HermiteSpline.js

/*global define*/
define([
        './Cartesian3',
        './Cartesian4',
        './defaultValue',
        './defined',
        './defineProperties',
        './DeveloperError',
        './LinearSpline',
        './Matrix4',
        './Spline',
        './TridiagonalSystemSolver'
    ], function(
        Cartesian3,
        Cartesian4,
        defaultValue,
        defined,
        defineProperties,
        DeveloperError,
        LinearSpline,
        Matrix4,
        Spline,
        TridiagonalSystemSolver) {
    'use strict';

    var scratchLower = [];
    var scratchDiagonal = [];
    var scratchUpper = [];
    var scratchRight = [];

    function generateClamped(points, firstTangent, lastTangent) {
        var l = scratchLower;
        var u = scratchUpper;
        var d = scratchDiagonal;
        var r = scratchRight;

        l.length = u.length = points.length - 1;
        d.length = r.length = points.length;

        var i;
        l[0] = d[0] = 1.0;
        u[0] = 0.0;

        var right = r[0];
        if (!defined(right)) {
            right = r[0] = new Cartesian3();
        }
        Cartesian3.clone(firstTangent, right);

        for (i = 1; i < l.length - 1; ++i) {
            l[i] = u[i] = 1.0;
            d[i] = 4.0;

            right = r[i];
            if (!defined(right)) {
                right = r[i] = new Cartesian3();
            }
            Cartesian3.subtract(points[i + 1], points[i - 1], right);
            Cartesian3.multiplyByScalar(right, 3.0, right);
        }

        l[i] = 0.0;
        u[i] = 1.0;
        d[i] = 4.0;

        right = r[i];
        if (!defined(right)) {
            right = r[i] = new Cartesian3();
        }
        Cartesian3.subtract(points[i + 1], points[i - 1], right);
        Cartesian3.multiplyByScalar(right, 3.0, right);

        d[i + 1] = 1.0;
        right = r[i + 1];
        if (!defined(right)) {
            right = r[i + 1] = new Cartesian3();
        }
        Cartesian3.clone(lastTangent, right);

        return TridiagonalSystemSolver.solve(l, d, u, r);
    }

    function generateNatural(points){
        var l = scratchLower;
        var u = scratchUpper;
        var d = scratchDiagonal;
        var r = scratchRight;

        l.length = u.length = points.length - 1;
        d.length = r.length = points.length;

        var i;
        l[0] = u[0] = 1.0;
        d[0] = 2.0;

        var right = r[0];
        if (!defined(right)) {
            right = r[0] = new Cartesian3();
        }
        Cartesian3.subtract(points[1], points[0], right);
        Cartesian3.multiplyByScalar(right, 3.0, right);

        for (i = 1; i < l.length; ++i) {
            l[i] = u[i] = 1.0;
            d[i] = 4.0;

            right = r[i];
            if (!defined(right)) {
                right = r[i] = new Cartesian3();
            }
            Cartesian3.subtract(points[i + 1], points[i - 1], right);
            Cartesian3.multiplyByScalar(right, 3.0, right);
        }

        d[i] = 2.0;

        right = r[i];
        if (!defined(right)) {
            right = r[i] = new Cartesian3();
        }
        Cartesian3.subtract(points[i], points[i - 1], right);
        Cartesian3.multiplyByScalar(right, 3.0, right);

        return TridiagonalSystemSolver.solve(l, d, u, r);
    }

    /**
     * A Hermite spline is a cubic interpolating spline. Points, incoming tangents, outgoing tangents, and times
     * must be defined for each control point. The outgoing tangents are defined for points [0, n - 2] and the incoming
     * tangents are defined for points [1, n - 1]. For example, when interpolating a segment of the curve between <code>points[i]</code> and
     * <code>points[i + 1]</code>, the tangents at the points will be <code>outTangents[i]</code> and <code>inTangents[i]</code>,
     * respectively.
     *
     * @alias HermiteSpline
     * @constructor
     *
     * @param {Object} options Object with the following properties:
     * @param {Number[]} options.times An array of strictly increasing, unit-less, floating-point times at each point.
     *                The values are in no way connected to the clock time. They are the parameterization for the curve.
     * @param {Cartesian3[]} options.points The array of {@link Cartesian3} control points.
     * @param {Cartesian3[]} options.inTangents The array of {@link Cartesian3} incoming tangents at each control point.
     * @param {Cartesian3[]} options.outTangents The array of {@link Cartesian3} outgoing tangents at each control point.
     *
     * @exception {DeveloperError} points.length must be greater than or equal to 2.
     * @exception {DeveloperError} times.length must be equal to points.length.
     * @exception {DeveloperError} inTangents and outTangents must have a length equal to points.length - 1.
     *
     *
     * @example
     * // Create a G<sup>1</sup> continuous Hermite spline
     * var times = [ 0.0, 1.5, 3.0, 4.5, 6.0 ];
     * var spline = new Cesium.HermiteSpline({
     *     times : times,
     *     points : [
     *         new Cesium.Cartesian3(1235398.0, -4810983.0, 4146266.0),
     *         new Cesium.Cartesian3(1372574.0, -5345182.0, 4606657.0),
     *         new Cesium.Cartesian3(-757983.0, -5542796.0, 4514323.0),
     *         new Cesium.Cartesian3(-2821260.0, -5248423.0, 4021290.0),
     *         new Cesium.Cartesian3(-2539788.0, -4724797.0, 3620093.0)
     *     ],
     *     outTangents : [
     *         new Cesium.Cartesian3(1125196, -161816, 270551),
     *         new Cesium.Cartesian3(-996690.5, -365906.5, 184028.5),
     *         new Cesium.Cartesian3(-2096917, 48379.5, -292683.5),
     *         new Cesium.Cartesian3(-890902.5, 408999.5, -447115)
     *     ],
     *     inTangents : [
     *         new Cesium.Cartesian3(-1993381, -731813, 368057),
     *         new Cesium.Cartesian3(-4193834, 96759, -585367),
     *         new Cesium.Cartesian3(-1781805, 817999, -894230),
     *         new Cesium.Cartesian3(1165345, 112641, 47281)
     *     ]
     * });
     *
     * var p0 = spline.evaluate(times[0]);
     * 
     * @see CatmullRomSpline
     * @see LinearSpline
     * @see QuaternionSpline
     */
    function HermiteSpline(options) {
        options = defaultValue(options, defaultValue.EMPTY_OBJECT);

        var points = options.points;
        var times = options.times;
        var inTangents = options.inTangents;
        var outTangents = options.outTangents;

        //>>includeStart('debug', pragmas.debug);
        if (!defined(points) || !defined(times) || !defined(inTangents) || !defined(outTangents)) {
            throw new DeveloperError('times, points, inTangents, and outTangents are required.');
        }
        if (points.length < 2) {
            throw new DeveloperError('points.length must be greater than or equal to 2.');
        }
        if (times.length !== points.length) {
            throw new DeveloperError('times.length must be equal to points.length.');
        }
        if (inTangents.length !== outTangents.length || inTangents.length !== points.length - 1) {
            throw new DeveloperError('inTangents and outTangents must have a length equal to points.length - 1.');
        }
        //>>includeEnd('debug');

        this._times = times;
        this._points = points;
        this._inTangents = inTangents;
        this._outTangents = outTangents;

        this._lastTimeIndex = 0;
    }

    defineProperties(HermiteSpline.prototype, {
        /**
         * An array of times for the control points.
         *
         * @memberof HermiteSpline.prototype
         *
         * @type {Number[]}
         * @readonly
         */
        times : {
            get : function() {
                return this._times;
            }
        },

        /**
         * An array of {@link Cartesian3} control points.
         *
         * @memberof HermiteSpline.prototype
         *
         * @type {Cartesian3[]}
         * @readonly
         */
        points : {
            get : function() {
                return this._points;
            }
        },

        /**
         * An array of {@link Cartesian3} incoming tangents at each control point.
         *
         * @memberof HermiteSpline.prototype
         *
         * @type {Cartesian3[]}
         * @readonly
         */
        inTangents : {
            get : function() {
                return this._inTangents;
            }
        },

        /**
         * An array of {@link Cartesian3} outgoing tangents at each control point.
         *
         * @memberof HermiteSpline.prototype
         *
         * @type {Cartesian3[]}
         * @readonly
         */
        outTangents : {
            get : function() {
                return this._outTangents;
            }
        }
    });

    /**
     * Creates a spline where the tangents at each control point are the same.
     * The curves are guaranteed to be at least in the class C<sup>1</sup>.
     *
     * @param {Object} options Object with the following properties:
     * @param {Number[]} options.times The array of control point times.
     * @param {Cartesian3[]} options.points The array of control points.
     * @param {Cartesian3[]} options.tangents The array of tangents at the control points.
     * @returns {HermiteSpline} A hermite spline.
     *
     * @exception {DeveloperError} points, times and tangents are required.
     * @exception {DeveloperError} points.length must be greater than or equal to 2.
     * @exception {DeveloperError} times, points and tangents must have the same length.
     *
     * @example
     * var points = [
     *     new Cesium.Cartesian3(1235398.0, -4810983.0, 4146266.0),
     *     new Cesium.Cartesian3(1372574.0, -5345182.0, 4606657.0),
     *     new Cesium.Cartesian3(-757983.0, -5542796.0, 4514323.0),
     *     new Cesium.Cartesian3(-2821260.0, -5248423.0, 4021290.0),
     *     new Cesium.Cartesian3(-2539788.0, -4724797.0, 3620093.0)
     * ];
     *
     * // Add tangents
     * var tangents = new Array(points.length);
     * tangents[0] = new Cesium.Cartesian3(1125196, -161816, 270551);
     * var temp = new Cesium.Cartesian3();
     * for (var i = 1; i < tangents.length - 1; ++i) {
     *     tangents[i] = Cesium.Cartesian3.multiplyByScalar(Cesium.Cartesian3.subtract(points[i + 1], points[i - 1], temp), 0.5, new Cesium.Cartesian3());
     * }
     * tangents[tangents.length - 1] = new Cesium.Cartesian3(1165345, 112641, 47281);
     *
     * var spline = Cesium.HermiteSpline.createC1({
     *     times : times,
     *     points : points,
     *     tangents : tangents
     * });
     */
    HermiteSpline.createC1 = function(options) {
        options = defaultValue(options, defaultValue.EMPTY_OBJECT);

        var times = options.times;
        var points = options.points;
        var tangents = options.tangents;

        //>>includeStart('debug', pragmas.debug);
        if (!defined(points) || !defined(times) || !defined(tangents)) {
            throw new DeveloperError('points, times and tangents are required.');
        }
        if (points.length < 2) {
            throw new DeveloperError('points.length must be greater than or equal to 2.');
        }
        if (times.length !== points.length || times.length !== tangents.length) {
            throw new DeveloperError('times, points and tangents must have the same length.');
        }
        //>>includeEnd('debug');

        var outTangents = tangents.slice(0, tangents.length - 1);
        var inTangents = tangents.slice(1, tangents.length);

        return new HermiteSpline({
            times : times,
            points : points,
            inTangents : inTangents,
            outTangents : outTangents
        });
    };

    /**
     * Creates a natural cubic spline. The tangents at the control points are generated
     * to create a curve in the class C<sup>2</sup>.
     *
     * @param {Object} options Object with the following properties:
     * @param {Number[]} options.times The array of control point times.
     * @param {Cartesian3[]} options.points The array of control points.
     * @returns {HermiteSpline|LinearSpline} A hermite spline or a linear spline if less than 3 control points were given.
     *
     * @exception {DeveloperError} points and times are required.
     * @exception {DeveloperError} points.length must be greater than or equal to 2.
     * @exception {DeveloperError} times.length must be equal to points.length.
     *
     * @example
     * // Create a natural cubic spline above the earth from Philadelphia to Los Angeles.
     * var spline = Cesium.HermiteSpline.createNaturalCubic({
     *     times : [ 0.0, 1.5, 3.0, 4.5, 6.0 ],
     *     points : [
     *         new Cesium.Cartesian3(1235398.0, -4810983.0, 4146266.0),
     *         new Cesium.Cartesian3(1372574.0, -5345182.0, 4606657.0),
     *         new Cesium.Cartesian3(-757983.0, -5542796.0, 4514323.0),
     *         new Cesium.Cartesian3(-2821260.0, -5248423.0, 4021290.0),
     *         new Cesium.Cartesian3(-2539788.0, -4724797.0, 3620093.0)
     *     ]
     * });
     */
    HermiteSpline.createNaturalCubic = function(options) {
        options = defaultValue(options, defaultValue.EMPTY_OBJECT);

        var times = options.times;
        var points = options.points;

        //>>includeStart('debug', pragmas.debug);
        if (!defined(points) || !defined(times)) {
            throw new DeveloperError('points and times are required.');
        }
        if (points.length < 2) {
            throw new DeveloperError('points.length must be greater than or equal to 2.');
        }
        if (times.length !== points.length) {
            throw new DeveloperError('times.length must be equal to points.length.');
        }
        //>>includeEnd('debug');

        if (points.length < 3) {
            return new LinearSpline({
                points : points,
                times : times
            });
        }

        var tangents = generateNatural(points);
        var outTangents = tangents.slice(0, tangents.length - 1);
        var inTangents = tangents.slice(1, tangents.length);

        return new HermiteSpline({
            times : times,
            points : points,
            inTangents : inTangents,
            outTangents : outTangents
        });
    };

    /**
     * Creates a clamped cubic spline. The tangents at the interior control points are generated
     * to create a curve in the class C<sup>2</sup>.
     *
     * @param {Object} options Object with the following properties:
     * @param {Number[]} options.times The array of control point times.
     * @param {Cartesian3[]} options.points The array of control points.
     * @param {Cartesian3} options.firstTangent The outgoing tangent of the first control point.
     * @param {Cartesian3} options.lastTangent The incoming tangent of the last control point.
     * @returns {HermiteSpline|LinearSpline} A hermite spline or a linear spline if less than 3 control points were given.
     *
     * @exception {DeveloperError} points, times, firstTangent and lastTangent are required.
     * @exception {DeveloperError} points.length must be greater than or equal to 2.
     * @exception {DeveloperError} times.length must be equal to points.length.
     *
     * @example
     * // Create a clamped cubic spline above the earth from Philadelphia to Los Angeles.
     * var spline = Cesium.HermiteSpline.createClampedCubic({
     *     times : [ 0.0, 1.5, 3.0, 4.5, 6.0 ],
     *     points : [
     *         new Cesium.Cartesian3(1235398.0, -4810983.0, 4146266.0),
     *         new Cesium.Cartesian3(1372574.0, -5345182.0, 4606657.0),
     *         new Cesium.Cartesian3(-757983.0, -5542796.0, 4514323.0),
     *         new Cesium.Cartesian3(-2821260.0, -5248423.0, 4021290.0),
     *         new Cesium.Cartesian3(-2539788.0, -4724797.0, 3620093.0)
     *     ],
     *     firstTangent : new Cesium.Cartesian3(1125196, -161816, 270551),
     *     lastTangent : new Cesium.Cartesian3(1165345, 112641, 47281)
     * });
     */
    HermiteSpline.createClampedCubic = function(options) {
        options = defaultValue(options, defaultValue.EMPTY_OBJECT);

        var times = options.times;
        var points = options.points;
        var firstTangent = options.firstTangent;
        var lastTangent = options.lastTangent;

        //>>includeStart('debug', pragmas.debug);
        if (!defined(points) || !defined(times) || !defined(firstTangent) || !defined(lastTangent)) {
            throw new DeveloperError('points, times, firstTangent and lastTangent are required.');
        }
        if (points.length < 2) {
            throw new DeveloperError('points.length must be greater than or equal to 2.');
        }
        if (times.length !== points.length) {
            throw new DeveloperError('times.length must be equal to points.length.');
        }
        //>>includeEnd('debug');

        if (points.length < 3) {
            return new LinearSpline({
                points : points,
                times : times
            });
        }

        var tangents = generateClamped(points, firstTangent, lastTangent);
        var outTangents = tangents.slice(0, tangents.length - 1);
        var inTangents = tangents.slice(1, tangents.length);

        return new HermiteSpline({
            times : times,
            points : points,
            inTangents : inTangents,
            outTangents : outTangents
        });
    };

    HermiteSpline.hermiteCoefficientMatrix = new Matrix4(
             2.0, -3.0,  0.0,  1.0,
            -2.0,  3.0,  0.0,  0.0,
             1.0, -2.0,  1.0,  0.0,
             1.0, -1.0,  0.0,  0.0);

    /**
     * Finds an index <code>i</code> in <code>times</code> such that the parameter
     * <code>time</code> is in the interval <code>[times[i], times[i + 1]]</code>.
     * @function
     *
     * @param {Number} time The time.
     * @returns {Number} The index for the element at the start of the interval.
     *
     * @exception {DeveloperError} time must be in the range <code>[t<sub>0</sub>, t<sub>n</sub>]</code>, where <code>t<sub>0</sub></code>
     *                             is the first element in the array <code>times</code> and <code>t<sub>n</sub></code> is the last element
     *                             in the array <code>times</code>.
     */
    HermiteSpline.prototype.findTimeInterval = Spline.prototype.findTimeInterval;

    var scratchTimeVec = new Cartesian4();
    var scratchTemp = new Cartesian3();

    /**
     * Evaluates the curve at a given time.
     *
     * @param {Number} time The time at which to evaluate the curve.
     * @param {Cartesian3} [result] The object onto which to store the result.
     * @returns {Cartesian3} The modified result parameter or a new instance of the point on the curve at the given time.
     *
     * @exception {DeveloperError} time must be in the range <code>[t<sub>0</sub>, t<sub>n</sub>]</code>, where <code>t<sub>0</sub></code>
     *                             is the first element in the array <code>times</code> and <code>t<sub>n</sub></code> is the last element
     *                             in the array <code>times</code>.
     */
    HermiteSpline.prototype.evaluate = function(time, result) {
        if (!defined(result)) {
            result = new Cartesian3();
        }
        var points = this.points;
        var times = this.times;
        var inTangents = this.inTangents;
        var outTangents = this.outTangents;

        var i = this._lastTimeIndex = this.findTimeInterval(time, this._lastTimeIndex);
        var u = (time - times[i]) / (times[i + 1] - times[i]);

        var timeVec = scratchTimeVec;
        timeVec.z = u;
        timeVec.y = u * u;
        timeVec.x = timeVec.y * u;
        timeVec.w = 1.0;

        var coefs = Matrix4.multiplyByVector(HermiteSpline.hermiteCoefficientMatrix, timeVec, timeVec);

        result = Cartesian3.multiplyByScalar(points[i], coefs.x, result);
        Cartesian3.multiplyByScalar(points[i + 1], coefs.y, scratchTemp);
        Cartesian3.add(result, scratchTemp, result);
        Cartesian3.multiplyByScalar(outTangents[i], coefs.z, scratchTemp);
        Cartesian3.add(result, scratchTemp, result);
        Cartesian3.multiplyByScalar(inTangents[i], coefs.w, scratchTemp);
        return Cartesian3.add(result, scratchTemp, result);
    };

    return HermiteSpline;
});