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After The Fieldwork

The drawing of the map

(Page's URL: after_the_fieldwork.htm)

These pages will show you how you could make a map. Yourself. No Google. No electronics.

This material started with my page "Make a Map Yourself"
If you tried what was presented there, you will have some data collected in the field. My other page speaks a bit about what you can do with it. This page will look at the question in more detail.

What you have so far

From your work in the field, you should have a number: The length of your baseline. (The line between the two places from which you captured angles.)

For the purposes of this essay, we will assume that you were only interested in "mapping" one point. Of course, such a "map" would be pretty boring, but to build a more interesting, more useful map, you just repeately put another point on it... by the techniques I will be discussing here.

As before...

As before: at every step in the process, be asking yourself "how critical is this bit, to getting "the right answer"? What can you do to minimize the effect of the inevitable small errors in measurements. Can you think of another way to achieve what is being done... a way less likely to introduce errors.

What should you have from your fieldwork? And we'll "name parts" here, too..."

You had your "pizza box" (or large sheet of paper, previously streched flat on some surface)...

A "pizza box" with two sets of three or four pinholes in it. Here's a not-to-scale sketch of what you come in from the fieldwork with...

That, from left to right, is meant to show:

The pinholes are the small green dots inside the small green circles.

Those pinholes "capture" two angles, angle BAC and angle ABC. I hope "the angles" are clear, from the diagram above?

To make an accurate map

To make an accurate map from the fieldwork, you need to have put the pinholes in the right places! You need to have measured the baseline's length accurately.

With that data...

You put two dots on the map you are making. They can be "anywhere", but with experience you will choose where to put them well.

If you want you map to have an easy to use scale, you do something like this...

Let's say your baseline was, as mine was in the exercise that led to this webpage, 4311cm long. (You might want to critize that for having significance implied to too many digits.)

In that case, if, (on the piece of paper that will become your map), you made the distance between where you put A and B 43 cm, then your map would have a SCALE of 1cm = 1m, wouldn't you?

We're not going to spend much time on it here, but, in passing... if you want "North"... magnetic, or "true" to be at the "top" of your page, you can still put "A" "anywhere"... but where "B" is, relative to "A" determines the direction of both "Norths" for your map. (In my exercise, I measured and found that my baseline ran pretty nearly N/S. The north end pointed 14 degrees east of magnetic north. (That datum is not on the "fieldwork" as presented above.)


So! We have the start of our map! We have a piece of paper with "A" and "B" shown on it.

"All" we need to do is to copy the angles from the fieldwork to the map, and extend the lines from "A" and from "B" towards "C", until those lines cross. Where they cross is "C"'s place on our map!

If you work very carefully, it is easy enough to transfer the pinholes across to the map. Depending on the sizes of things, you may need to add extra paper to the map, but that's not difficult.

Alternatively, you can use a protrator to "copy" the two angles from fieldwork to map.


If the distance between A and B is not much smaller than the distance from them to C, you can get pretty good results this way. Especially if you have a lot of craftsmanship, and do the drawing parts very well.

Perversely, it is exactly when the distance to C is much bigger than the length of the baseline that the whole exercise is the most fun!

However, in those circumstances, if either of your lines... the ones from the angles out to "C"... is even a TINY bit "off", it changes the position of "C" radically. (Think about it. You should see what I mean.)

By the simple techniques given above, I got a distance to C (from A) of 1104m on 9 June 2022... for something that maps.google.com suggested should be 1950m. Yes, I was disappointed. But not down-hearted. It could have been worse. At least my angles dictated lines which converged. If they'd been even a bit "off" in the other direction, I wouldn't have had a triangle, "C" would have "been" infinitly distant from my baseline.

But! Never Fear

Other approaches, though not so easy to "follow" exist, to get you around this difficulty!

I said "never fear". Don't fear that there is no answer to the problem. (You may allow a little fear of the mathematics involved, but I am going to take you gently by the hand, lead you through them.)

"The angles"

For the next few sections, I am going to talk repeatedly about "the angles". I will be talking about the angles shown above as angle BAC and angle ABC... The angles at the ends of the baseline, the angles between baseline and the distant "thing" at "C" that you can add to your map after you have decided where "A" and "B" are on your map.

Copying angles

First... copying angles...

There are various ways to directly copy the angles from the fieldwork to the map. You should be thinking all the time, while reading this. If you are thinking, I suspect I need say little more here, as far as the direct copying approach is concerned. I've discussed it a lot in other pages, anyway.

I haven't mentioned an easy low tech answer: Use a protractor. Two problems: It is hard to measure very precisely with a small protractor. It is hard to get a good big protractor. And for each angle, you have to "read" it from the fieldwork, and then "write" it onto the map... TWO chances... for each angle... for errors to creep in. But "by protractor" WILL WORK. Work quite well, to "map" things in a field.

Two "no angles" ways!

And then there are the "no angles" ways.

Well... it all still hinges on the angles. But you don't "draw" them on the map quite as you would if using the more obvious approach. In fact, in the first "no angles" method, you don't draw a map at all, of all you want to know is how far it is to C.

Both are started by drawing a line across the arms of each of "the angles". ("The angles" as defined above.) You convert an open "V" to a triangle, by adding a side opposite the corner the angle is in. You put it as far "out" as you can.

Now measure the length of each side.

(Wikipedia Law Of Cosines)

(My thanks to Wikipedia for the diagram above.)

From those numbers, you can... with "law of cosines"... calculate the size of the angles in the two triangles you have created. With angle BAC and angle ABC, and the length of the base you can, using the law of sines, calculate either AC or BC (or both)... I'll try to come back to this someday, and give you the details. (Law of cosine can be used in other ways, too, but that's the trick useful to us here.)

Alternatvely, you can use the lengths of the sides of the triangles to "construct" the same triangles on your map. Lay them down right, and you will have angle BAC and angle ABC where you need them to define the lines which eventually cross, and determine the right place on your map for "C". (this is another "to be expanded later" thing! WHAT BIT do YOU "need" most, soonest? Contact details at bottom, and please cite "" if you write.)

Use a theolodite

Or use a theolodite!...

No pizza box required! But, of course, this approach starts with "catch your theolodite". (I'll address that in a moment.)

(A thelodite is the instrument that is usually mounted on a tripod, and is used to measure the angular "distance" between two bearings.)

You set the the theolodite up at "A", measure angle BAC.

Then you set it up at "B", measure angle ABC.

And measure the length of the baseline as before.

Fieldwork done! If you have a theolodite.

First catch your theolodite

I've been interested in all this for many years. At an auction along the way, I was lucky to pick up a gorgeous old theolodite in perfect working order, for a fraction of what it would have cost "back in the day". It took a bit of head scratching to figure everything out, but I think I now understand the essentials.

Using my antique, I measured the same angles as I captured with my pizza box. Instead of trying to draw a map, I plugged the numbers into the law of sines, and came out with a distance BC of about 1400m... to the same "C" that my earlier work calculated to be 1104m away, and which Google suggests is more like 1950m.

Curious. Pretty good agreement between my two determinations, considering the shortness of my base, relative to AC. What could explain that?


... answer in a moment... don't scroll 'til you finish trying to figure out where I might have gone wrong, and get those results....



Did you "get it"? One possible thing that would give those results would be a mistake over the length of the baseline. Alas, Google gave me the same figures for the baseline, within the limitations of that source, as I had recorded when I was doing my fieldwork. Just one of life's little mysteries, I guess. (I haven't redone all the sums... yet. I may have a happy surprize in due course, I suppose.

Going back to "catch your theolodite": You could, of course, make your own! I've done a guide to this. Try to enter into the spirit of how the guide was written. What's the fun of just being told? The guide invites you to figure the problems out for yourself... but gives you hints, and eventually gets you all the way from concept to design.

As I said... I've been having fun with all of this for a Very Long Time. Pehaps the best "index" to my stuff, at the moment, is...Flat Earth Academy- Geography Topics.

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