calculating distance with compass only

[Harvestman]

Basically, the further away something is, the more difficult the calculations are to get accurate.

As a rule of thumb, by the time you have done the calculations to your satisfaction, you could have walked to your target object anyway

 

[Joonsy]

Hi Harvestman, yes i accept your point, however i asked if anyone knew a simple method that didn't involve excessive maths as when i have been walking with a pal or two someone has often said ''how far do you think that hill/or whatever is'' and i wondered if there was a way to give a reasonably quick estimate without referring to the map instead of just making a wild guess, of course it's not important to the days walk as i carry a map anyway it's just a bit of pleasant banter on route and i find navigation interesting anyway. thanks for your input.

 

[Harvestman]

Joonsy, my remark wasn't to be taken seriously

 

[Joonsy]

Joonsy, my remark wasn't to be taken seriously

and it wasn't thanks

 

[Noddy]

...............

 

[Joonsy]

Hi Noddy, thankyou for your input.

 

[superc0ntra]

Take the bearing, move a known distance at a 90 degree angle to the object. Take a new bearing to measure the angle between positions 1 and 2.
When you know the distance between positions 1 and 2 and the angle in respect to the distant object, it's just a matter of trigonometry:
Distance to object = Distance between 1 and 2 * tan(angle).

 

[Thoth]

You can't reliably measure distance with just a compass from a single fixed point Joonsy. It can be done with a theodolite, or some levels (which are basically a telescope mounted on a protractor). BUT you need a surveying staff at the object point! The cross hairs include additional marks allowing the height at which the marks meet the staff to be measured. The height difference between the marks at the object point is now known & the angle known for the particular instrument being used so allowing the distance to be calculated using simple trig. Not possible with just a compass though. Sorry!

 

[Joonsy]

thankyou thoth & supercontra for your replies, well i have at least learned that it can't be done simply with a compass only so my curiosity has been answered and i will just refer to the map as normal, thankyou all for your helpful replies.

 

[Nomad]

This can be done using the basic trigonometry principle described by supercontra, and it can be done without using a calculator, or trig tables, or even a pre-printed lookup table. All you need is a compass and a way to work out the distance to the second point. Before I go further, I need to offer a correction: it’s the difference in degrees divided by tangent of the angle, not multiplied.

The idea is to use a method to set things out, such that the amount of calculation done in the field is minimised. At first, I thought of using a way to pace out 100m perpendicular to the first bearing to the target, and then seeing if there was a simplified way to to get an estimate of the distance. The numbers for 100m weren’t particularly good, so I played around and came up with a magic number that makes it easy.

The magic number is 17.5.

Here’s how it works...

At point 1, take a bearing to the target.
Move at 90 degrees to this bearing in multiples of 17.5 meters.
As you go...
- Count the number of times you do a hop of 17.5 meters.
- Take new bearings to the target and calculate the difference in degrees.
Keep going until the difference between the two bearings is of a reasonable size.
Get the distance in kilometres from point 1 to target by doing: number of hops divided by degrees.


What is a reasonable size for the difference between the two bearings? This depends on how accurately you can measure the bearings with your compass, and on how accurately you can move to point 2. If you reckon that the best you can do with a protractor compass is +/- 1 degree, then you have to do enough hops to get a difference in bearings that reduces that error to an acceptable level.

For example, if I aim to cover enough hops to give me a difference of 10 degrees, I could be out by 1 degree at point 1, and out by 1 degree in the other direction at point 2 – an error of 2 degrees, or 20%. The greater the error, the less accurate the estimate of distance to the target will be. For larger distances, reducing the error as much as possible is desirable, especially if you intend to walk to the target. A good sighting compass and careful technique can make a big difference.

Covering the 17.5m hops can be done with reasonable accuracy by using a 17.5m length of string with a stick or tent peg at one end. Poke the stick in the ground at point 1, and follow your course at 90 degrees to the first bearing. When the string is taut, give it a yank to pull the stick out of the ground and then gather it all up, poke the stick into the ground where you’re standing, and follow the course for another 17.5m, etc. If you put something at point 1 that you can see from your eventual location at point 2, you can verify the overall accuracy of your series of hops (ie, reduce the zig-zag effect) by taking a back-bearing to that object. Note that, if you put something at point 1 that you don’t want to lose, like your rucksack, you will have to go back and get it.

A couple of examples...

Example 1.

I see a copse of trees that I want to make for. I take a bearing to it, and get 11 degrees. There is open ground to my right, so I decide to head in that direction perpendicular to 11 degrees: 11 + 90 = 101 degrees. I put something at point 1 that is large enough for me to see from a distance (tall stick with some leaves on the top, whatever). I put the peg on my surveying cord into the ground and move along 101 degrees until the string is taut, pull the string, reset the peg, follow the course, etc, taking new bearings as I go. when I get to a bearing of 5 degrees, giving a difference between the two bearings of 6 degrees, I decide to stop and do the numbers. I’ve covered 10 hops of 17.5m, so I do 10 divided by 6, and get 1.67km – the distance from point 1 to the copse of trees.

This can be verified mathematically...

tangent of 6 degrees = 0.105
10 x 17.5m hops = 175

distance to point 2 divided by tangent of degrees = distance to target in metres

175 / 0.105 = 1666.67, or 1.67 km



Example 2.

I’m on a moor, standing next to a lone scraggy tree. There is a mountain in the distance that I want to make for. I grab my nice, accurate sighting compass and take a bearing to a prominent feature on the mountain. I get 3 degrees. The moor is open on both sides, so I look at the overall terrain and work out which direction to head in for my second bearing such that I’m set up for my preferred approach. There’s a marsh to the right, and reasonable looking terrain to the left, so I choose to go left: my peg and string hops follow a course of 273 degrees. I use the scraggy tree as my back-bearing marker for point 1 to check that my course isn’t zig-zagging too much. After many hops, I get a new bearing to the mountain of 8 degrees, giving a difference between the two of 5 degrees. I covered a total of 37 hops, so I do 37 divided by 5 = 7.4km from point 1 to the target.

Mathematical verification...

tangent of 5 degrees = 0.087
37 x 17.5m hops = 647.5

distance to point 2 divided by tangent of degrees = distance to target in metres

647.5 / 0.087 = 7400.96 or 7.4 km

Note that I’m not heading back to the tree – I picked a perpendicular course that is going to take me more or less towards the route I want to follow. That means that I will be following a route that is slightly longer than the calculated one, but not by a great deal due to the small angle of 5 degrees. This new course is also less than the course I would follow if I then walked back to point 1 and then headed for the target.


Effect of errors.

The biggest potential for error comes from the compass readings. For shorter distances, an error of +/- 1 degree isn’t too bad because you can readily cover enough hops to get a reasonable value for the difference in the two bearings, but it can be significant for longer distances because the distance to point 2 can be quite large to get the same difference - it might be tempting to stop sooner and work with a smaller difference.

Note that the error in the compass readings does not translate into the same percentage error in the calculated distance. Here's the effect of this on the two examples...

Example 1
Readings were 11 and 5 degrees, giving a difference of 6 degrees.
Being out by +/- 2 degrees here gives an error of 33% - 4 or 8 degrees worst case.
Number of hops to point 2 was 10.

10 hops / 6 = 1.67 km
10 hops / 4 = 2.5 km, or 50% more
10 hops / 8 = 1.25 km, or 25% less



Example 2
Readings were 3 and 8 degrees, giving a difference of 5 degrees.
Being out by +/- 2 degrees here gives an error of 40% - 3 or 7 degrees worst case.
Number of hops to point 2 was 37.

37 hops / 5 degrees = 7.4 km
37 hops / 3 degrees = 12.33 km, or 66% more
37 hops / 7 degrees = 5.29 km, or 28% less

Here is example 2 with sighting compasses that give readings to +/- 0.5 and +/- 0.25 degrees...

Two readings with +/- 0.5 gives a max error of 1 degree = 20%.
37 hops / 5 degrees = 7.4 km
37 hops / 4 degrees = 9.25 km, or 25% more
37 hops / 6 degrees = 6.17 km, or 17% less

Two readings with +/- 0.25 gives a max error of 0.5 degree = 10%.
37 hops / 5 degrees = 7.4 km
37 hops / 4.5 degrees = 8.22 km, or 11% more
37 hops / 5.5 degrees = 6.72 km, or 9% less

Clearly, accurate compass work is vital to getting a good estimate, especially when dealing with targets that are long way away. On the other hand, the further you go to establish point 2, the more you can reduce the error. If you covered enough distance to get a difference of 20 degrees, your 2 degree error would equate to the 10% seen with the +/- 0.25 sighting compass. The downside is that the distance needed to get a 20 degree difference can be rather long – for our mountain at 7.4km, you’d need to go about 2.7km sideways (and count 154 hops). If both legs were as the crow flies, the total distance covered would be 2.7 + 7.9 = about 10.6km.


How useful is this?

For general navigation, not much. For longer distances, where the error is magnified, or you chose to cover a lot of ground to get to point 2, it is surely easier, quicker, and more accurate, to get the map out and measure the distance, provided you have a good idea of where you are. It’s also questionable for things at shorter distances – with these, we are generally okay at estimating distances by the apparent size of the object (tree, building, etc).

For situations where you don’t have a map, and are perhaps in unfamiliar territory (like a survival situation that you've chosen to walk out of), it does have some potential to be useful. If you knew roughly where you wanted to go (for example, knew there was a village near the mountain in example 2), being able to estimate the distance could make the difference in being able to say whether you can walk there before dark.

As a thing to do while camping or otherwise spending time in an area, it could be interesting, if only to check on a map later to see how accurate your estimates were. It’s not really what I would call navigation – more like surveying using triangulation. If nothing else, it might be a good exercise in compass usage to see just how accurate you (and your compass) really are.

 

[superc0ntra]

... it’s the difference in degrees divided by tangent of the angle, not multiplied.

Sorry, my bad. A bit rusty at trig

 

[Nomad]

Sorry, my bad. A bit rusty at trig

I had to use a spreadsheet to see which calculation produced a distance that matched a scale drawing.

 

[Elen Sentier]

Basically, the further away something is, the more difficult the calculations are to get accurate.

As a rule of thumb, by the time you have done the calculations to your satisfaction, you could have walked to your target object anyway

As I got lost after the first sentence I think I'll stick to the "walk there" method ... maths and I are like the same poles of a magnet !!!

 

[bearbait]

There is a way that this problem can be solved using a compass and a hand-held GPS unit with a graphical display - and wiv no sums. (I know it doesn't exactly answer the OP's question but it is possibly of interest.)

X marks the spot we are interested in.

A marks the spot we are at now. Mark a waypoint, WPT_A. Take a bearing of the point X using your compass or the GPS's compass. Project a waypoint, say WPT_AX, on this bearing from your position to some distance beyond point X, e.g. if you estimate the distance to X is 5KM project the waypoint to, say, 10KM.

Set off hiking/biking/whatever to point B where the included angle AXB is a min. of say 30 degrees. (45 is good to aim for but 90 degrees would be best. To achieve this end the compass bearing to point X from your new position B should be at least 30 degrees more or less than the original bearing from point A to X.) Create WPT_B. As at point A take a bearing of point X and project a waypoint, WPT_BX on this bearing to some way beyond X.

Now create a route in the GPS from your current position: WPT_B to WPT_BX to WPT_AX to WPT_A. On the screen of the GPS you will note that the route crosses at one point. This is point X. Navigate the cursor to this point and create a waypoint, WPT_X. Now you can use the GPS to give you a distance to this point. Job done.

What you have effectively done here is to have fixed the approximate position of unknown point X by intersection of two position lines, from two known points, point A and point B. The accuracy of X's determined position will vary according to the quality of the bearings taken, the distance to point X, and the resulting angle between the position lines.

You could probably also solve the problem by means of equal right-angled triangles and the compass, given clear open country so that you can pace count and see markers that you have set down. See, in various outdoor and survival publications, a technique for determining the width of a river without crossing it for the solution.

 

[Nomad]

Been out a couple of times to try the compass and string method, with varying accuracy...

First time was a tall outcrop of rock not very far away. I did five 17.5m hops and got a difference of 4 degrees, giving 1.25km. Checking the map, it was actually 1.5km. Second was sighting on a hill a good distance away. From my start at point 1, I got seven hops before some trees got in the way, forcing me to use that distance. I was a bit more careful with the compass readings this time, and got 0.5 degrees, giving a distance of 14km. On the map, the actual distance was 12.3km.

I wouldn't like to say whether my compass technique had improved massively between the first and second attempts, such that the second one is going to be representative of the accuracy that I can acheive. I was a little surprised that it was so close, but I have to put it down to a possible fluke, at least until I try a few more times.

The compass I used (and hadn't used before) was a Plastimo Iris 50, which is a marine hand bearing compass. Its design does allow for easy reading while sighting on the target, and one can probably place the index line to within 0.25 degrees. It's easy to see if it's on a degree line, and mid-way between two degree lines, but a little harder to say whether it is at the quarter positions either side of the mid point. My compass has a little modification to make it easier to align accurately, which I'll describe in a separate post.

The conclusion so far is that more practice is needed.