Matrix3.js

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

    /**
     * A 3x3 matrix, indexable as a column-major order array.
     * Constructor parameters are in row-major order for code readability.
     * @alias Matrix3
     * @constructor
     *
     * @param {Number} [column0Row0=0.0] The value for column 0, row 0.
     * @param {Number} [column1Row0=0.0] The value for column 1, row 0.
     * @param {Number} [column2Row0=0.0] The value for column 2, row 0.
     * @param {Number} [column0Row1=0.0] The value for column 0, row 1.
     * @param {Number} [column1Row1=0.0] The value for column 1, row 1.
     * @param {Number} [column2Row1=0.0] The value for column 2, row 1.
     * @param {Number} [column0Row2=0.0] The value for column 0, row 2.
     * @param {Number} [column1Row2=0.0] The value for column 1, row 2.
     * @param {Number} [column2Row2=0.0] The value for column 2, row 2.
     *
     * @see Matrix3.fromColumnMajorArray
     * @see Matrix3.fromRowMajorArray
     * @see Matrix3.fromQuaternion
     * @see Matrix3.fromScale
     * @see Matrix3.fromUniformScale
     * @see Matrix2
     * @see Matrix4
     */
    var Matrix3 = function(column0Row0, column1Row0, column2Row0,
                           column0Row1, column1Row1, column2Row1,
                           column0Row2, column1Row2, column2Row2) {
        this[0] = defaultValue(column0Row0, 0.0);
        this[1] = defaultValue(column0Row1, 0.0);
        this[2] = defaultValue(column0Row2, 0.0);
        this[3] = defaultValue(column1Row0, 0.0);
        this[4] = defaultValue(column1Row1, 0.0);
        this[5] = defaultValue(column1Row2, 0.0);
        this[6] = defaultValue(column2Row0, 0.0);
        this[7] = defaultValue(column2Row1, 0.0);
        this[8] = defaultValue(column2Row2, 0.0);
    };

    /**
     * Duplicates a Matrix3 instance.
     *
     * @param {Matrix3} matrix The matrix to duplicate.
     * @param {Matrix3} [result] The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter or a new Matrix3 instance if one was not provided. (Returns undefined if matrix is undefined)
     */
    Matrix3.clone = function(values, result) {
        if (!defined(values)) {
            return undefined;
        }
        if (!defined(result)) {
            return new Matrix3(values[0], values[3], values[6],
                               values[1], values[4], values[7],
                               values[2], values[5], values[8]);
        }
        result[0] = values[0];
        result[1] = values[1];
        result[2] = values[2];
        result[3] = values[3];
        result[4] = values[4];
        result[5] = values[5];
        result[6] = values[6];
        result[7] = values[7];
        result[8] = values[8];
        return result;
    };

    /**
     * Creates a Matrix3 from 9 consecutive elements in an array.
     *
     * @param {Number[]} array The array whose 9 consecutive elements correspond to the positions of the matrix.  Assumes column-major order.
     * @param {Number} [startingIndex=0] The offset into the array of the first element, which corresponds to first column first row position in the matrix.
     * @param {Matrix3} [result] The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter or a new Matrix3 instance if one was not provided.
     *
     * @example
     * // Create the Matrix3:
     * // [1.0, 2.0, 3.0]
     * // [1.0, 2.0, 3.0]
     * // [1.0, 2.0, 3.0]
     *
     * var v = [1.0, 1.0, 1.0, 2.0, 2.0, 2.0, 3.0, 3.0, 3.0];
     * var m = Cesium.Matrix3.fromArray(v);
     *
     * // Create same Matrix3 with using an offset into an array
     * var v2 = [0.0, 0.0, 1.0, 1.0, 1.0, 2.0, 2.0, 2.0, 3.0, 3.0, 3.0];
     * var m2 = Cesium.Matrix3.fromArray(v2, 2);
     */
    Matrix3.fromArray = 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 Matrix3();
        }

        result[0] = array[startingIndex];
        result[1] = array[startingIndex + 1];
        result[2] = array[startingIndex + 2];
        result[3] = array[startingIndex + 3];
        result[4] = array[startingIndex + 4];
        result[5] = array[startingIndex + 5];
        result[6] = array[startingIndex + 6];
        result[7] = array[startingIndex + 7];
        result[8] = array[startingIndex + 8];
        return result;
    };

    /**
     * Creates a Matrix3 instance from a column-major order array.
     *
     * @param {Number[]} values The column-major order array.
     * @param {Matrix3} [result] The object in which the result will be stored, if undefined a new instance will be created.
     * @returns The modified result parameter, or a new Matrix3 instance if one was not provided.
     */
    Matrix3.fromColumnMajorArray = function(values, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(values)) {
            throw new DeveloperError('values parameter is required');
        }
        //>>includeEnd('debug');

        return Matrix3.clone(values, result);
    };

    /**
     * Creates a Matrix3 instance from a row-major order array.
     * The resulting matrix will be in column-major order.
     *
     * @param {Number[]} values The row-major order array.
     * @param {Matrix3} [result] The object in which the result will be stored, if undefined a new instance will be created.
     * @returns The modified result parameter, or a new Matrix3 instance if one was not provided.
     */
    Matrix3.fromRowMajorArray = function(values, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(values)) {
            throw new DeveloperError('values is required.');
        }
        //>>includeEnd('debug');

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

    /**
     * Computes a 3x3 rotation matrix from the provided quaternion.
     *
     * @param {Quaternion} quaternion the quaternion to use.
     * @returns {Matrix3} The 3x3 rotation matrix from this quaternion.
     */
    Matrix3.fromQuaternion = function(quaternion, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(quaternion)) {
            throw new DeveloperError('quaternion is required');
        }
        //>>includeEnd('debug');

        var x2 = quaternion.x * quaternion.x;
        var xy = quaternion.x * quaternion.y;
        var xz = quaternion.x * quaternion.z;
        var xw = quaternion.x * quaternion.w;
        var y2 = quaternion.y * quaternion.y;
        var yz = quaternion.y * quaternion.z;
        var yw = quaternion.y * quaternion.w;
        var z2 = quaternion.z * quaternion.z;
        var zw = quaternion.z * quaternion.w;
        var w2 = quaternion.w * quaternion.w;

        var m00 = x2 - y2 - z2 + w2;
        var m01 = 2.0 * (xy - zw);
        var m02 = 2.0 * (xz + yw);

        var m10 = 2.0 * (xy + zw);
        var m11 = -x2 + y2 - z2 + w2;
        var m12 = 2.0 * (yz - xw);

        var m20 = 2.0 * (xz - yw);
        var m21 = 2.0 * (yz + xw);
        var m22 = -x2 - y2 + z2 + w2;

        if (!defined(result)) {
            return new Matrix3(m00, m01, m02,
                               m10, m11, m12,
                               m20, m21, m22);
        }
        result[0] = m00;
        result[1] = m10;
        result[2] = m20;
        result[3] = m01;
        result[4] = m11;
        result[5] = m21;
        result[6] = m02;
        result[7] = m12;
        result[8] = m22;
        return result;
    };

    /**
     * Computes a Matrix3 instance representing a non-uniform scale.
     *
     * @param {Cartesian3} scale The x, y, and z scale factors.
     * @param {Matrix3} [result] The object in which the result will be stored, if undefined a new instance will be created.
     * @returns The modified result parameter, or a new Matrix3 instance if one was not provided.
     *
     * @example
     * // Creates
     * //   [7.0, 0.0, 0.0]
     * //   [0.0, 8.0, 0.0]
     * //   [0.0, 0.0, 9.0]
     * var m = Cesium.Matrix3.fromScale(new Cesium.Cartesian3(7.0, 8.0, 9.0));
     */
    Matrix3.fromScale = function(scale, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(scale)) {
            throw new DeveloperError('scale is required.');
        }
        //>>includeEnd('debug');

        if (!defined(result)) {
            return new Matrix3(
                scale.x, 0.0,     0.0,
                0.0,     scale.y, 0.0,
                0.0,     0.0,     scale.z);
        }

        result[0] = scale.x;
        result[1] = 0.0;
        result[2] = 0.0;
        result[3] = 0.0;
        result[4] = scale.y;
        result[5] = 0.0;
        result[6] = 0.0;
        result[7] = 0.0;
        result[8] = scale.z;
        return result;
    };

    /**
     * Computes a Matrix3 instance representing a uniform scale.
     *
     * @param {Number} scale The uniform scale factor.
     * @param {Matrix3} [result] The object in which the result will be stored, if undefined a new instance will be created.
     * @returns The modified result parameter, or a new Matrix3 instance if one was not provided.
     *
     * @example
     * // Creates
     * //   [2.0, 0.0, 0.0]
     * //   [0.0, 2.0, 0.0]
     * //   [0.0, 0.0, 2.0]
     * var m = Cesium.Matrix3.fromUniformScale(2.0);
     */
    Matrix3.fromUniformScale = function(scale, result) {
        //>>includeStart('debug', pragmas.debug);
        if (typeof scale !== 'number') {
            throw new DeveloperError('scale is required.');
        }
        //>>includeEnd('debug');

        if (!defined(result)) {
            return new Matrix3(
                scale, 0.0,   0.0,
                0.0,   scale, 0.0,
                0.0,   0.0,   scale);
        }

        result[0] = scale;
        result[1] = 0.0;
        result[2] = 0.0;
        result[3] = 0.0;
        result[4] = scale;
        result[5] = 0.0;
        result[6] = 0.0;
        result[7] = 0.0;
        result[8] = scale;
        return result;
    };

    /**
     * Computes a Matrix3 instance representing the cross product equivalent matrix of a Cartesian3 vector.
     *
     * @param {Cartesian3} the vector on the left hand side of the cross product operation.
     * @param {Matrix3} [result] The object in which the result will be stored, if undefined a new instance will be created.
     * @returns The modified result parameter, or a new Matrix3 instance if one was not provided.
     *
     * @example
     * // Creates
     * //   [0.0, -9.0,  8.0]
     * //   [9.0,  0.0, -7.0]
     * //   [-8.0, 7.0,  0.0]
     * var m = Cesium.Matrix3.fromCrossProduct(new Cesium.Cartesian3(7.0, 8.0, 9.0));
     */
    Matrix3.fromCrossProduct = function(vector, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(vector)) {
            throw new DeveloperError('vector is required.');
        }
        //>>includeEnd('debug');

        if (!defined(result)) {
            return new Matrix3(
                      0.0, -vector.z,  vector.y,
                 vector.z,       0.0, -vector.x,
                -vector.y,  vector.x,       0.0);
        }

        result[0] = 0.0;
        result[1] = vector.z;
        result[2] = -vector.y;
        result[3] = -vector.z;
        result[4] = 0.0;
        result[5] = vector.x;
        result[6] = vector.y;
        result[7] = -vector.x;
        result[8] = 0.0;
        return result;
    };

    /**
     * Creates a rotation matrix around the x-axis.
     *
     * @param {Number} angle The angle, in radians, of the rotation.  Positive angles are counterclockwise.
     * @param {Matrix3} [result] The object in which the result will be stored, if undefined a new instance will be created.
     * @returns The modified result parameter, or a new Matrix3 instance if one was not provided.
     *
     * @example
     * // Rotate a point 45 degrees counterclockwise around the x-axis.
     * var p = new Cesium.Cartesian3(5, 6, 7);
     * var m = Cesium.Matrix3.fromRotationX(Cesium.Math.toRadians(45.0));
     * var rotated = Cesium.Matrix3.multiplyByVector(m, p, new Cesium.Cartesian3());
     */
    Matrix3.fromRotationX = function(angle, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(angle)) {
            throw new DeveloperError('angle is required.');
        }
        //>>includeEnd('debug');

        var cosAngle = Math.cos(angle);
        var sinAngle = Math.sin(angle);

        if (!defined(result)) {
            return new Matrix3(
                1.0, 0.0, 0.0,
                0.0, cosAngle, -sinAngle,
                0.0, sinAngle, cosAngle);
        }

        result[0] = 1.0;
        result[1] = 0.0;
        result[2] = 0.0;
        result[3] = 0.0;
        result[4] = cosAngle;
        result[5] = sinAngle;
        result[6] = 0.0;
        result[7] = -sinAngle;
        result[8] = cosAngle;

        return result;
    };

    /**
     * Creates a rotation matrix around the y-axis.
     *
     * @param {Number} angle The angle, in radians, of the rotation.  Positive angles are counterclockwise.
     * @param {Matrix3} [result] The object in which the result will be stored, if undefined a new instance will be created.
     * @returns The modified result parameter, or a new Matrix3 instance if one was not provided.
     *
     * @example
     * // Rotate a point 45 degrees counterclockwise around the y-axis.
     * var p = new Cesium.Cartesian3(5, 6, 7);
     * var m = Cesium.Matrix3.fromRotationY(Cesium.Math.toRadians(45.0));
     * var rotated = Cesium.Matrix3.multiplyByVector(m, p, new Cesium.Cartesian3());
     */
    Matrix3.fromRotationY = function(angle, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(angle)) {
            throw new DeveloperError('angle is required.');
        }
        //>>includeEnd('debug');

        var cosAngle = Math.cos(angle);
        var sinAngle = Math.sin(angle);

        if (!defined(result)) {
            return new Matrix3(
                cosAngle, 0.0, sinAngle,
                0.0, 1.0, 0.0,
                -sinAngle, 0.0, cosAngle);
        }

        result[0] = cosAngle;
        result[1] = 0.0;
        result[2] = -sinAngle;
        result[3] = 0.0;
        result[4] = 1.0;
        result[5] = 0.0;
        result[6] = sinAngle;
        result[7] = 0.0;
        result[8] = cosAngle;

        return result;
    };

    /**
     * Creates a rotation matrix around the z-axis.
     *
     * @param {Number} angle The angle, in radians, of the rotation.  Positive angles are counterclockwise.
     * @param {Matrix3} [result] The object in which the result will be stored, if undefined a new instance will be created.
     * @returns The modified result parameter, or a new Matrix3 instance if one was not provided.
     *
     * @example
     * // Rotate a point 45 degrees counterclockwise around the z-axis.
     * var p = new Cesium.Cartesian3(5, 6, 7);
     * var m = Cesium.Matrix3.fromRotationZ(Cesium.Math.toRadians(45.0));
     * var rotated = Cesium.Matrix3.multiplyByVector(m, p, new Cesium.Cartesian3());
     */
    Matrix3.fromRotationZ = function(angle, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(angle)) {
            throw new DeveloperError('angle is required.');
        }
        //>>includeEnd('debug');

        var cosAngle = Math.cos(angle);
        var sinAngle = Math.sin(angle);

        if (!defined(result)) {
            return new Matrix3(
                cosAngle, -sinAngle, 0.0,
                sinAngle, cosAngle, 0.0,
                0.0, 0.0, 1.0);
        }

        result[0] = cosAngle;
        result[1] = sinAngle;
        result[2] = 0.0;
        result[3] = -sinAngle;
        result[4] = cosAngle;
        result[5] = 0.0;
        result[6] = 0.0;
        result[7] = 0.0;
        result[8] = 1.0;

        return result;
    };

    /**
     * Creates an Array from the provided Matrix3 instance.
     * The array will be in column-major order.
     *
     * @param {Matrix3} matrix The matrix to use..
     * @param {Number[]} [result] The Array onto which to store the result.
     * @returns {Number[]} The modified Array parameter or a new Array instance if one was not provided.
     */
    Matrix3.toArray = function(matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        //>>includeEnd('debug');

        if (!defined(result)) {
            return [matrix[0], matrix[1], matrix[2], matrix[3], matrix[4], matrix[5], matrix[6], matrix[7], matrix[8]];
        }
        result[0] = matrix[0];
        result[1] = matrix[1];
        result[2] = matrix[2];
        result[3] = matrix[3];
        result[4] = matrix[4];
        result[5] = matrix[5];
        result[6] = matrix[6];
        result[7] = matrix[7];
        result[8] = matrix[8];
        return result;
    };

    /**
     * Computes the array index of the element at the provided row and column.
     *
     * @param {Number} row The zero-based index of the row.
     * @param {Number} column The zero-based index of the column.
     * @returns {Number} The index of the element at the provided row and column.
     *
     * @exception {DeveloperError} row must be 0, 1, or 2.
     * @exception {DeveloperError} column must be 0, 1, or 2.
     *
     * @example
     * var myMatrix = new Cesium.Matrix3();
     * var column1Row0Index = Cesium.Matrix3.getElementIndex(1, 0);
     * var column1Row0 = myMatrix[column1Row0Index]
     * myMatrix[column1Row0Index] = 10.0;
     */
    Matrix3.getElementIndex = function(column, row) {
        //>>includeStart('debug', pragmas.debug);
        if (typeof row !== 'number' || row < 0 || row > 2) {
            throw new DeveloperError('row must be 0, 1, or 2.');
        }
        if (typeof column !== 'number' || column < 0 || column > 2) {
            throw new DeveloperError('column must be 0, 1, or 2.');
        }
        //>>includeEnd('debug');

        return column * 3 + row;
    };

    /**
     * Retrieves a copy of the matrix column at the provided index as a Cartesian3 instance.
     *
     * @param {Matrix3} matrix The matrix to use.
     * @param {Number} index The zero-based index of the column to retrieve.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     *
     * @exception {DeveloperError} index must be 0, 1, or 2.
     */
    Matrix3.getColumn = function(matrix, index, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required.');
        }

        if (typeof index !== 'number' || index < 0 || index > 2) {
            throw new DeveloperError('index must be 0, 1, or 2.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        var startIndex = index * 3;
        var x = matrix[startIndex];
        var y = matrix[startIndex + 1];
        var z = matrix[startIndex + 2];

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

    /**
     * Computes a new matrix that replaces the specified column in the provided matrix with the provided Cartesian3 instance.
     *
     * @param {Matrix3} matrix The matrix to use.
     * @param {Number} index The zero-based index of the column to set.
     * @param {Cartesian3} cartesian The Cartesian whose values will be assigned to the specified column.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     *
     * @exception {DeveloperError} index must be 0, 1, or 2.
     */
    Matrix3.setColumn = function(matrix, index, cartesian, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian is required');
        }
        if (typeof index !== 'number' || index < 0 || index > 2) {
            throw new DeveloperError('index must be 0, 1, or 2.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        result = Matrix3.clone(matrix, result);
        var startIndex = index * 3;
        result[startIndex] = cartesian.x;
        result[startIndex + 1] = cartesian.y;
        result[startIndex + 2] = cartesian.z;
        return result;
    };

    /**
     * Retrieves a copy of the matrix row at the provided index as a Cartesian3 instance.
     *
     * @param {Matrix3} matrix The matrix to use.
     * @param {Number} index The zero-based index of the row to retrieve.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     *
     * @exception {DeveloperError} index must be 0, 1, or 2.
     */
    Matrix3.getRow = function(matrix, index, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required.');
        }
        if (typeof index !== 'number' || index < 0 || index > 2) {
            throw new DeveloperError('index must be 0, 1, or 2.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        var x = matrix[index];
        var y = matrix[index + 3];
        var z = matrix[index + 6];

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

    /**
     * Computes a new matrix that replaces the specified row in the provided matrix with the provided Cartesian3 instance.
     *
     * @param {Matrix3} matrix The matrix to use.
     * @param {Number} index The zero-based index of the row to set.
     * @param {Cartesian3} cartesian The Cartesian whose values will be assigned to the specified row.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     *
     * @exception {DeveloperError} index must be 0, 1, or 2.
     */
    Matrix3.setRow = function(matrix, index, cartesian, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian is required');
        }
        if (typeof index !== 'number' || index < 0 || index > 2) {
            throw new DeveloperError('index must be 0, 1, or 2.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        result = Matrix3.clone(matrix, result);
        result[index] = cartesian.x;
        result[index + 3] = cartesian.y;
        result[index + 6] = cartesian.z;
        return result;
    };

    var scratchColumn = new Cartesian3();

    /**
     * Extracts the non-uniform scale assuming the matrix is an affine transformation.
     *
     * @param {Matrix3} matrix The matrix.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Matrix3.getScale = function(matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required.');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        result.x = Cartesian3.magnitude(Cartesian3.fromElements(matrix[0], matrix[1], matrix[2], scratchColumn));
        result.y = Cartesian3.magnitude(Cartesian3.fromElements(matrix[3], matrix[4], matrix[5], scratchColumn));
        result.z = Cartesian3.magnitude(Cartesian3.fromElements(matrix[6], matrix[7], matrix[8], scratchColumn));
        return result;
    };

    var scratchScale = new Cartesian3();

    /**
     * Computes the maximum scale assuming the matrix is an affine transformation.
     * The maximum scale is the maximum length of the column vectors.
     *
     * @param {Matrix3} matrix The matrix.
     * @returns {Number} The maximum scale.
     */
    Matrix3.getMaximumScale = function(matrix) {
        Matrix3.getScale(matrix, scratchScale);
        return Cartesian3.maximumComponent(scratchScale);
    };

    /**
     * Computes the product of two matrices.
     *
     * @param {Matrix3} left The first matrix.
     * @param {Matrix3} right The second matrix.
     * @param {Matrix3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     */
    Matrix3.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 column0Row0 = left[0] * right[0] + left[3] * right[1] + left[6] * right[2];
        var column0Row1 = left[1] * right[0] + left[4] * right[1] + left[7] * right[2];
        var column0Row2 = left[2] * right[0] + left[5] * right[1] + left[8] * right[2];

        var column1Row0 = left[0] * right[3] + left[3] * right[4] + left[6] * right[5];
        var column1Row1 = left[1] * right[3] + left[4] * right[4] + left[7] * right[5];
        var column1Row2 = left[2] * right[3] + left[5] * right[4] + left[8] * right[5];

        var column2Row0 = left[0] * right[6] + left[3] * right[7] + left[6] * right[8];
        var column2Row1 = left[1] * right[6] + left[4] * right[7] + left[7] * right[8];
        var column2Row2 = left[2] * right[6] + left[5] * right[7] + left[8] * right[8];

        result[0] = column0Row0;
        result[1] = column0Row1;
        result[2] = column0Row2;
        result[3] = column1Row0;
        result[4] = column1Row1;
        result[5] = column1Row2;
        result[6] = column2Row0;
        result[7] = column2Row1;
        result[8] = column2Row2;
        return result;
    };

    /**
     * Computes the sum of two matrices.
     *
     * @param {Matrix3} left The first matrix.
     * @param {Matrix3} right The second matrix.
     * @param {Matrix3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     */
    Matrix3.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[0] = left[0] + right[0];
        result[1] = left[1] + right[1];
        result[2] = left[2] + right[2];
        result[3] = left[3] + right[3];
        result[4] = left[4] + right[4];
        result[5] = left[5] + right[5];
        result[6] = left[6] + right[6];
        result[7] = left[7] + right[7];
        result[8] = left[8] + right[8];
        return result;
    };

    /**
     * Computes the difference of two matrices.
     *
     * @param {Matrix3} left The first matrix.
     * @param {Matrix3} right The second matrix.
     * @param {Matrix3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     */
    Matrix3.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[0] = left[0] - right[0];
        result[1] = left[1] - right[1];
        result[2] = left[2] - right[2];
        result[3] = left[3] - right[3];
        result[4] = left[4] - right[4];
        result[5] = left[5] - right[5];
        result[6] = left[6] - right[6];
        result[7] = left[7] - right[7];
        result[8] = left[8] - right[8];
        return result;
    };

    /**
     * Computes the product of a matrix and a column vector.
     *
     * @param {Matrix3} matrix The matrix.
     * @param {Cartesian3} cartesian The column.
     * @param {Cartesian3} result The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter.
     */
    Matrix3.multiplyByVector = function(matrix, cartesian, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        if (!defined(cartesian)) {
            throw new DeveloperError('cartesian is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        var vX = cartesian.x;
        var vY = cartesian.y;
        var vZ = cartesian.z;

        var x = matrix[0] * vX + matrix[3] * vY + matrix[6] * vZ;
        var y = matrix[1] * vX + matrix[4] * vY + matrix[7] * vZ;
        var z = matrix[2] * vX + matrix[5] * vY + matrix[8] * vZ;

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

    /**
     * Computes the product of a matrix and a scalar.
     *
     * @param {Matrix3} matrix The matrix.
     * @param {Number} scalar The number to multiply by.
     * @param {Matrix3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     */
    Matrix3.multiplyByScalar = function(matrix, scalar, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        if (typeof scalar !== 'number') {
            throw new DeveloperError('scalar must be a number');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        result[0] = matrix[0] * scalar;
        result[1] = matrix[1] * scalar;
        result[2] = matrix[2] * scalar;
        result[3] = matrix[3] * scalar;
        result[4] = matrix[4] * scalar;
        result[5] = matrix[5] * scalar;
        result[6] = matrix[6] * scalar;
        result[7] = matrix[7] * scalar;
        result[8] = matrix[8] * scalar;
        return result;
    };

    /**
     * Creates a negated copy of the provided matrix.
     *
     * @param {Matrix3} matrix The matrix to negate.
     * @param {Matrix3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     */
    Matrix3.negate = function(matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        result[0] = -matrix[0];
        result[1] = -matrix[1];
        result[2] = -matrix[2];
        result[3] = -matrix[3];
        result[4] = -matrix[4];
        result[5] = -matrix[5];
        result[6] = -matrix[6];
        result[7] = -matrix[7];
        result[8] = -matrix[8];
        return result;
    };

    /**
     * Computes the transpose of the provided matrix.
     *
     * @param {Matrix3} matrix The matrix to transpose.
     * @param {Matrix3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     */
    Matrix3.transpose = function(matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        var column0Row0 = matrix[0];
        var column0Row1 = matrix[3];
        var column0Row2 = matrix[6];
        var column1Row0 = matrix[1];
        var column1Row1 = matrix[4];
        var column1Row2 = matrix[7];
        var column2Row0 = matrix[2];
        var column2Row1 = matrix[5];
        var column2Row2 = matrix[8];

        result[0] = column0Row0;
        result[1] = column0Row1;
        result[2] = column0Row2;
        result[3] = column1Row0;
        result[4] = column1Row1;
        result[5] = column1Row2;
        result[6] = column2Row0;
        result[7] = column2Row1;
        result[8] = column2Row2;
        return result;
    };

    function computeFrobeniusNorm(matrix) {
        var norm = 0.0;
        for (var i = 0; i < 9; ++i) {
            var temp = matrix[i];
            norm += temp * temp;
        }

        return Math.sqrt(norm);
    }

    var rowVal = [1, 0, 0];
    var colVal = [2, 2, 1];

    function offDiagonalFrobeniusNorm(matrix) {
        // Computes the "off-diagonal" Frobenius norm.
        // Assumes matrix is symmetric.

        var norm = 0.0;
        for (var i = 0; i < 3; ++i) {
            var temp = matrix[Matrix3.getElementIndex(colVal[i], rowVal[i])];
            norm += 2.0 * temp * temp;
        }

        return Math.sqrt(norm);
    }

    function shurDecomposition(matrix, result) {
        // This routine was created based upon Matrix Computations, 3rd ed., by Golub and Van Loan,
        // section 8.4.2 The 2by2 Symmetric Schur Decomposition.
        //
        // The routine takes a matrix, which is assumed to be symmetric, and
        // finds the largest off-diagonal term, and then creates
        // a matrix (result) which can be used to help reduce it

        var tolerance = CesiumMath.EPSILON15;

        var maxDiagonal = 0.0;
        var rotAxis = 1;

        // find pivot (rotAxis) based on max diagonal of matrix
        for (var i = 0; i < 3; ++i) {
            var temp = Math.abs(matrix[Matrix3.getElementIndex(colVal[i], rowVal[i])]);
            if (temp > maxDiagonal) {
                rotAxis = i;
                maxDiagonal = temp;
            }
        }

        var c = 1.0;
        var s = 0.0;

        var p = rowVal[rotAxis];
        var q = colVal[rotAxis];

        if (Math.abs(matrix[Matrix3.getElementIndex(q, p)]) > tolerance) {
            var qq = matrix[Matrix3.getElementIndex(q, q)];
            var pp = matrix[Matrix3.getElementIndex(p, p)];
            var qp = matrix[Matrix3.getElementIndex(q, p)];

            var tau = (qq - pp) / 2.0 / qp;
            var t;

            if (tau < 0.0) {
                t = -1.0 / (-tau + Math.sqrt(1.0 + tau * tau));
            } else {
                t = 1.0 / (tau + Math.sqrt(1.0 + tau * tau));
            }

            c = 1.0 / Math.sqrt(1.0 + t * t);
            s = t * c;
        }

        result = Matrix3.clone(Matrix3.IDENTITY, result);

        result[Matrix3.getElementIndex(p, p)] = result[Matrix3.getElementIndex(q, q)] = c;
        result[Matrix3.getElementIndex(q, p)] = s;
        result[Matrix3.getElementIndex(p, q)] = -s;

        return result;
    }

    var jMatrix = new Matrix3();
    var jMatrixTranspose = new Matrix3();

    /**
     * Computes the eigenvectors and eigenvalues of a symmetric matrix.
     * <p>
     * Returns a diagonal matrix and unitary matrix such that:
     * <code>matrix = unitary matrix * diagonal matrix * transpose(unitary matrix)</code>
     * </p>
     * <p>
     * The values along the diagonal of the diagonal matrix are the eigenvalues. The columns
     * of the unitary matrix are the corresponding eigenvectors.
     * </p>
     *
     * @param {Matrix3} matrix The matrix to decompose into diagonal and unitary matrix. Expected to be symmetric.
     * @param {Object} [result] An object with unitary and diagonal properties which are matrices onto which to store the result.
     * @returns {Object} An object with unitary and diagonal properties which are the unitary and diagonal matrices, respectively.
     *
     * @example
     * var a = //... symetric matrix
     * var result = {
     *     unitary : new Cesium.Matrix3(),
     *     diagonal : new Cesium.Matrix3()
     * };
     * Cesium.Matrix3.computeEigenDecomposition(a, result);
     *
     * var unitaryTranspose = Cesium.Matrix3.transpose(result.unitary, new Cesium.Matrix3());
     * var b = Cesium.Matrix3.multiply(result.unitary, result.diagonal, new Cesium.Matrix3());
     * Cesium.Matrix3.multiply(b, unitaryTranspose, b); // b is now equal to a
     *
     * var lambda = Cesium.Matrix3.getColumn(result.diagonal, 0, new Cesium.Cartesian3()).x;  // first eigenvalue
     * var v = Cesium.Matrix3.getColumn(result.unitary, 0, new Cesium.Cartesian3());          // first eigenvector
     * var c = Cesium.Cartesian3.multiplyByScalar(v, lambda, new Cesium.Cartesian3());        // equal to Cesium.Matrix3.multiplyByVector(a, v)
     */
    Matrix3.computeEigenDecomposition = function(matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required.');
        }
        //>>includeEnd('debug');

        // This routine was created based upon Matrix Computations, 3rd ed., by Golub and Van Loan,
        // section 8.4.3 The Classical Jacobi Algorithm

        var tolerance = CesiumMath.EPSILON20;
        var maxSweeps = 10;

        var count = 0;
        var sweep = 0;

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

        var unitaryMatrix = result.unitary = Matrix3.clone(Matrix3.IDENTITY, result.unitary);
        var diagMatrix = result.diagonal = Matrix3.clone(matrix, result.diagonal);

        var epsilon = tolerance * computeFrobeniusNorm(diagMatrix);

        while (sweep < maxSweeps && offDiagonalFrobeniusNorm(diagMatrix) > epsilon) {
            shurDecomposition(diagMatrix, jMatrix);
            Matrix3.transpose(jMatrix, jMatrixTranspose);
            Matrix3.multiply(diagMatrix, jMatrix, diagMatrix);
            Matrix3.multiply(jMatrixTranspose, diagMatrix, diagMatrix);
            Matrix3.multiply(unitaryMatrix, jMatrix, unitaryMatrix);

            if (++count > 2) {
                ++sweep;
                count = 0;
            }
        }

        return result;
    };

    /**
     * Computes a matrix, which contains the absolute (unsigned) values of the provided matrix's elements.
     *
     * @param {Matrix3} matrix The matrix with signed elements.
     * @param {Matrix3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     */
    Matrix3.abs = function(matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        result[0] = Math.abs(matrix[0]);
        result[1] = Math.abs(matrix[1]);
        result[2] = Math.abs(matrix[2]);
        result[3] = Math.abs(matrix[3]);
        result[4] = Math.abs(matrix[4]);
        result[5] = Math.abs(matrix[5]);
        result[6] = Math.abs(matrix[6]);
        result[7] = Math.abs(matrix[7]);
        result[8] = Math.abs(matrix[8]);

        return result;
    };

    /**
     * Computes the determinant of the provided matrix.
     *
     * @param {Matrix3} matrix The matrix to use.
     * @returns {Number} The value of the determinant of the matrix.
     */
    Matrix3.determinant = function(matrix) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        //>>includeEnd('debug');

        var m11 = matrix[0];
        var m21 = matrix[3];
        var m31 = matrix[6];
        var m12 = matrix[1];
        var m22 = matrix[4];
        var m32 = matrix[7];
        var m13 = matrix[2];
        var m23 = matrix[5];
        var m33 = matrix[8];

        return m11 * (m22 * m33 - m23 * m32) + m12 * (m23 * m31 - m21 * m33) + m13 * (m21 * m32 - m22 * m31);
    };

    /**
     * Computes the inverse of the provided matrix.
     *
     * @param {Matrix3} matrix The matrix to invert.
     * @param {Matrix3} result The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter.
     *
     * @exception {DeveloperError} matrix is not invertible.
     */
    Matrix3.inverse = function(matrix, result) {
        //>>includeStart('debug', pragmas.debug);
        if (!defined(matrix)) {
            throw new DeveloperError('matrix is required');
        }
        if (!defined(result)) {
            throw new DeveloperError('result is required,');
        }
        //>>includeEnd('debug');

        var m11 = matrix[0];
        var m21 = matrix[1];
        var m31 = matrix[2];
        var m12 = matrix[3];
        var m22 = matrix[4];
        var m32 = matrix[5];
        var m13 = matrix[6];
        var m23 = matrix[7];
        var m33 = matrix[8];

        var determinant = Matrix3.determinant(matrix);

        if (Math.abs(determinant) <= CesiumMath.EPSILON15) {
            throw new DeveloperError('matrix is not invertible');
        }

        result[0] = m22 * m33 - m23 * m32;
        result[1] = m23 * m31 - m21 * m33;
        result[2] = m21 * m32 - m22 * m31;
        result[3] = m13 * m32 - m12 * m33;
        result[4] = m11 * m33 - m13 * m31;
        result[5] = m12 * m31 - m11 * m32;
        result[6] = m12 * m23 - m13 * m22;
        result[7] = m13 * m21 - m11 * m23;
        result[8] = m11 * m22 - m12 * m21;

       var scale = 1.0 / determinant;
       return Matrix3.multiplyByScalar(result, scale, result);
    };

    /**
     * Compares the provided matrices componentwise and returns
     * <code>true</code> if they are equal, <code>false</code> otherwise.
     *
     * @param {Matrix3} [left] The first matrix.
     * @param {Matrix3} [right] The second matrix.
     * @returns {Boolean} <code>true</code> if left and right are equal, <code>false</code> otherwise.
     */
    Matrix3.equals = function(left, right) {
        return (left === right) ||
               (defined(left) &&
                defined(right) &&
                left[0] === right[0] &&
                left[1] === right[1] &&
                left[2] === right[2] &&
                left[3] === right[3] &&
                left[4] === right[4] &&
                left[5] === right[5] &&
                left[6] === right[6] &&
                left[7] === right[7] &&
                left[8] === right[8]);
    };

    /**
     * Compares the provided matrices componentwise and returns
     * <code>true</code> if they are within the provided epsilon,
     * <code>false</code> otherwise.
     *
     * @param {Matrix3} [left] The first matrix.
     * @param {Matrix3} [right] The second matrix.
     * @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.
     */
    Matrix3.equalsEpsilon = function(left, right, epsilon) {
        //>>includeStart('debug', pragmas.debug);
        if (typeof epsilon !== 'number') {
            throw new DeveloperError('epsilon must be a number');
        }
        //>>includeEnd('debug');

        return (left === right) ||
                (defined(left) &&
                defined(right) &&
                Math.abs(left[0] - right[0]) <= epsilon &&
                Math.abs(left[1] - right[1]) <= epsilon &&
                Math.abs(left[2] - right[2]) <= epsilon &&
                Math.abs(left[3] - right[3]) <= epsilon &&
                Math.abs(left[4] - right[4]) <= epsilon &&
                Math.abs(left[5] - right[5]) <= epsilon &&
                Math.abs(left[6] - right[6]) <= epsilon &&
                Math.abs(left[7] - right[7]) <= epsilon &&
                Math.abs(left[8] - right[8]) <= epsilon);
    };

    /**
     * An immutable Matrix3 instance initialized to the identity matrix.
     *
     * @type {Matrix3}
     * @constant
     */
    Matrix3.IDENTITY = freezeObject(new Matrix3(1.0, 0.0, 0.0,
                                                0.0, 1.0, 0.0,
                                                0.0, 0.0, 1.0));

    /**
     * The index into Matrix3 for column 0, row 0.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN0ROW0 = 0;

    /**
     * The index into Matrix3 for column 0, row 1.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN0ROW1 = 1;

    /**
     * The index into Matrix3 for column 0, row 2.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN0ROW2 = 2;

    /**
     * The index into Matrix3 for column 1, row 0.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN1ROW0 = 3;

    /**
     * The index into Matrix3 for column 1, row 1.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN1ROW1 = 4;

    /**
     * The index into Matrix3 for column 1, row 2.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN1ROW2 = 5;

    /**
     * The index into Matrix3 for column 2, row 0.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN2ROW0 = 6;

    /**
     * The index into Matrix3 for column 2, row 1.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN2ROW1 = 7;

    /**
     * The index into Matrix3 for column 2, row 2.
     *
     * @type {Number}
     * @constant
     */
    Matrix3.COLUMN2ROW2 = 8;

    /**
     * Duplicates the provided Matrix3 instance.
     *
     * @param {Matrix3} [result] The object onto which to store the result.
     * @returns {Matrix3} The modified result parameter or a new Matrix3 instance if one was not provided.
     */
    Matrix3.prototype.clone = function(result) {
        return Matrix3.clone(this, result);
    };

    /**
     * Compares this matrix to the provided matrix componentwise and returns
     * <code>true</code> if they are equal, <code>false</code> otherwise.
     *
     * @param {Matrix3} [right] The right hand side matrix.
     * @returns {Boolean} <code>true</code> if they are equal, <code>false</code> otherwise.
     */
    Matrix3.prototype.equals = function(right) {
        return Matrix3.equals(this, right);
    };

    /**
     * Compares this matrix to the provided matrix componentwise and returns
     * <code>true</code> if they are within the provided epsilon,
     * <code>false</code> otherwise.
     *
     * @param {Matrix3} [right] The right hand side matrix.
     * @param {Number} epsilon The epsilon to use for equality testing.
     * @returns {Boolean} <code>true</code> if they are within the provided epsilon, <code>false</code> otherwise.
     */
    Matrix3.prototype.equalsEpsilon = function(right, epsilon) {
        return Matrix3.equalsEpsilon(this, right, epsilon);
    };

    /**
     * Creates a string representing this Matrix with each row being
     * on a separate line and in the format '(column0, column1, column2)'.
     *
     * @returns {String} A string representing the provided Matrix with each row being on a separate line and in the format '(column0, column1, column2)'.
     */
    Matrix3.prototype.toString = function() {
        return '(' + this[0] + ', ' + this[3] + ', ' + this[6] + ')\n' +
               '(' + this[1] + ', ' + this[4] + ', ' + this[7] + ')\n' +
               '(' + this[2] + ', ' + this[5] + ', ' + this[8] + ')';
    };

    return Matrix3;
});