cartesian.x * sinThetaOverTheta;
        result.y = cartesian.y * sinThetaOverTheta;
        result.z = cartesian.z * sinThetaOverTheta;
        result.w = Math.cos(theta);

        return result;
    };

    var squadScratchCartesian0 = new Cartesian3();
    var squadScratchCartesian1 = new Cartesian3();
    var squadScratchQuaternion0 = new Quaternion();
    var squadScratchQuaternion1 = new Quaternion();

    /**
     * Computes an inner quadrangle point.
     * <p>This will compute quaternions that ensure a squad curve is C<sup>1</sup>.</p>
     *
     * @param {Quaternion} q0 The first quaternion.
     * @param {Quaternion} q1 The second quaternion.
     * @param {Quaternion} q2 The third quaternion.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     *
     * @see Quaternion#squad
     */
    Quaternion.computeInnerQuadrangle = function(q0, q1, q2, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(q0) || !defined(q1) || !defined(q2)) {
            throw new DeveloperError('q0, q1, and q2 are required.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var qInv = Quaternion.conjugate(q1, squadScratchQuaternion0);
        Quaternion.multiply(qInv, q2, squadScratchQuaternion1);
        var cart0 = Quaternion.log(squadScratchQuaternion1, squadScratchCartesian0);

        Quaternion.multiply(qInv, q0, squadScratchQuaternion1);
        var cart1 = Quaternion.log(squadScratchQuaternion1, squadScratchCartesian1);

        Cartesian3.add(cart0, cart1, cart0);
        Cartesian3.multiplyByScalar(cart0, 0.25, cart0);
        Cartesian3.negate(cart0, cart0);
        Quaternion.exp(cart0, squadScratchQuaternion0);

        return Quaternion.multiply(q1, squadScratchQuaternion0, result);
    };

    /**
     * Computes the spherical quadrangle interpolation between quaternions.
     *
     * @param {Quaternion} q0 The first quaternion.
     * @param {Quaternion} q1 The second quaternion.
     * @param {Quaternion} s0 The first inner quadrangle.
     * @param {Quaternion} s1 The second inner quadrangle.
     * @param {Number} t The time in [0,1] used to interpolate.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     *
     *
     * @example
     * // 1. compute the squad interpolation between two quaternions on a curve
     * var s0 = Cesium.Quaternion.computeInnerQuadrangle(quaternions[i - 1], quaternions[i], quaternions[i + 1], new Cesium.Quaternion());
     * var s1 = Cesium.Quaternion.computeInnerQuadrangle(quaternions[i], quaternions[i + 1], quaternions[i + 2], new Cesium.Quaternion());
     * var q = Cesium.Quaternion.squad(quaternions[i], quaternions[i + 1], s0, s1, t, new Cesium.Quaternion());
     *
     * // 2. compute the squad interpolation as above but where the first quaternion is a end point.
     * var s1 = Cesium.Quaternion.computeInnerQuadrangle(quaternions[0], quaternions[1], quaternions[2], new Cesium.Quaternion());
     * var q = Cesium.Quaternion.squad(quaternions[0], quaternions[1], quaternions[0], s1, t, new Cesium.Quaternion());
     *
     * @see Quaternion#computeInnerQuadrangle
     */
    Quaternion.squad = function(q0, q1, s0, s1, t, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(q0) || !defined(q1) || !defined(s0) || !defined(s1)) {
            throw new DeveloperError('q0, q1, s0, and s1 are required.');
        }
        if (typeof t !== 'number') {
            throw new DeveloperError('t is required and must be a number.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var slerp0 = Quaternion.slerp(q0, q1, t, squadScratchQuaternion0);
        var slerp1 = Quaternion.slerp(s0, s1, t, squadScratchQuaternion1);
        return Quaternion.slerp(slerp0, slerp1, 2.0 * t * (1.0 - t), result);
    };

    var fastSlerpScratchQuaternion = new Quaternion();
    var opmu = 1.90110745351730037;
    var u = FeatureDetection.supportsTypedArrays() ? new Float32Array(8) : [];
    var v = FeatureDetection.supportsTypedArrays() ? new Float32Array(8) : [];
    var bT = FeatureDetection.supportsTypedArrays() ? new Float32Array(8) : [];
    var bD = FeatureDetection.supportsTypedArrays() ? new Float32Array(8) : [];

    for (var i = 0; i < 7; ++i) {
        var s = i + 1.0;
        var t = 2.0 * s + 1.0;
        u[i] = 1.0 / (s * t);
        v[i] = s / t;
    }

    u[7] = opmu / (8.0 * 17.0);
    v[7] = opmu * 8.0 / 17.0;

    /**
     * Computes the spherical linear interpolation or extrapolation at t using the provided quaternions.
     * This implementation is faster than {@link Quaternion#slerp}, but is only accurate up to 10<sup>-6</sup>.
     *
     * @param {Quaternion} start The value corresponding to t at 0.0.
     * @param {Quaternion} end The value corresponding to t at 1.0.
     * @param {Number} t The point along t at which to interpolate.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter.
     *
     * @see Quaternion#slerp
     */
    Quaternion.fastSlerp = function(start, end, t, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(start)) {
            throw new DeveloperError('start is required.');
        }
        if (!defined(end)) {
            throw new DeveloperError('end is required.');
        }
        if (typeof t !== 'number') {
            throw new DeveloperError('t is required and must be a number.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var x = Quaternion.dot(start, end);

        var sign;
        if (x >= 0) {
            sign = 1.0;
        } else {
            sign = -1.0;
            x = -x;
        }

        var xm1 = x - 1.0;
        var d = 1.0 - t;
        var sqrT = t * t;
        var sqrD = d * d;

        for (var i = 7; i >= 0; --i) {
            bT[i] = (u[i] * sqrT - v[i]) * xm1;
            bD[i] = (u[i] * sqrD - v[i]) * xm1;
        }

        var cT = sign * t * (
            1.0 + bT[0] * (1.0 + bT[1] * (1.0 + bT[2] * (1.0 + bT[3] * (
            1.0 + bT[4] * (1.0 + bT[5] * (1.0 + bT[6] * (1.0 + bT[7]))))))));
        var cD = d * (
            1.0 + bD[0] * (1.0 + bD[1] * (1.0 + bD[2] * (1.0 + bD[3] * (
            1.0 + bD[4] * (1.0 + bD[5] * (1.0 + bD[6] * (1.0 + bD[7]))))))));

        var temp = Quaternion.multiplyByScalar(start, cD, fastSlerpScratchQuaternion);
        Quaternion.multiplyByScalar(end, cT, result);
        return Quaternion.add(temp, result, result);
    };

    /**
     * Computes the spherical quadrangle interpolation between quaternions.
     * An implementation that is faster than {@link Quaternion#squad}, but less accurate.
     *
     * @param {Quaternion} q0 The first quaternion.
     * @param {Quaternion} q1 The second quaternion.
     * @param {Quaternion} s0 The first inner quadrangle.
     * @param {Quaternion} s1 The second inner quadrangle.
     * @param {Number} t The time in [0,1] used to interpolate.
     * @param {Quaternion} result The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter or a new instance if none was provided.
     *
     * @see Quaternion#squad
     */
    Quaternion.fastSquad = function(q0, q1, s0, s1, t, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(q0) || !defined(q1) || !defined(s0) || !defined(s1)) {
            throw new DeveloperError('q0, q1, s0, and s1 are required.');
        }
        if (typeof t !== 'number') {
            throw new DeveloperError('t is required and must be a number.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required');
        }
        //>>includeEnd('debug');

        var slerp0 = Quaternion.fastSlerp(q0, q1, t, squadScratchQuaternion0);
        var slerp1 = Quaternion.fastSlerp(s0, s1, t, squadScratchQuaternion1);
        return Quaternion.fastSlerp(slerp0, slerp1, 2.0 * t * (1.0 - t), result);
    };

    /**
     * Compares the provided quaternions componentwise and returns
     * <code>true</code> if they are equal, <code>false</code> otherwise.
     *
     * @param {Quaternion} [left] The first quaternion.
     * @param {Quaternion} [right] The second quaternion.
     * @returns {Boolean} <code>true</code> if left and right are equal, <code>false</code> otherwise.
     */
    Quaternion.equals = function(left, right) {
        return (left === right) ||
               ((defined(left)) &&
                (defined(right)) &&
                (left.x === right.x) &&
                (left.y === right.y) &&
                (left.z === right.z) &&
                (left.w === right.w));
    };

    /**
     * Compares the provided quaternions componentwise and returns
     * <code>true</code> if they are within the provided epsilon,
     * <code>false</code> otherwise.
     *
     * @param {Quaternion} [left] The first quaternion.
     * @param {Quaternion} [right] The second quaternion.
     * @param {Number} epsilon The epsilon to use for equality testing.
     * @returns {Boolean} <code>true</code> if left and right are within the provided epsilon, <code>false</code> otherwise.
     */
    Quaternion.equalsEpsilon = function(left, right, epsilon) {
        //>>includeStart('debug', pragmas.debug);
        if (typeof epsilon !== 'number') {
            throw new DeveloperError('epsilon is required and must be a number.');
        }
        //>>includeEnd('debug');

        return (left === right) ||
               ((defined(left)) &&
                (defined(right)) &&
                (Math.abs(left.x - right.x) <= epsilon) &&
                (Math.abs(left.y - right.y) <= epsilon) &&
                (Math.abs(left.z - right.z) <= epsilon) &&
                (Math.abs(left.w - right.w) <= epsilon));
    };

    /**
     * An immutable Quaternion instance initialized to (0.0, 0.0, 0.0, 0.0).
     *
     * @type {Quaternion}
     * @constant
     */
    Quaternion.ZERO = freezeObject(new Quaternion(0.0, 0.0, 0.0, 0.0));

    /**
     * An immutable Quaternion instance initialized to (0.0, 0.0, 0.0, 1.0).
     *
     * @type {Quaternion}
     * @constant
     */
    Quaternion.IDENTITY = freezeObject(new Quaternion(0.0, 0.0, 0.0, 1.0));

    /**
     * Duplicates this Quaternion instance.
     *
     * @param {Quaternion} [result] The object onto which to store the result.
     * @returns {Quaternion} The modified result parameter or a new Quaternion instance if one was not provided.
     */
    Quaternion.prototype.clone = function(result) {
        return Quaternion.clone(this, result);
    };

    /**
     * Compares this and the provided quaternion componentwise and returns
     * <code>true</code> if they are equal, <code>false</code> otherwise.
     *
     * @param {Quaternion} [right] The right hand side quaternion.
     * @returns {Boolean} <code>true</code> if left and right are equal, <code>false</code> otherwise.
     */
    Quaternion.prototype.equals = function(right) {
        return Quaternion.equals(this, right);
    };

    /**
     * Compares this and the provided quaternion componentwise and returns
     * <code>true</code> if they are within the provided epsilon,
     * <code>false</code> otherwise.
     *
     * @param {Quaternion} [right] The right hand side quaternion.
     * @param {Number} epsilon The epsilon to use for equality testing.
     * @returns {Boolean} <code>true</code> if left and right are within the provided epsilon, <code>false</code> otherwise.
     */
    Quaternion.prototype.equalsEpsilon = function(right, epsilon) {
        return Quaternion.equalsEpsilon(this, right, epsilon);
    };

    /**
     * Returns a string representing this quaternion in the format (x, y, z, w).
     *
     * @returns {String} A string representing this Quaternion.
     */
    Quaternion.prototype.toString = function() {
        return '(' + this.x + ', ' + this.y + ', ' + this.z + ', ' + this.w + ')';
    };

    return Quaternion;
});