An exact method of area calculation using UTM coordinates

Question

I have a list of lat/long coordinates that I would like to use to calculate an area of a polygon. I can get exact in many cases, but the larger the polygon gets, the higher chance for error.

I am first converting the coordinates to UTM using http://www.ibm.com/developerworks/java/library/j-coordconvert/

From there, I am using http://www.mathopenref.com/coordpolygonarea2.html to calculate the area of the UTM coordinates.

private Double polygonArea(int[] x, int[] y) {      
    Double area = 0.0;
    int j = x.length-1;
    for(int i = 0; i < x.length; i++) {
        area = area + (x[j]+x[i]) * (y[j]-y[i]);
        j = i;
    }
    area = area/2;
    if (area < 0)
        area = area * -1;
    return area;
}

I compare these areas to the same coordinates I put into Microsoft SQL server and ArcGIS, but I cannot seem to match them exactly all the time. Does anyone know of a more exact method than this?

Thanks in advance.

EDIT 1

Thank you for the comments. Here is my code for getting the area (CoordinateConversion code is listed above on the IBM link):

private Map<Integer, GeoPoint> vertices;

private Double getArea() {
    List<Integer> xpoints = new ArrayList<Integer>();
    List<Integer> ypoints = new ArrayList<Integer>();
    CoordinateConversion cc = new CoordinateConversion();
    for(Entry<Integer, GeoPoint> itm : vertices.entrySet()) {
        GeoPoint pnt = itm.getValue();
        String temp = cc.latLon2MGRUTM(pnt.getLatitudeE6()/1E6, pnt.getLongitudeE6()/1E6);
        // Example return from CC: 02CNR0634657742
        String easting = temp.substring(5, 10);
        String northing = temp.substring(10, 15);
        xpoints.add(Integer.parseInt(easting));
        ypoints.add(Integer.parseInt(northing));
    }

    int[] x = toIntArray(xpoints);
    int[] y = toIntArray(ypoints); 
    return polygonArea(x,y);
}

Here is an example list of points:

44.80016800 -106.40808100
44.80016800 -106.72123800
44.75016800 -106.72123800
44.75016800 -106.80123800
44.56699100 -106.80123800

In ArcGIS and MS SQL server I get 90847.0 Acres. Using the code above I get 90817.4 Acres.

Another example list of points:

45.78412600 -108.51506700
45.78402600 -108.67972100
45.75512200 -108.67949400
45.75512200 -108.69962300
45.69795400 -108.69929400

In ArcGIS and MS SQL server I get 15732.9 Acres. Using the code above I get 15731.9 Acres.

Answer 1

It turns out that UTM just isn't able to get the extreme accuracy I was looking for. Switching projection systems to something more accurate like Albers or State Plane provided a much more accurate calculation.

Answer 2

The area formula you are using is valid only on a flat plane. As the polygon gets larger, the Earth's curvature starts to have an effect, making the area larger than what you calculate with this formula. You need to find a formula that works on a the surface of a sphere.

A simple Google search for "area of polygon on spherical surface" turns up a bunch of hits, of which the most interesting is Wolfram MathWorld Spherical Polygon