Without using Bearings
For this method we use two times two GPS coordinates of points that form two straight lines with the distant point for which we want to calculate the exact position. This method is not the most simple, but it is very precise, as we do not use bearings (very difficult to get them with high precision) and we do not need to estimate the distance to the remote point. If you have a cell phone that allows you to read the GPS coordinates of your actual position, you can use this method.
Straight lines
A straight line is fully determined by two points on the line. The general equation of a straight line is (y-y1)/(x-x1)=(y2-y1)/(x2-x1). We can also write the equation for a straight line as A * x + B * y + C = 0. When we know two points on the line, we can calculate A, B and C and if we know A, B and C for two different straight lines, we can calculate the intersection of these two lines.
The actual situation
Point P is a steeple. We want to know its GPS coordinates, but can not or are not allowed to go to this spot. M2 and M4 are two other steeples (could be any points, visible over a longer distance) that we can visit, so we can determine their coordinates. On the small road to the SW of M2 we choose a position that brings P - M2 - ourselves in one straight line and we note the coordinates of our position (M1). Now we go to M2, along any road, and note the GPS coordinates of this point.
On the unpaved road near M3 we choose such a position that P - M4 - ourselves are on a straight line and note the coordinates (M3). Now we go to M4 and determine the position of this visible object. The exact positions of these four points allow us to calculate the position of point P.
Map extract copyright © National Geographic Institute at www.ngi.be
UTM coordinates
If your cell phone, or whichever device you use to determine the GPS coordinates of the four points, can only show these values in Lat/Lon, you will have to translate them to UTM coordinates with Eastings and Northings.
Here are the simple Excel VBA functions that calculate the Easting and Northing of the unknown point P, with the Eastings and Northings of the four points as input.
Test of Excel functions
We have tested our Excel functions with round coordinates and the result shows that all goes well (bold numbers are calculated).
The End Result
The next table shows the GPS coordinates of points M1 - M4, as well as the Easting and Northing of Point P (in bold). "Dist(m)" indicates the distance of the four points to point P in meters (calculated). Right to the bold results are the real values for Point P. We also see that the error in the calculated coordinates is 37 m for the Easting and 15 m for the Northing. We have measured the points M1 - M4 on an electronic map on our PC, where one pixel equaled 6 m.
Accuracy
It must be possible to extend the reach of this method to much greater distances, as long as you respect the straight line positions. And two straight lines perpendicular to each other will give the best result.
Overview
On the pocketPC the five points looked like in the shot below.
