Some formulas to computes coordinates, distance and bearing on a rhumb line.

rhumbline.js

/* jshint indent: 4 */

/* General functions */
function modulo180(val) {
    if (Math.abs(val) > 180) {
        val = -(val % 180);
    }
    return val;
}

function inc_lat(lat) {
    var result = Math.log(Math.tan(toRad(45 + lat / 2)));
    return toDeg(result);
}

function toDeg(rad) {
    return 180 * rad / Math.PI;
}

function toRad(deg) {
    return Math.PI * deg / 180;
}

/**
 * Returns the distance from two points, in M, travelling along a rhumb line
 *
 * Exact formula (longitude W negative)
 *
 * @param   {lat, lon} point: Latitude/longitude of departure point
 *          {lat, lon} point: Latitude/longitude of arrival point
 * @returns {Number} Distance in M between this point and destination point
 *          {Number} Bearing of the segment
 */
function RhumbLine(pointA, pointB) {
    // Differences latLon
    var dlat = pointB.lat - pointA.lat,
        dlon = modulo180(parseFloat(-pointB.lon) + parseFloat(pointA.lon));

    var bearing = 0,
        distance = 0;

    // Computes the bearing
    // Increasing latitudes
    var inc_latA = inc_lat(pointA.lat),
        inc_latB = inc_lat(pointB.lat);
    // Special cases
    if (dlat === 0) {
        // East-West track
        bearing = 90;
    } else if (dlon === 0) {
        // North-South track
        bearing = 0;
    } else {
        // Default case
        bearing = toDeg(Math.atan(dlon / (inc_latA - inc_latB)));
    }

    // Applying correction depending of relative position of A vs. B
    if ((dlon > 0) and (bearing <0) {
        // Case North West
        bearing += 360;
    } else {
        bearing += 180;
    }

    // Computes the distance
    // No horizontal track
    if (dlat !== 0) {
        distance = Math.abs(dlat / Math.cos(toRad(bearing)));
    } else {
        // Horizontal track
        distance = Math.abs(dlon * Math.cos(toRad(pointA.lat)));
    }
    distance *= 60;

    // Return
    return {
        bearing: bearing,
        distance: distance
    };
}

/**
 * Returns the coordinates of the arrival point, in M, travelling along a rhumb line.
 *
 * Exact formula (longitude W negative)
 *
 * @param   {distance} Distance of the track
 *          {bearing} Bearing of the track
 *          {bearing} point: Latitude/longitude of departure point
 * @returns {lat, lon} point: Latitude/longitude of arrival point
 */
function ArrivalPoint(bearing, distance, start) {
    // Computing the arrival latitude
    var point = {};
    var dlat = distance * Math.cos(toRad(bearing)) / 60;
    point.lat = start.lat + dlat;

    // Computing the arrival longitude
    var dlon = 0;
    if (dlat !== 0) {
        // Increasing latitudes
        var inc_latA = inc_lat(start.lat),
            inc_latB = inc_lat(point.lat);
        dlon = (inc_latA - inc_latB) * Math.tan(toRad(bearing));
    } else {
        // East-West track
        dlon = distance / Math.cos(toRad(start.lat));
    }
    // Special case: South departure point
    if (start.lat < 0) {
        dlon *= -1;
    }

    point.lon = start.lon - dlon;
    return point;
}

Changed correction of bearing thanks to @navugo remarks.