## ERROR ELLIPSES

**Description:
**

An error ellipse is a convenient way of expressing the uncertainty of the position of a point in a graphical format. Absolute error ellipses give a measure of the uncertainty of a point relative to the position of the fixed points in a network and relative error ellipses show the uncertainty of one defined point with respect to another defined point in the network.

**Absolute error ellipses:**

An absolute error ellipse is a figure that describes the uncertainty of the computed position of a point. If the eastings and northings of points are successive elements in an x vector, then their variances will appear as successive elements on the leading diagonal of the variance-covariance matrix of the parameters. The square roots of these variances give the standard errors of the individual northings and eastings of the points. For descriptive purposes it might be imagined that a rectangular box could be drawn about the computed point, with sides of length of 2σE in the east–west direction and 2σN in the north–south direction and the centre of the box at the point. Such a box, it might be supposed, would describe the error in the computed coordinates of the point. Attractive as such a simple description may be, it is inadequate on two counts.

Firstly, an ellipse better describes a bivariate distribution and secondly, there is no reason why that Absolute error ellipses. An absolute error ellipse is a figure that describes the uncertainty of the computed position of a point. If the eastings and northings of points are successive elements in an x vector, then their variances will appear as successive elements on the leading diagonal of the variance-covariance matrix of the parameters. The square roots of these variances give the standard errors of the individual northings and eastings of the points. For descriptive purposes it might be imagined that a rectangular box could be drawn about the computed point, with sides of length of 2σE in the east–west direction and 2σN in the north–south direction and the centre of the box at the point. Such a box, it might be supposed, would describe the error in the computed coordinates of the point. Attractive as such a simple description may be, it is inadequate on two counts. Firstly, an ellipse better describes a bivariate distribution and secondly, there is no reason why that

From the Gauss propagation of error law, the variance-covariance matrix of the coordinates with respect to the m and n axes is related to the variance-covariance matrix of the coordinates with respect to the E and N axes by:

When multiplied out this gives:

To find the direction of the major axis, find the maximum value of σ2 n as a changes. This will be when a’s rate of change is zero.

**Coordinate system rotation**