How to convert a bing maps quadkey to mercator coordinates

Question

How do you convert a bing maps quad key, such as "03200320023" to coordinates in mercator virtual earth (epsg 3857) coordinate system?

I've got an algorithm to convert to coordinates in latitude longitude (e.g. the quad key above comes to "-84.19921875,33.7243385314941,-84.0234375,33.8704147338867"). I could then convert from latitude longitude to mercator coordinate, but that smells far too expensive in processing power.

The mercator virtual earth result I'm looking for given the above quadkey is: "-9373014,4011415,-9353446,4030983"

Also a question on stackoverflow

Answer 1

The code below comes close. Maybe this can get you started. Update: notice the amount of error equals the last offset size.

quadkey: 03200320023
0 offset: 10018754.1713946 lod 0
3 offset: 5009377.08569731 lod 1
2 offset: 2504688.54284866 lod 2
0 offset: 1252344.27142433 lod 3
0 offset: 626172.135712164 lod 4
3 offset: 313086.067856082 lod 5
2 offset: 156543.033928041 lod 6
0 offset: 78271.5169640205 lod 7
0 offset: 39135.7584820102 lod 8
2 offset: 19567.8792410051 lod 9
3 offset: 9783.93962050256 lod 10
Mercator: x -9363230.21682095, y 4001631.30478555
error x 9783.78317905217 y-9783.69521445315

Here's the code:

private void BingTest()
{
    const double EarthRadius = 6378137;

    string qdkey = "03200320023";
    double x = 0;
    double y = 0;
    double offset = EarthRadius * Math.PI / 2.0;
    Debug.Print("quadkey: {0}", qdkey);
    for (int i = 0; i < qdkey.Length; i++)
    {
        string s = qdkey.Substring(i, 1);
        Debug.Print("{0} offset: {1} lod {2}",s, offset,i);
        switch (s)
        {
            case "0":  x -= offset; y += offset; break;
            case "1":  x += offset; y += offset; break;
            case "2":  x -= offset; y -= offset; break;
            case "3":  x += offset; y -= offset; break;
            default: throw new Exception("bad quadkey");
        }
        offset *= 0.5;
    }

    // compare with expected result
    double x0 = -9373014;
    double y0 = 4011415;
    Debug.Print("Mercator: x {0}, y {1}", x, y);
    Debug.Print("error x {0} y{1}", x - x0, y - y0);            
}