Lab #1

Triangulation

Preparing for this Lab

Read over the lab in advance.  Make sure you understand what you should bring.  If weather is inclement check with the instructor in advance.  Do not assume class will be cancelled.  Switching a lecture and proficiencies session indoors for a lab outdoors is an option.  Do not assume class is cancelled unless the University is shut down.  (Here's one to make you feel real important - we wouldn't want to hurt the instruments but you - we'll let you get wet down and frozen - sometime surveyors do get to rough it a little).

Review the procedures for the lab.  Review the suggested style of taking notes.

What Will I Be Doing in this Lab

You will lay out a "baseline" of two known points.  You will turn angles from instruments on each end of the baseline and center on a person at an unknown point in the field.  You will use your instrument readings and trigonometry to figure out the location of the unknown point.  This surveying technique is called triangulation.

What You Will Need

(Things Provided by the Department and Instructor)

1- A Transit with a compass and plumb bob

2- A Wild T2 Theodolite

3- A Surveyors Chain and two plumb bobs

4- Wood Stakes

5- Nails

6- Hammer

7- A Philadelphia Rod

(Things You Should Bring)

1- Clothing compatible with cold, rain, wind, sleet, or snow (unless the weather and forcast clearly indicate you will not need them)

2- Paper for writing down survey notes including some sort of backing to make the paper rigid enough to write on.

3- A writing instrument

4- A tape measure at least 6 feet long.

5- A magnifying glass (or a very very good set of eyeballs - for reading transit vernier)

What You Will Do - Detailed Instructions

(Work to be done by both survey crews)

Go to the north side of the Engineering Building near the east - around where the funny looking blue artwork thingy is located.

Pick a point that looks nice and name that point 0,0.  Make sure the point has a reasonably clear view to the lawn in front of Neckers.  (This is an example of doing a local survey where your initial point known point is selected arbitrarily).

Set up the transit and level it.  Hang a plumb bob from the transit.  Drive a wooden stake into the ground and put a nail in the stake directly beneath the plumb bob.  (This is a way of marking your points in the field).

Release the compass and aim the telescope of the transit to magnetic north (the direction the compass arrow points).  Using the telescope of the transit to keep the a crew member on track have the crew member pace off about 300 feet while staying in line with the magnetic north line of site.  The crew member can adjust the exact distance he/she paces to reach a point that has a clear view of the lawn in front of Neckers.  Have the crew member drive in another stake and nail at this point.  Use the transit telescope to make sure the point is marked directly on the magnetic north line of site.  (This is an example of using a compass and magnetic north to establish a line of known direction).

(Work to be done by Survey Crew #1)

Set up the theodolite over the second point and level the instrument.  A theodolite has an optical plumb bob.

(Work to be done by Survey Crew #2)

While Survey Crew #1 is setting up the theodolite, measure the height of your instrument and record the result in your note book.  (The height of your instrument is measured from the ground to the point on the instrument where the telescope turns up or down).  Zero the transit vernier on the magnetic north line of site.  (This is an example of setting your instrument on a backsight).

Have a member of your crew take a Philadelphia Rod and go to a point of their own choosing on the lawn in front of Neckers.  Have them mark the spot with a stake (the rod is large enough that centering it on a nail is not realistic).  Have them hold the rod up as straight and vertical as possible.

Releasing the upper motion of the transit, turn the angle to the right and center the vertical cross hair on the Philadelphia Rod.  (This is an example of turning an angle right from a backsight to a foresight).

Read the angle on the transit and make notes of your data.

(Work to be done by Survey Crew #1)

Depending on speed and efficiency you may have the theodolite set up by now.

Measure the height of your instrument and record the result in your notebook.  (The height of your instrument is measured from the ground to the point on the instrument where the telescope turns up or down).

Have a member of your crew take a stake and Philadelphia Rod and pick a point on the lawn in front of Neckers.  Have your crew member mark the point with a stake and then hold the rod vertical over the point they set.

Turn the telescope of the theodolite and center the vertical cross hair on the nail in the stake beneath the transit.  Set the horizontal angle to 0 while aiming at the nail beneath the transit.  (This is an example of setting your instrument on a backsight).

Release the horizontal motion on the theodolite and turn the angle to the right (it will be a long turn most of the way around a circle in all likelihood but remember to always turn your angles to the right).  Turn the telescope and center the vertical cross hair on the middle of the Philadelphia Rod.  (This is an example of turning an angle right from a backsight to a foresight).

Read the angle on your theodolite and record the result in your notes.

Take down the theodolite.

(Work to be done by Survey Crew #2)

Hopefully you will have finished taking the transit reading on your person out in the field by Neckers.  At this point assuming reasonably similar pacing of both crews the other survey crew is now measuring angles to their person out in the field by Neckers using the theodolite.

While the other survey crew is using the theodolite, take the surveyors chain and begin measuring the distance from the point under the transit to the point under the theodolite.  Use a plumb bob to center the zero of the chain over the point under the transit.  Pull the chain tight and the person on the other end of the chain will hang down his/her plumb bob and mark the spot with a nail.  Since you will have to use the chain many times to measure the distance this will allow the person on the rear to center the chain over the same point where you ended.  Use the telescope of the transit to make sure you stay on a straight line while you measure.  Unless you have a freak event it will not be an even number of chain lengths to the point under the theodolite.  For the last leg of the distance the rear end of the chain should be over the last point marked.  The other end of the chain will pass over the point under the theodolite.  Using a plumb bob count the number of links to the point under the theodolite.  Record the number of chains and links in your field notebook.  (This is an example of chaining - the old way of measuring distances).  Old Survey crews laying out the original land surveys used Gunter's Chain which was exactly 66 ft long and would measure miles etc in an even number of chains - which was the reason for the curious length selection of 66 feet.

(Work to be done by Survey Crew #1).

Move from the theodolite down to the transit.  The other crew may well be using the surveyors chain at this point.

Use the upper motion of the transit to zero the verneir reading.

Use the lower motion to backsight the point where the theodolite was set up.

Send a person back out with the Philidelphia Rod to the same point they were at when you sighted them with the theodolite.  (You should be rotating jobs within your crews so that everyone gets a chance to run the instruments).

Use the upper motion of the transit to turn the angle right and sight your crew member on the unknown point with the Philadelphia Rod.

Record the angle and height of instrument in your notes.

Using the same procedure specified for Survey Crew #2 above, chain the distance from the transit point to the theodolite point.

(Work to be done by Survey Crew #2)

Using the same instructions as were given for Survey Crew #1 set up the theodolite over the northern point and then measure the instrument height and angle right from the baseline to your unknown point in the field in front of Neckers.

(For Both Survey Crews)

At this point you should have all measured the distance between the two instruments and used both instruments to turn the angle right from the baseline to the unknown point in the field.  This should give you the data to use trig to figure the coordinates of the third point in the field.

Comments on this Lab

While the types of things you have done in this lab are typical of traditional surveying they do not necessarily represent the way the pro's do things.  This will probably be the only time in your life you measure something with a surveyor's chain (just a little nostalgia so you can say you've done it).  I did not have you measure vertical angles.  We treated all measurements across the ground as if everything really were a level plain (don't worry we'll make things harder as we go along).  Assuming everything is level introduces unacceptable errors into a real life survey.  You were ask to measure the height of your instrument, but because we assume everything is level in this exercise you never use the information.  This will get you into practice measuring the height of your instrument.  You will work with one baseline and one triangle.  Real Surveyors layout networks of triangles and set up instruments at all points of the triangle and measure all sides of the triangles each time.  Real surveyors triangulation is used for very high precision work and measuring all sides and angles provide multiple measurements and trigonometric checks on everything.  You switched instruments while doing your survey.  Real surveyors don't trade instruments every few moves.  Each instrument is a little different and switching back and forth is an invitation to make a mistake.  In this class we switch instruments to ensure that you get experience working with many types of instruments so that what ever you are handed in the future you will have some experience with something like it during this class. 

Taking Field Notes

A form similar to the one shown below is suggested for taking your field notes.

Finishing the Lab

Your group will turn in your lab notes and a map at the start of the next class.  One set of notes and map will cover your entire group.  You will use trig to calculate the coordinates of the 3rd point which you do not yet know.

How to Get an Azimuth from a Backsight Azimuth and an Angle to the Right

One of the reasons for always turning angles to the right is that the azimuth of your foresight line will always be your backsight azimuth plus your angle to the right.  For example, using the field notes above the azimuth of the line from Station A (the transit) to point C (the mystery point) is

0+ 63º 35' = 63º35'

The azimuth from Station B (the theodolite) to point C (mystery point) is

180º + 292º53'30" = 472º53'30"

since a value greater than 360º means one has gone around the circle we subtract 360.

thus the azimuth is

472º53'30"-360º = 112º53'30"

How to Convert Grads into Degrees Minutes and Seconds

Of course by now you know that I tried to drive you crazy by making you use a theodolite that was laid out in Grads (darn those crazy Europeans).  To convert your angle measured in grads (sometimes called gon {possibly because someone's brain was gone}) use the following conversion.

328.7138 gon * 0.9 = 295.8424º   the 0.9 factor converts grads to decimal degrees

now convert the decimal degrees to degrees minutes and seconds (like God intended it to be).

The 295 is already in degrees so take it now

295º  plus 0.8424 decimal degrees

Now multiply 0.8424 by 60 to get the decimal minutes

0.8424 * 60 = 50.544 decimal minutes

Take the 50 minutes

50'   plus 0.544 decimal minutes

Now multiply the 0.544 by 60 to get decimal seconds

0.544 * 60 = 32.64"

It is acceptable to have decimal seconds - the angle in this example is

295º 50' 32.64"

How to Use Your Baseline and Two Angles to Triangulate the Location of Point C

While there is more than one way to use trig to get the coordinates of the mystery point C the following is one way.

In a plane all the interior angles of a triangle add up to 180º

The angle right at Station A in our example is an interior angle  63º 35'

The angle we turned with our theodolite was an exterior angle.  I can get the interior angle by subtracting the exterior angle from 360º

360º - 292º 53' 30" = 67º 06' 30"

With two known angles we can subtract from 180º to get the 3rd interior angle of the triangle

180º - 63º 35' - 67º 06' 30" = 49º 18' 30"

At this point I now know all the interior angles and one side of the triangle.  I recall the law of sines

A/sin(a) = B/sin(b) = C/sin(c)  where the angles represented by the small letters represent the angles opposite the side represented by the capital letter.

49º 18' 30" represents the angle opposite our baseline (the length of which we chained).  In our example we know

289.3/sin(49º 18' 30") = 381.5469

We can now algebraically solve the length of the other triangle sides

B = 381.5469*sin(b)  and A = 381.5469*sin(a)   {of course 381.5469 is only right for our example}

Thus if side A is the line from the transit to point C then a is the interior angle we calculated from the theodolite reading

A= 381.5469 * sin(67º 06' 30") = 351.5

We know our transit sat at point 0,0 and the line to point C had a azimuth of 63º 35' and the line was 351.5 ft long.  We can now use our right triangle trig to get the coordinates of point C

We can see our Northern movement is

351.5 * cos (63º 35') = 156.38

Our Eastward movement is

351.5 * sin (63º 35') = 314.79

Adding these distances to our initial coordinates (which in this case happen to be 0,0) we get the coordinates of point C as

156.38, 314.79

Completing Your Lab Submission

To complete the assignment draw a map (preferentially using Autocad) that shows all three angles of your triangle.  The length of all sides of your triangle.  The coordinates of all three endpoints.  And the azimuth of all three sides (use an arrow on the side to indicate which way the azimuth is measured - remember the azimuth of a line going the other direction is 180º different).

Make a copy of your field notes making sure that the notes disclose the names of all members of your group.

(You may include neatly explained calculations sheets if you want partial credit in the event of a goof up but you are not required to submit this).

Submit this work to the instructor during the next class period.