Just wondering what level of orienteering abilities we might share out there. I've mentioned this before a while back, but does anyone know how to find themselves using only landmarks, a compass and no map?

Most orienteering is taught using a scaled map and an orienteering compass with a scale or scales calbrated to the map. This is a nice thing to have, but not absolutely necessary if you know how to triangulate your position without a but knowing a landmark. It does require you keep or memorize a sine table. The nice thing is if you take the time and effort to memorize the table, it never changes.

I've put this method to the test at least a dozen times, and I've only missed my mark on three occasions. Terrain was a factor in each miss, but I've learned to compensate for it since then. In each case, I still found my way to the destination by complementary means.
PS:

I certainly enjoy playing with map, compass, and GPS navigation skills, and I'm quite familiar with triangulation using a map and compass to identify one's position on a map (obtain bearings to three landmarks and draw direction lines on a map through the landmarks - location is where the three direction lines intersect).

Can you give more detail on the method you are talking about?? To use the sine function (or any other trig function) you'd need a right angle (90 degrees) for one of the corners and know the length of at least two sides of the triangle.

Ken K.

I once asked a stupid trigonometry question in front of Bill Engvall and he said, "Here's your sine".

Actually, you don't need right angles. What you need to know is that the side of any triangle divided by the sine of the angle opposite that side is equal to any other side of the same triangle divided by the sine of it's opposite angle (the reciprocal is also valid). The law of sines is thus:

sin a/A=sin b/B=sin c/C where the lower case a, b, and c are all sides of the same triangle, and the upper case A, B, and C are all three angles of the same triangle respectively. If you know a landmark and can get a decent bearing on it relative to north, then you can move in a direction for a given distance (paces) and shoot another bearing and if the two points are relatively far enough apart to generate a significant angular difference, you can then figure out how far you are from the landmark and in which direction. If you know where you want to be (ie base camp) and it's relative location to the landmark, you now have a bearing and rough distance to get there.

Essentially, all you need is two sides and one angle of the triangle, or one side and two angles. You don't need a map except at the very beginning to determine base camp relative to the lanmark. You can also do this if you have two landmarks and no other reference point (including a memorized map location).

Trigonometry is pretty cool. Even if I hadn't been an electronics tech, I would still have found a use for that once boring class.

Here's an excellent link that demonstrates how to use the law of sines, including the deductive arithmetic that starts to make it work with just a compass.

http://www.wisc-online.com/objects/index_tj.asp?objid=TMH2702

If you need more help or haven't gone through a trig class (I highly recommend it), you should contact a math professor and see if you can get in on a class.

Very cool!! I'm actually a biologist and statistician by trade (yeah, weird combination) and have had all sorts of algebra, calculus, and advanced calculus classes (and forgot much of what I'd learned), but I'll agree that the most fun class I had was my 10th grade geometry class. BUT I don't recall learning about the Law of Sines. It would be fun to find an advanced geometric class at my local junior college.

Just to clarify, the Law of Sines is

d = a / SIN(A) = b / SIN(B) = c / SIN(C)
where a, b, c are side lengths and A, B, C are the angles opposite of the smaller case sides. On top of that, d is the diameter of the circle that can be drawn around the triangle touching all three corners (real cool!!).

OK, lets see if I can walk us through a scenario:

There are three points of interest:
1-My location at potition #1
2-My location at position #2
3-Landmark

1. While at #1 I take a bearing on the landmark and get 332 degrees (kind of NNW).

2. I then walk on a bearing of 33 degrees (ENE) for 1.5 miles and then take another bearing on the landmark, and get 301 degrees.

The first triangle angle is (360-332)+33=61 degrees.
(360-332)=28 gives the angle west from north, and then 33 gives the angle on the east side of north.

The second triangle corner is 180-(360-301)-33 = 88 degrees.
(360-301) = 59 gives the angle west of north, 33 is the angle east of north, and these two angles plus the angle for the second triangle corner have to add up to 180 degrees. That is the half arc of a circle along the line of my 33 degree bearing line of travel: 88+59+33 = 180.

So far I know two angles for the triangle: 61 degrees and 88 degrees. The third angle must be 180-61-88 = 31 degrees.

So using the Law of Sines we have

1.5 / SIN(31) = b / SIN(88) = c / SIN(61)

where b is the distance from the landmark to postion #1 and c is the distance from the landmark to position #2.

Using some algebra:
b = 1.5 x SIN(88) / SIN(31) = 2.91 miles from landmark to #1

c = 1.5 x SIN(61) / SIN(31) = 2.55 miles from landmark to #2

Lessons learned - besides the Law of Sines itself:
1 - geometry skills are useful for geometry
2 - bring a decent pad of paper and a good pen/pencil
3 - bring a small solar-powered calculator that does trig functions in degrees (as opposed to radians)
4 - having a good 6" unbreakable ruler would help too

Thanks Benjammin!

P.S. My wife has been watching me with pieces of paper, a ruler, and a protractor on my lap drawing pictures & triangles and thinks (knows?) I am nuts. Nothing new. She has accepted my odd hobbies.

{I'd like to self-nominate this post for the "most math in an ETS post" award} <img src="/images/graemlins/blush.gif" alt="" />

Absolutely fantastic!!! <img src="/images/graemlins/grin.gif" alt="" /> <img src="/images/graemlins/grin.gif" alt="" /> <img src="/images/graemlins/grin.gif" alt="" />

That is exactly how it gets done. Now if your math skills are properly honed, and you either memorize the sine table or bring a piece of paper with the table printed on it with you, you can do the math without the calculator. If you have a good medium (like snow, sand, or dirt) and a stick, you can draw or sketch the triangulation out without the pad of paper and pencil. It is rough, but it does work. Instead of using miles as a measure, I count paces, which is easier to evaluate, but if you're good at measuring distance travelled, then miles will certainly work.

Once you have your location figured out relative to the landmark, if you know the location of any other point relative to the landmark, you can calculate the bearing and distance to that location, all without a map in hand, using just a compass. Isn't that just handy as all getup?

I'll admit, the one thing I have to have (besides the compass, of course) is the sine table with me. I've memorized it and forgot it so many times my mind is no longer reliable enough to trust. Did I mention I also wear a calculator watch as my EDC everywhere I go? <img src="/images/graemlins/smirk.gif" alt="" />

Have you tried this with a slide rule because I EDC a pilot watch with the slide rule bezel.
I wouldnt have the foggiest idea how to do sine on it though.

Wow, I haven't played with a slide rule in 25 years. I can't recall what all you can do with a slide rule, but I just bet there's a way to calc the sine of an angle on one.

http://www.sphere.bc.ca/test/howto.html

More than I ever wanted to know about slide rules. Some can do multiplications (and divisions?) of sines.

Ouch! Even used slide rules can be pricey. I think I'd stick with the Texas Instruments TI-36X Solar Scientific calculator for about $20.

Being a Palm PDA user, I wish they'd make an AA or AAA battery-powered Palm, but not anymore. I guess the brainy chips use too much power for the alkalines. They all use rechargables and the solar chargers are just too pricey for me.