Mathematical Definition Of Error
Position error
- If (X,Y,Z) is the true value of a point and (XA, YA, ZA) the approximate value.
- Use the Euclidean metric E2 = [(X- XA)2 + (Y- YA)2 + (Z- ZA)2] to determine an error ball of radius E. For two dimensional systems, set the Zs to 0.
Angular error
- There two types of geodetic points: (lat, lon, h) or for the map projections (lat, lon, 0).
- Except for UTM, the forward transformations are exact.
General approach
- Generate a known set of points {(lat, lon, h)}.
- When the exact transformation is available, generate the corresponding exact set of points {(X,Y,Z)}.
- E in terms of position errors can always be calculated in two or three dimensions.
UTM is a special case because there is no exact transformation in either direction
- Angular measures can be converted to distance measures using s = r•ø.
- Again, start with a known set of exact points {(lat, lon)}.
- Given the approximate point (latA, lonA) compute e2 = [(lat - latA)RM]2 + [( lon - lonA)RN]2
- Where RN is the radius of curvature in the prime vertical and RM is the radius of curvature in the meridian.
- e is the (approximate) radius of the positional error ball.
- When the angular errors are small, the error measure e is nearly E.