Trilateration is an alternative to triangulation that relies upon distance measurements only. Electronic distance measurement technologies make trilateration a cost-effective positioning technique for control surveys. Not only is it used by land surveyors, trilateration is also used to determine location coordinates with Global Positioning System satellites and receivers.
Trilateration networks commence the same way as triangulation nets. If only one existing control point is available, a second point (B) is established by open traverse. Using a total station equipped with an electronic distance measurement device, the survey team measures the azimuth α and baseline distance AB. The total station operator may set up her instrument over point A, while her assistant holds a reflector mounted on a shoulder-high pole as steadily as he can over point B. Depending on the requirements of the control survey, the accuracy of the calculated position B may be confirmed by astronomical observation.
Next, the survey team uses the electronic distance measurement feature of the total station to measure the distances AC and BC. Both forward and backward measurements are taken. After the measurements are reduced from slope distances to horizontal distances, the law of cosines can be employed to calculate interior angles, and the coordinates of position C can be fixed. The accuracy of the fix is then checked by plotting triangle ABC and evaluating the error of closure.
Next, the trilateration network is extended by measuring the distances CD and BD, and fixing point D in a plane coordinate system.
Use trilateration to determine a control point location
Trilateration is a technique land surveyors use to calculate an undetermined position in a plane coordinate system by measuring distances from two known positions. As you will see later in this chapter, trilateration is also the technique that GPS receivers use to calculate their positions on the Earth's surface, relative to the positions of three or more satellite transmitters. The purpose of this exercise is to make sure you understand how trilateration works. (Estimated time to complete: 5 minutes.)
Note: You will need to have the Adobe Flash player installed in order to complete this exercise. If you do not already have the Flash player, you can download it for free from Adobe.
- Display a coordinate system grid: In this exercise, you will interact with a coordinate system grid. First, display the coordinate system grid in a separate window so that you can interact with it while you read these instructions. Arrange the coordinate system grid window and this window so that you can easily view both. You may need to make this window more narrow. Two control points, A and B, are plotted in the coordinate system grid. A survey crew has measured distances from the control points to another point, point C, whose coordinates are unknown. Your job is to fix the position of point C. You will find point C at the intersection of two circles centered on control points A and B, where the radii of the two circles equals the measured distances from the control points to point C.
- Plot the distance from control point A to point C: On the coordinate system grid, click on control point A to display the data entry form. (You'll need to click on the actual point, not the "A".) The form consists of a text field in which you can type in a distance and a button that plots a circle centered on point A. The radius of the circle will be the distance you specify. According to the surveyors' measurements, the distance between control point A and point C is 9400 feet. Enter that distance now, and click Plot to plot the circle. [View result of Step 2]
- Plot the distance from control point B to point C: The measured distance from point B to point C is 7000 feet. Click on point B (on the actual point, not the "B"), enter that distance, and plot a circle. [View result of Step 3]
- Plot point C: Now click within the coordinate grid to reveal the position of point C. You may have to hunt for it, but you should know where to look based on the intersection of the circles. [View result of Step 4]
- Extend the control network further: Now, continue extending the control network by plotting a fourth point, point D, in the coordinate system grid. First, plot new circles at points A and C. The measured distance from point A to point D is 9600 feet. The measured distance from point C to point D is 8000 feet. (You may wish to set the radius of the circle centered upon point B to 0.) Finally, click in the coordinate system grid to plot point D. [View result of Step 5]
Once you have finished viewing the grid, close the popup window.