MercatorDistance.cs

// Calcluate airline distance for two mercator points.
// This approximation formula is sufficiently accurate for 
// our needs for distances of up to 600 km and 
// 80° latitude (the error is never more than 5% even for extreme values). 
using System;

public class Program
{
    public static void Main()
    {
        // our latitude/longitude values for test
        double lat1 = 49.0;
        double lon1 = 8.0;
        double lat2 = 50.0;
        double lon2 = 9.0;

        // convert to PTV Mercator, so assume we have Mercator as input
        double x1, y1, x2, y2;
        LatLonToSphereMercator(lat1, lon1, out x1, out y1);
        LatLonToSphereMercator(lat2, lon2, out x2, out y2);

        // calclate mercator distance with Pythagoras
        double mercDist = Math.Sqrt(Math.Pow(x2 - x1, 2) + Math.Pow(y2 - y1, 2));

        // the center of the y values to apply the correction factor
        double centerY = (y1 + y2) / 2;

        // calculate the corresponding latitude
        double centerLat = LatFromMercator(centerY);

        // the latitude determines our correction factor
        double correctionFactor = Math.Cos(centerLat * Math.PI / 180.0);

        // apply the correction factor to real distance
        double realDist = mercDist * correctionFactor;

        Console.WriteLine("Distance Between {0}/{1} and {2}/{3}", x1, y1, x2, y2);
        Console.WriteLine("Pythagoras: {0}", mercDist);
        Console.WriteLine("Correction Factor: {0}", correctionFactor);
        Console.WriteLine("Real: {0}", realDist);
    }

    // Project a geographic coordinate on the (spherical) Mercator map. 
    // Using the mean between major and minor axis here ("PTV standard"). 
    // You could also use the major axis 6378137 ("Google standard").  
    public static void LatLonToSphereMercator(
        double latitude, double longitude, out double x, out double y)
    {
        x = 6371000.0 * longitude * Math.PI / 180;
        y = 6371000.0 * Math.Log(Math.Tan(Math.PI / 4 + latitude * Math.PI / 360));
    }

    // Caclculate the latitude for a mercator Y
    public static double LatFromMercator(double y)
    {
        return (360 / Math.PI) * (Math.Atan(Math.Exp(y / 6371000.0)) - (Math.PI / 4));
    }
}