Tuesday, September 7, 2010

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.
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.

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