Learn basic concepts of surveying, especially triangulation.
Most people often don’t think about it, but without surveyors, not much would happen in a modern world. Houses, banks, roads, bridges, airports–in short, any structure–relies on accurate surveying. Property ownership is also surveyed
, and when it’s not done properly expensive legal conflicts are usually unavoidable. Surveying is important and is a key part of Western society. The mathematical foundation of surveying is Euclidean geometry. This is the oldest geometry in the Western world and the one that approximates very well our actual experience of distance relationships.
In this exercise, you will learn a little about one of the oldest and most reliable surveying techniques, the plane table survey, by preparing a simple survey and doing some geometric evaluation. While the results of this exercise will not be accurate
, the technique you will learn, when conducted with the appropriate instruments and robust procedures, will allow you to survey many things. And you will have a new insight into why mathematics is important for maps.
Plane table (Figure from Jerry Davis, http://bss.sfsu.edu/jdavis/geog602/)
Many technologies are used in surveying. The plane table survey is an incredibly simple technology for fairly accurate surveys. The technology usually uses a large 2' by 2' board positioned over reference points. Using an accurate instrument for sighting points (one type is called an alidade) and keeping a consistent scale for measuring distances
, a surveyor measures angles that bisect accurately position points. In other words, a traditional surveyor uses angles to determine distances and rarely measures distances directly. (With computer-based surveying devices using laser range finders surveyors now are likely to measure distances.)
For this exercise, you will be using a simple piece of paper as a pseudo-plane table working indoors (making it possible to do this exercise any time of year). In comparison to working with a plane table, this diminishes the accuracy of the survey
, however, the concepts and techniques remain the same. Using measurements of angles, you will be applying Euclidean geometry’s law of sines
to construct a locational survey of items in the classroom. (See the attached sheet for more information.) Make sure to look at law of sines example before starting with the exercise to remind yourself.
Exercise Steps and Questions
You need to have a piece of paper (called a worksheet) for drawing your measurements and making the basic calculations, which you should place on a pad or spiral notebook. You will also need a straight edge and a few colored pencils. Your instructor will provide you with a protractor which you will use to measure angles.
We have created several base lines in the lab room. Each line is measured in centimeters. Each base line forms a side of the triangles you will construct to locate objects in the room.
What to do
First, form a group of 4 to 5 people. Write your name, section number, and the names of the other people in your group on your worksheet. Prepare your lab instruments and get one protractor for each group. Each person is responsible for handing in a sheet indicating all measures, constructions, and calculated distances.
You will be surveying and determining the angles from the base lines to five objects
in the room. Example objects are:
To start with your survey, go to one of the base lines. Two points are indicated, one labeled A and the other C. Draw a line to scale on your worksheet positioned to fit the surveyed elements and label the points. If your base line is in the front of the room, put it at the bottom of the page. No matter where you are in the room, remember to always keep the orientation of your page. To figure the scale, set up a conversion ratio, for example, 1 inch on paper = 100 cm in the room.
Put point A on your worksheet directly over the corresponding point A on a baseline in the room. Accuracy here is very important. Make sure to keep the paper stable after you found the right position. Now, take your straight edge and point it from the point on your worksheet to the object you will survey, clock, thermostat, etc.. Make sure you are very accurate in drawing the line with one of your lighter colored color pencils. Draw a straight line as far as you can, but at least long enough to cover the distance in scale (this will get better with experience). Move to the second point (point C), reposition your paper so that point C on the paper coincides with point C on the ground, and repeat the sighting with your straight edge and drawing of a line. The two lines should meet forming a triangle. Help the next person set up, checking to see if they are also following the correct procedure.
Now, use the protractor to determine the angles of the triangle you have just drawn. The protractor has two degree indications. Negative angles run from left to right, positive angles run from right to left. Use the indications that correspond to the direction of the angle, or direction of the ‘base’ of the angle, e.g., if the base of the angle (also called initial side) points to the right, use the angle indicators that run from right to left. Write the angle measurements on your worksheet together with the figures.
You can now use the law of sines to determine the distances from the baseline to the surveyed object. You only need to calculate one distance, but calculating both distances will be helpful.
Repeat steps 1, 2, and 3 for the other four objects making sure to position your worksheet accurately and measure angles very carefully. Put all your measures and the results of your calculations down on your worksheet. Work together with other people in your group to make sure everybody has the same (or almost the same) measures.
When you have finished surveying and calculating distances answer the following questions on a separate piece of paper
(don’t forget your name and lab section number on each page).
Draw a line in another color connecting your surveyed objects on your worksheet. Does it look like a straight line approximating the wall? Compare your measurements and calculations for each of the five objects you surveyed. How accurate were your measurements and calculations? What is the difference between your calculated positions and measurements? What explains the difference? (12)
You surveyed in only two dimensions. Would adding a third dimension for height make your survey less accurate or more accurate? Why? What about for more precise surveying work in general? What is the name of the process a surveyor conducts to assure accurate height measurements? (7)
Campbell discusses surveying in chapter 2. What is the name of the traditional survey network that you also constructed for this exercise? (4)
to hand in your calculations for each point
, a drawing of the five points you surveyed (based on calculations!) and a line approximating the room wall. On a separate sheet of paper
hand in your answers to questions 1, 2, and 3 answered on a separate sheet of paper.
The Law of Sines and Euclidean Geometry
In this exercise, you will be using Euclidean geometry after the mathematician Euclid who lived around 300 B.C. and who wrote 13 books about mathematics called Euclid’s Elements
. It is the most established approach to codify perceptions of space and motion. Euclidean geometry is also called classical geometry, because many other people contributed to it and added to it over the centuries. Euclid’s geometry consists of several axioms for fundamental geometrical relationships, such as the sum of angles in a triangle always equals 180°.
Even though Euclidean geometry is very old and physics have disproved its validity for many phenomena (think of Einstein’s theory of relavitity and quantum dynamics for example), Euclidean geometry is very important for many modern activities ranging from surveying to computer aided design
, computer vision, and robotics. If you have ever played, or seen, a new video game and been amazed by the graphics, a large proportion of the math behind those graphics is Euclidean geometry.
The law of sines is one of the most fundamental parts of Euclidean geometry used by surveyors. It expresses the relationship between an angle and its opposite side. In right angle triangles
, the sine is the relationship between the opposite side and the hypotenuse. In any triangle the ratio of one side to its opposite angle is the same as the ratio of any other side to its opposite angle. Expressed mathematically:
The law of sines are related to the law of cosines and the law of tangents. These are more complicated formulas for solving for the lengths of sides and size of unknown angles.
If you want to find more information about the law of sines or Euclidean geometery:
Euclid’s Elements (with interactive demonstrations)
Geometry Reference Materials
Using the Law of Sines in Surveying
The law of sines is used to solve for the length of an unknown side when you know the length of one side and two angles. In this example, I go through the steps to find out the length of c in this figure.
In the law of sines, all ratios are equivalent. If we know any three terms from two ratios we can use basic algebra to solve for the unknown term. In this case:
Now substituting the known terms
then using a sine table or first three digits of a calculator’s sine:
finally solving for c by multiplying both sides by .985