Cartesian3.js

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

    /**
     * A 3D Cartesian point.
     * @alias Cartesian3
     * @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.
     *
     * @see Cartesian2
     * @see Cartesian4
     * @see Packable
     */
    var Cartesian3 = function(x, y, z) {
        /**
         * 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);
    };

    /**
     * Converts the provided Spherical into Cartesian3 coordinates.
     *
     * @param {Spherical} spherical The Spherical to be converted to Cartesian3.
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if one was not provided.
     */
    Cartesian3.fromSpherical = function(spherical, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(spherical)) {
            throw new DeveloperError('spherical is required');
        }
        //>>includeEnd('debug');

        if (!defined(result)) {
            result = new Cartesian3();
        }

        var clock = spherical.clock;
        var cone = spherical.cone;
        var magnitude = defaultValue(spherical.magnitude, 1.0);
        var radial = magnitude * Math.sin(cone);
        result.x = radial * Math.cos(clock);
        result.y = radial * Math.sin(clock);
        result.z = magnitude * Math.cos(cone);
        return result;
    };

    /**
     * Creates a Cartesian3 instance from x, y and z coordinates.
     *
     * @param {Number} x The x coordinate.
     * @param {Number} y The y coordinate.
     * @param {Number} z The z coordinate.
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if one was not provided.
     */
    Cartesian3.fromElements = function(x, y, z, result) {
        if (!defined(result)) {
            return new Cartesian3(x, y, z);
        }

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

    /**
     * Duplicates a Cartesian3 instance.
     *
     * @param {Cartesian3} cartesian The Cartesian to duplicate.
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if one was not provided. (Returns undefined if cartesian is undefined)
     */
    Cartesian3.clone = function(cartesian, result) {
        if (!defined(cartesian)) {
            return undefined;
        }
        if (!defined(result)) {
            return new Cartesian3(cartesian.x, cartesian.y, cartesian.z);
        }

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

    /**
     * Creates a Cartesian3 instance from an existing Cartesian4.  This simply takes the
     * x, y, and z properties of the Cartesian4 and drops w.
     * @function
     *
     * @param {Cartesian4} cartesian The Cartesian4 instance to create a Cartesian3 instance from.
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if one was not provided.
     */
    Cartesian3.fromCartesian4 = Cartesian3.clone;

    /**
     * The number of elements used to pack the object into an array.
     * @type {Number}
     */
    Cartesian3.packedLength = 3;

    /**
     * Stores the provided instance into the provided array.
     *
     * @param {Cartesian3} 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.
     */
    Cartesian3.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;
    };

    /**
     * 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 {Cartesian3} [result] The object into which to store the result.
     * @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if one was not provided.
     */
    Cartesian3.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 Cartesian3();
        }
        result.x = array[startingIndex++];
        result.y = array[startingIndex++];
        result.z = array[startingIndex];
        return result;
    };

    /**
     * Creates a Cartesian3 from three consecutive elements in an array.
     * @function
     *
     * @param {Number[]} array The array whose three consecutive elements correspond to the x, y, and z components, respectively.
     * @param {Number} [startingIndex=0] The offset into the array of the first element, which corresponds to the x component.
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if one was not provided.
     *
     * @example
     * // Create a Cartesian3 with (1.0, 2.0, 3.0)
     * var v = [1.0, 2.0, 3.0];
     * var p = Cesium.Cartesian3.fromArray(v);
     *
     * // Create a Cartesian3 with (1.0, 2.0, 3.0) using an offset into an array
     * var v2 = [0.0, 0.0, 1.0, 2.0, 3.0];
     * var p2 = Cesium.Cartesian3.fromArray(v2, 2);
     */
    Cartesian3.fromArray = Cartesian3.unpack;

    /**
     * Computes the value of the maximum component for the supplied Cartesian.
     *
     * @param {Cartesian3} cartesian The cartesian to use.
     * @returns {Number} The value of the maximum component.
     */
    Cartesian3.maximumComponent = function(cartesian) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian is required');
        }
        //>>includeEnd('debug');

        return Math.max(cartesian.x, cartesian.y, cartesian.z);
    };

    /**
     * Computes the value of the minimum component for the supplied Cartesian.
     *
     * @param {Cartesian3} cartesian The cartesian to use.
     * @returns {Number} The value of the minimum component.
     */
    Cartesian3.minimumComponent = function(cartesian) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian is required');
        }
        //>>includeEnd('debug');

        return Math.min(cartesian.x, cartesian.y, cartesian.z);
    };

    /**
     * Compares two Cartesians and computes a Cartesian which contains the minimum components of the supplied Cartesians.
     *
     * @param {Cartesian3} first A cartesian to compare.
     * @param {Cartesian3} second A cartesian to compare.
     * @param {Cartesian3} result The object into which to store the result.
     * @returns {Cartesian3} A cartesian with the minimum components.
     */
    Cartesian3.minimumByComponent = function(first, second, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(first)) {
            throw new DeveloperError('first is required.');
        }
        if (!defined(second)) {
            throw new DeveloperError('second is required.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required.');
        }
        //>>includeEnd('debug');

        result.x = Math.min(first.x, second.x);
        result.y = Math.min(first.y, second.y);
        result.z = Math.min(first.z, second.z);

        return result;
    };

    /**
     * Compares two Cartesians and computes a Cartesian which contains the maximum components of the supplied Cartesians.
     *
     * @param {Cartesian3} first A cartesian to compare.
     * @param {Cartesian3} second A cartesian to compare.
     * @param {Cartesian3} result The object into which to store the result.
     * @returns {Cartesian3} A cartesian with the maximum components.
     */
    Cartesian3.maximumByComponent = function(first, second, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(first)) {
            throw new DeveloperError('first is required.');
        }
        if (!defined(second)) {
            throw new DeveloperError('second is required.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required.');
        }
        //>>includeEnd('debug');

        result.x = Math.max(first.x, second.x);
        result.y = Math.max(first.y, second.y);
        result.z = Math.max(first.z, second.z);
        return result;
    };

    /**
     * Computes the provided Cartesian's squared magnitude.
     *
     * @param {Cartesian3} cartesian The Cartesian instance whose squared magnitude is to be computed.
     * @returns {Number} The squared magnitude.
     */
    Cartesian3.magnitudeSquared = function(cartesian) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian is required');
        }
        //>>includeEnd('debug');

        return cartesian.x * cartesian.x + cartesian.y * cartesian.y + cartesian.z * cartesian.z;
    };

    /**
     * Computes the Cartesian's magnitude (length).
     *
     * @param {Cartesian3} cartesian The Cartesian instance whose magnitude is to be computed.
     * @returns {Number} The magnitude.
     */
    Cartesian3.magnitude = function(cartesian) {
        return Math.sqrt(Cartesian3.magnitudeSquared(cartesian));
    };

    var distanceScratch = new Cartesian3();

    /**
     * Computes the distance between two points.
     *
     * @param {Cartesian3} left The first point to compute the distance from.
     * @param {Cartesian3} right The second point to compute the distance to.
     * @returns {Number} The distance between two points.
     *
     * @example
     * // Returns 1.0
     * var d = Cesium.Cartesian3.distance(new Cesium.Cartesian3(1.0, 0.0, 0.0), new Cesium.Cartesian3(2.0, 0.0, 0.0));
     */
    Cartesian3.distance = function(left, right) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(left) || !defined(right)) {
            throw new DeveloperError('left and right are required.');
        }
        //>>includeEnd('debug');

        Cartesian3.subtract(left, right, distanceScratch);
        return Cartesian3.magnitude(distanceScratch);
    };

    /**
     * Computes the squared distance between two points.  Comparing squared distances
     * using this function is more efficient than comparing distances using {@link Cartesian3#distance}.
     *
     * @param {Cartesian3} left The first point to compute the distance from.
     * @param {Cartesian3} right The second point to compute the distance to.
     * @returns {Number} The distance between two points.
     *
     * @example
     * // Returns 4.0, not 2.0
     * var d = Cesium.Cartesian3.distanceSquared(new Cesium.Cartesian3(1.0, 0.0, 0.0), new Cesium.Cartesian3(3.0, 0.0, 0.0));
     */
    Cartesian3.distanceSquared = function(left, right) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(left) || !defined(right)) {
            throw new DeveloperError('left and right are required.');
        }
        //>>includeEnd('debug');

        Cartesian3.subtract(left, right, distanceScratch);
        return Cartesian3.magnitudeSquared(distanceScratch);
    };

    /**
     * Computes the normalized form of the supplied Cartesian.
     *
     * @param {Cartesian3} cartesian The Cartesian to be normalized.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.normalize = 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 magnitude = Cartesian3.magnitude(cartesian);

        result.x = cartesian.x / magnitude;
        result.y = cartesian.y / magnitude;
        result.z = cartesian.z / magnitude;
        return result;
    };

    /**
     * Computes the dot (scalar) product of two Cartesians.
     *
     * @param {Cartesian3} left The first Cartesian.
     * @param {Cartesian3} right The second Cartesian.
     * @returns {Number} The dot product.
     */
    Cartesian3.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;
    };

    /**
     * Computes the componentwise product of two Cartesians.
     *
     * @param {Cartesian3} left The first Cartesian.
     * @param {Cartesian3} right The second Cartesian.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.multiplyComponents = 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;
        return result;
    };

    /**
     * Computes the componentwise sum of two Cartesians.
     *
     * @param {Cartesian3} left The first Cartesian.
     * @param {Cartesian3} right The second Cartesian.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.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;
        return result;
    };

    /**
     * Computes the componentwise difference of two Cartesians.
     *
     * @param {Cartesian3} left The first Cartesian.
     * @param {Cartesian3} right The second Cartesian.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.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;
        return result;
    };

    /**
     * Multiplies the provided Cartesian componentwise by the provided scalar.
     *
     * @param {Cartesian3} cartesian The Cartesian to be scaled.
     * @param {Number} scalar The scalar to multiply with.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.multiplyByScalar = function(cartesian, scalar, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian 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 = cartesian.x * scalar;
        result.y = cartesian.y * scalar;
        result.z = cartesian.z * scalar;
        return result;
    };

    /**
     * Divides the provided Cartesian componentwise by the provided scalar.
     *
     * @param {Cartesian3} cartesian The Cartesian to be divided.
     * @param {Number} scalar The scalar to divide by.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.divideByScalar = function(cartesian, scalar, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian 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 = cartesian.x / scalar;
        result.y = cartesian.y / scalar;
        result.z = cartesian.z / scalar;
        return result;
    };

    /**
     * Negates the provided Cartesian.
     *
     * @param {Cartesian3} cartesian The Cartesian to be negated.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.negate = 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');

        result.x = -cartesian.x;
        result.y = -cartesian.y;
        result.z = -cartesian.z;
        return result;
    };

    /**
     * Computes the absolute value of the provided Cartesian.
     *
     * @param {Cartesian3} cartesian The Cartesian whose absolute value is to be computed.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.abs = 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');

        result.x = Math.abs(cartesian.x);
        result.y = Math.abs(cartesian.y);
        result.z = Math.abs(cartesian.z);
        return result;
    };

    var lerpScratch = new Cartesian3();
    /**
     * Computes the linear interpolation or extrapolation at t using the provided cartesians.
     *
     * @param {Cartesian3} start The value corresponding to t at 0.0.
     * @param {Cartesian3} end The value corresponding to t at 1.0.
     * @param {Number} t The point along t at which to interpolate.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Cartesian3.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');

        Cartesian3.multiplyByScalar(end, t, lerpScratch);
        result = Cartesian3.multiplyByScalar(start, 1.0 - t, result);
        return Cartesian3.add(lerpScratch, result, result);
    };

    var angleBetweenScratch = new Cartesian3();
    var angleBetweenScratch2 = new Cartesian3();
    /**
     * Returns the angle, in radians, between the provided Cartesians.
     *
     * @param {Cartesian3} left The first Cartesian.
     * @param {Cartesian3} right The second Cartesian.
     * @returns {Number} The angle between the Cartesians.
     */
    Cartesian3.angleBetween = 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');

        Cartesian3.normalize(left, angleBetweenScratch);
        Cartesian3.normalize(right, angleBetweenScratch2);
        var cosine = Cartesian3.dot(angleBetweenScratch, angleBetweenScratch2);
        var sine = Cartesian3.magnitude(Cartesian3.cross(angleBetweenScratch, angleBetweenScratch2, angleBetweenScratch));
        return Math.atan2(sine, cosine);
    };

    var mostOrthogonalAxisScratch = new Cartesian3();
    /**
     * Returns the axis that is most orthogonal to the provided Cartesian.
     *
     * @param {Cartesian3} cartesian The Cartesian on which to find the most orthogonal axis.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The most orthogonal axis.
     */
    Cartesian3.mostOrthogonalAxis = 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 f = Cartesian3.normalize(cartesian, mostOrthogonalAxisScratch);
        Cartesian3.abs(f, f);

        if (f.x <= f.y) {
            if (f.x <= f.z) {
                result = Cartesian3.clone(Cartesian3.UNIT_X, result);
            } else {
                result = Cartesian3.clone(Cartesian3.UNIT_Z, result);
            }
        } else {
            if (f.y <= f.z) {
                result = Cartesian3.clone(Cartesian3.UNIT_Y, result);
            } else {
                result = Cartesian3.clone(Cartesian3.UNIT_Z, result);
            }
        }

        return result;
    };

    /**
     * Compares the provided Cartesians componentwise and returns
     * <code>true</code> if they are equal, <code>false</code> otherwise.
     *
     * @param {Cartesian3} [left] The first Cartesian.
     * @param {Cartesian3} [right] The second Cartesian.
     * @returns {Boolean} <code>true</code> if left and right are equal, <code>false</code> otherwise.
     */
    Cartesian3.equals = function(left, right) {
            return (left === right) ||
              ((defined(left)) &&
               (defined(right)) &&
               (left.x === right.x) &&
               (left.y === right.y) &&
               (left.z === right.z));
    };

    /**
     * @private
     */
    Cartesian3.equalsArray = function(cartesian, array, offset) {
        return cartesian.x === array[offset] &&
               cartesian.y === array[offset + 1] &&
               cartesian.z === array[offset + 2];
    };

    /**
     * Compares the provided Cartesians componentwise and returns
     * <code>true</code> if they pass an absolute or relative tolerance test,
     * <code>false</code> otherwise.
     *
     * @param {Cartesian3} [left] The first Cartesian.
     * @param {Cartesian3} [right] The second Cartesian.
     * @param {Number} relativeEpsilon The relative epsilon tolerance to use for equality testing.
     * @param {Number} [absoluteEpsilon=relativeEpsilon] The absolute epsilon tolerance to use for equality testing.
     * @returns {Boolean} <code>true</code> if left and right are within the provided epsilon, <code>false</code> otherwise.
     */
    Cartesian3.equalsEpsilon = function(left, right, relativeEpsilon, absoluteEpsilon) {
        return (left === right) ||
               (defined(left) &&
                defined(right) &&
                CesiumMath.equalsEpsilon(left.x, right.x, relativeEpsilon, absoluteEpsilon) &&
                CesiumMath.equalsEpsilon(left.y, right.y, relativeEpsilon, absoluteEpsilon) &&
                CesiumMath.equalsEpsilon(left.z, right.z, relativeEpsilon, absoluteEpsilon));
    };

    /**
     * Computes the cross (outer) product of two Cartesians.
     *
     * @param {Cartesian3} left The first Cartesian.
     * @param {Cartesian3} right The second Cartesian.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The cross product.
     */
    Cartesian3.cross = 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 rightX = right.x;
        var rightY = right.y;
        var rightZ = right.z;

        var x = leftY * rightZ - leftZ * rightY;
        var y = leftZ * rightX - leftX * rightZ;
        var z = leftX * rightY - leftY * rightX;

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

    /**
     * Returns a Cartesian3 position from longitude and latitude values given in degrees.
     *
     * @param {Number} longitude The longitude, in degrees
     * @param {Number} latitude The latitude, in degrees
     * @param {Number} [height=0.0] The height, in meters, above the ellipsoid.
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid on which the position lies.
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The position
     *
     * @example
     * var position = Cesium.Cartesian3.fromDegrees(-115.0, 37.0);
     */
    Cartesian3.fromDegrees = function(longitude, latitude, height, ellipsoid, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(longitude)) {
            throw new DeveloperError('longitude is required');
        }
        if (!defined(latitude)) {
            throw new DeveloperError('latitude is required');
        }
        //>>includeEnd('debug');

        var lon = CesiumMath.toRadians(longitude);
        var lat = CesiumMath.toRadians(latitude);
        return Cartesian3.fromRadians(lon, lat, height, ellipsoid, result);
    };

    var scratchN = new Cartesian3();
    var scratchK = new Cartesian3();
    var wgs84RadiiSquared = new Cartesian3(6378137.0 * 6378137.0, 6378137.0 * 6378137.0, 6356752.3142451793 * 6356752.3142451793);

    /**
     * Returns a Cartesian3 position from longitude and latitude values given in radians.
     *
     * @param {Number} longitude The longitude, in radians
     * @param {Number} latitude The latitude, in radians
     * @param {Number} [height=0.0] The height, in meters, above the ellipsoid.
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid on which the position lies.
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The position
     *
     * @example
     * var position = Cesium.Cartesian3.fromRadians(-2.007, 0.645);
     */
    Cartesian3.fromRadians = function(longitude, latitude, height, ellipsoid, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(longitude)) {
            throw new DeveloperError('longitude is required');
        }
        if (!defined(latitude)) {
            throw new DeveloperError('latitude is required');
        }
        //>>includeEnd('debug');

        height = defaultValue(height, 0.0);
        var radiiSquared = defined(ellipsoid) ? ellipsoid.radiiSquared : wgs84RadiiSquared;

        var cosLatitude = Math.cos(latitude);
        scratchN.x = cosLatitude * Math.cos(longitude);
        scratchN.y = cosLatitude * Math.sin(longitude);
        scratchN.z = Math.sin(latitude);
        scratchN = Cartesian3.normalize(scratchN, scratchN);

        Cartesian3.multiplyComponents(radiiSquared, scratchN, scratchK);
        var gamma = Math.sqrt(Cartesian3.dot(scratchN, scratchK));
        scratchK = Cartesian3.divideByScalar(scratchK, gamma, scratchK);
        scratchN = Cartesian3.multiplyByScalar(scratchN, height, scratchN);

        if (!defined(result)) {
            result = new Cartesian3();
        }
        return Cartesian3.add(scratchK, scratchN, result);
    };

    /**
     * Returns an array of Cartesian3 positions given an array of longitude and latitude values given in degrees.
     *
     * @param {Number[]} coordinates A list of longitude and latitude values. Values alternate [longitude, latitude, longitude, latitude...].
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid on which the coordinates lie.
     * @param {Cartesian3[]} [result] An array of Cartesian3 objects to store the result.
     * @returns {Cartesian3[]} The array of positions.
     *
     * @example
     * var positions = Cesium.Cartesian3.fromDegreesArray([-115.0, 37.0, -107.0, 33.0]);
     */
    Cartesian3.fromDegreesArray = function(coordinates, ellipsoid, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(coordinates)) {
            throw new DeveloperError('positions is required.');
        }
        //>>includeEnd('debug');

        var pos = new Array(coordinates.length);
        for (var i = 0; i < coordinates.length; i++) {
            pos[i] = CesiumMath.toRadians(coordinates[i]);
        }

        return Cartesian3.fromRadiansArray(pos, ellipsoid, result);
    };

    /**
     * Returns an array of Cartesian3 positions given an array of longitude and latitude values given in radians.
     *
     * @param {Number[]} coordinates A list of longitude and latitude values. Values alternate [longitude, latitude, longitude, latitude...].
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid on which the coordinates lie.
     * @param {Cartesian3[]} [result] An array of Cartesian3 objects to store the result.
     * @returns {Cartesian3[]} The array of positions.
     *
     * @example
     * var positions = Cesium.Cartesian3.fromRadiansArray([-2.007, 0.645, -1.867, .575]);
     */
    Cartesian3.fromRadiansArray = function(coordinates, ellipsoid, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(coordinates)) {
            throw new DeveloperError('positions is required.');
        }
        if (coordinates.length < 2) {
            throw new DeveloperError('positions length cannot be less than 2.');
        }
        if (coordinates.length % 2 !== 0) {
            throw new DeveloperError('positions length must be a multiple of 2.');
        }
        //>>includeEnd('debug');

        var length = coordinates.length;
        if (!defined(result)) {
            result = new Array(length/2);
        } else {
            result.length = length/2;
        }

        for ( var i = 0; i < length; i+=2) {
            var lon = coordinates[i];
            var lat = coordinates[i+1];
            result[i/2] = Cartesian3.fromRadians(lon, lat, 0, ellipsoid, result[i/2]);
        }

        return result;
    };

    /**
     * Returns an array of Cartesian3 positions given an array of longitude, latitude and height values where longitude and latitude are given in degrees.
     *
     * @param {Number[]} coordinates A list of longitude, latitude and height values. Values alternate [longitude, latitude, height,, longitude, latitude, height...].
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid on which the position lies.
     * @param {Cartesian3[]} [result] An array of Cartesian3 objects to store the result.
     * @returns {Cartesian3[]} The array of positions.
     *
     * @example
     * var positions = Cesium.Cartesian3.fromDegreesArrayHeights([-115.0, 37.0, 100000.0, -107.0, 33.0, 150000.0]);
     */
    Cartesian3.fromDegreesArrayHeights = function(coordinates, ellipsoid, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(coordinates)) {
            throw new DeveloperError('positions is required.');
        }
        if (coordinates.length < 3) {
            throw new DeveloperError('positions length cannot be less than 3.');
        }
        if (coordinates.length % 3 !== 0) {
            throw new DeveloperError('positions length must be a multiple of 3.');
        }
        //>>includeEnd('debug');

        var pos = new Array(coordinates.length);
        for (var i = 0; i < coordinates.length; i+=3) {
            pos[i] = CesiumMath.toRadians(coordinates[i]);
            pos[i+1] = CesiumMath.toRadians(coordinates[i+1]);
            pos[i+2] = coordinates[i+2];
        }

        return Cartesian3.fromRadiansArrayHeights(pos, ellipsoid, result);
    };

    /**
     * Returns an array of Cartesian3 positions given an array of longitude, latitude and height values where longitude and latitude are given in radians.
     *
     * @param {Number[]} coordinates A list of longitude, latitude and height values. Values alternate [longitude, latitude, height,, longitude, latitude, height...].
     * @param {Ellipsoid} [ellipsoid=Ellipsoid.WGS84] The ellipsoid on which the position lies.
     * @param {Cartesian3[]} [result] An array of Cartesian3 objects to store the result.
     * @returns {Cartesian3[]} The array of positions.
     *
     * @example
     * var positions = Cesium.Cartesian3.fromRadiansArrayHeights([-2.007, 0.645, 100000.0, -1.867, .575, 150000.0]);
     */
    Cartesian3.fromRadiansArrayHeights = function(coordinates, ellipsoid, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(coordinates)) {
            throw new DeveloperError('positions is required.');
        }
        if (coordinates.length < 3) {
            throw new DeveloperError('positions length cannot be less than 3.');
        }
        if (coordinates.length % 3 !== 0) {
            throw new DeveloperError('positions length must be a multiple of 3.');
        }
        //>>includeEnd('debug');

        var length = coordinates.length;
        if (!defined(result)) {
            result = new Array(length/3);
        } else {
            result.length = length/3;
        }

        for ( var i = 0; i < length; i+=3) {
            var lon = coordinates[i];
            var lat = coordinates[i+1];
            var alt = coordinates[i+2];
            result[i/3] = Cartesian3.fromRadians(lon, lat, alt, ellipsoid, result[i/3]);
        }

        return result;
    };

    /**
     * An immutable Cartesian3 instance initialized to (0.0, 0.0, 0.0).
     *
     * @type {Cartesian3}
     * @constant
     */
    Cartesian3.ZERO = freezeObject(new Cartesian3(0.0, 0.0, 0.0));

    /**
     * An immutable Cartesian3 instance initialized to (1.0, 0.0, 0.0).
     *
     * @type {Cartesian3}
     * @constant
     */
    Cartesian3.UNIT_X = freezeObject(new Cartesian3(1.0, 0.0, 0.0));

    /**
     * An immutable Cartesian3 instance initialized to (0.0, 1.0, 0.0).
     *
     * @type {Cartesian3}
     * @constant
     */
    Cartesian3.UNIT_Y = freezeObject(new Cartesian3(0.0, 1.0, 0.0));

    /**
     * An immutable Cartesian3 instance initialized to (0.0, 0.0, 1.0).
     *
     * @type {Cartesian3}
     * @constant
     */
    Cartesian3.UNIT_Z = freezeObject(new Cartesian3(0.0, 0.0, 1.0));

    /**
     * Duplicates this Cartesian3 instance.
     *
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if one was not provided.
     */
    Cartesian3.prototype.clone = function(result) {
        return Cartesian3.clone(this, result);
    };

    /**
     * Compares this Cartesian against the provided Cartesian componentwise and returns
     * <code>true</code> if they are equal, <code>false</code> otherwise.
     *
     * @param {Cartesian3} [right] The right hand side Cartesian.
     * @returns {Boolean} <code>true</code> if they are equal, <code>false</code> otherwise.
     */
    Cartesian3.prototype.equals = function(right) {
        return Cartesian3.equals(this, right);
    };

    /**
     * Compares this Cartesian against the provided Cartesian componentwise and returns
     * <code>true</code> if they pass an absolute or relative tolerance test,
     * <code>false</code> otherwise.
     *
     * @param {Cartesian3} [right] The right hand side Cartesian.
     * @param {Number} relativeEpsilon The relative epsilon tolerance to use for equality testing.
     * @param {Number} [absoluteEpsilon=relativeEpsilon] The absolute epsilon tolerance to use for equality testing.
     * @returns {Boolean} <code>true</code> if they are within the provided epsilon, <code>false</code> otherwise.
     */
    Cartesian3.prototype.equalsEpsilon = function(right, relativeEpsilon, absoluteEpsilon) {
        return Cartesian3.equalsEpsilon(this, right, relativeEpsilon, absoluteEpsilon);
    };

    /**
     * Creates a string representing this Cartesian in the format '(x, y, z)'.
     *
     * @returns {String} A string representing this Cartesian in the format '(x, y, z)'.
     */
    Cartesian3.prototype.toString = function() {
        return '(' + this.x + ', ' + this.y + ', ' + this.z + ')';
    };

    return Cartesian3;
});