Calculating degree deltas for distances on the surface of the Earth

Here is the scenario: you've got your GPS coordinates (latitude & longitude in degrees) of your current position and you want to find the number of degrees north/south and east/west needed to contain an area of some size around you. I know, this sounds contrived, but it comes up if you use the iPhone's MapKit framework and want to zoom a MKMapView to a level where only a certain distance around a location is displayed. In that case, you need a MKCoordinateRegion to pass to the setRegion:animated: method.

One might think that if you know the rate of conversion between meters (or if you prefer, miles) and degrees, this would be a straightforward conversion. The problem is that it isn't so simple. Since the earth is a sphere, the number of meters in one degree of longitude depends on your latitude. For example, at the equator, there are 111.32 kilometers per degree of longitude; at the poles, however, there are 0 meters per degree.

A MKCoordinateRegion is comprised of two components: the coordinates of the center of the region and a span of latitudinal and longitudinal deltas. Let's assume the center is known; for example, it could be your user's current location. To calculate the span, here is a simple function that takes into account the curvature of the earth:

/*!
 * Calculate longitudinal and latitudinal deltas in
 * degrees for the given linear horizontal and vertical
 * distances in kilometers.  Longitudinal degrees per
 * kilometer vary with latitude, so a coordinate is
 * needed as a frame of reference.
 *
 * @param coord - point of reference.
 * @param xDistance - east-west distance in kilometers.
 * @param yDistance - north-south distance in kilometers.
 * @return MKCoordinateSpan representing the distances
 *         in degrees at the given coordinate.
 */
static
MKCoordinateSpan
spanForDistancesAtCoordinate(CLLocationCoordinate2D coord,
                             double xDistance,
                             double yDistance)
{
    const double kilometersPerDegree = 111.0;

    MKCoordinateSpan span;
    
    // Calculate the latitude and longitude deltas that
    // correspond to the distance (in kilometers) at
    // the given coordinate.  Note that the longitude
    // degrees calculation is complicated by virtue of
    // the fact that the number of meters per degree
    // varies depending on the coordinate's latitude.
    span.latitudeDelta = xDistance / kilometersPerDegree;
    span.longitudeDelta = yDistance / (kilometersPerDegree * cos(coord.latitude * M_PI / 180.0));
    return span;
}

The user's current location, which will become the center of the MKCoordinateRegion, should be passed as the point-of-reference coord argument. This is used to calculate the number of meters per degree at the user's current latitude.

The xDistance and yDistance parameters are the number of kilometers east-west and north-south, respectively, that define the region.

Note that the constant kilometersPerDegree represents the number of kilometers per degree of latitude or the number of kilometers per degree of longitude at the equator; it is only an estimate. Since the Earth isn't a perfect sphere, the actual number varies, but for the sake of most iPhone apps, the estimate of 111.0 kilometers/degree should be sufficient.