Getting area from GPS coordinates

Question

So, I've got a method that returns the area of a shape defined by its points (given in CCW or CW order, it doesn't really matter). I tested it with some shapes and it seems to be working.

The problem is, I want to use this method with GPS coordinates, and I want to return the result in m² or km², but that's definitly not what happens. In fact, I don't even know in which unit the result is when I use this method with that kind of coordinates.

So the question is, how to convert the result I have into m² or km² ? I tried some things, but either it does not work or it's inaccurate.

Here's my method, if you want to check :

public static double getArea(List<Vector2D> points) {

    double firstSum = 0, secondSum = 0;

    for (int i = 0 ; i < points.size()-1 ; i++) {
        firstSum += points.get(i).x * points.get(i+1).y;
        secondSum += points.get(i).y * points.get(i+1).x;
    }
    firstSum += points.get( points.size()-1 ).x * points.get(0).y;
    secondSum += points.get( points.size()-1 ).y * points.get(0).x;

    return Math.abs((firstSum-secondSum)/2);

}

( Vector2D is the class I use for points, with x as the latitude and y as the longitude)

Answer 1

The problem is that you're not taking the (approximately) spherical nature of the Earth into account. For a start, you need to take into account the radius of the Earth - you could have the same list of latitude and longitude on a smaller (or bigger) planet, but the radius would be different, and consequently the area would be different too.

You can use the approach in this question . It's trivial to convert that to Java:

public static double CalculatePolygonArea(List<Vector2D> coordinates)
{
    double area = 0;

    if (coordinates.size() > 2)
    {
        for (int i = 0; i < coordinates.size()-1; i++)
        {
            Vector2D p1, p2;
            p1 = coordinates.get(i);
            p2 = coordinates.get(i + 1);
            area += Math.toRadians(p2.x - p1.x) * (2 + Math.sin(Math.toRadians(p1.y))
               + Math.sin(Math.toRadians(p2.y)));

        }
        area = area * R * R / 2;
    }

    return Math.abs(area);
}

(assuming Vector2D.x is the longitude and Vector2D.y is the latitude).

R is the radius of the Earth. Use a value in the unit you want the area result to be in (eg 6_371_000 metres for square metres, 6_371 km for square km, 3_959 miles for square miles...)