The (east, north) coordinates, in the electronic total station (ETS) frame of reference, of a target point may differ from the relative coordinates in the UTM (Universal Transverse Mercator) frame of reference. True north is difficult to determine to an accuracy of better than a couple of degrees, and, true north is not necessarily the same as north on our part of the UTM grid.

In the figure above, let X and Y represent the ETS frame of reference (X = east, Y = north), while X’ and Y’ represent the map (UTM) frame of reference (map east and north respectively). The ETS measures a coordinate (x,y) in the (X,Y) frame of reference. How do we convert this (x,y) into its correct (x’,y’) coordinate in the map (X’,Y’) frame of reference?

Let θ = angle (measured counterclockwise) of the rotation needed for axis X to be in the same position as axis X’. It can be shown that coordinates (x’,y’) are found by calculating

The coefficients needed to rotate coordinates from (X,Y) to (X’,Y’) can be expressed as the rotation matrix A

Let P(x,y) = the matrix consisting of pairs of coordinates (east and north respectively) from an EDM (x,y,z) survey.  The P' matrix is the product of P*A (multiplying matrix P with matrix A).  This can be done using the array function mmult in Excel.  The rule for matrix multiplication is, the number of columns in the first matrix equals the number of rows in the second matrix.  The result of the multiplication yields a matrix with the same number of rows as the first matrix and the same number of columns in the second matrix.

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