# Who knows geometry

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I'm preparing a cache that requires the finder to mathematically triangulate the location of the main cache from three reference points. It would help me to know if a typical finder can do this. This new poll option is perfect for this.

Given three reference points with distances ...

Hey, this sounds like a good idea!

Even if someone isn't mathematically inclined, one could get a group of three geocachers together (or one geocacher with three GPSRs!) and have each one put in one of the three points as a waypoint, then try to move together as a group to the location where all the distances match up.

I might steal your idea for one of my own future caches...

quote:
Originally posted by DisQuoi:

I'm preparing a cache that requires the finder to mathematically triangulate the location of the main cache from three reference points. It would help me to know if a typical finder can do this. This new poll option is perfect for this.

Given three reference points with distances ...

Being a math teacher, this makes me giggle. If you know how many times, "when are we ever going to use this?". I guess you just never know when you're going to use this.

You should have paid attention to your math teacher and then you could all choose the first.

george

Are the reference points going to be great distances away from each other? Otherwise why would you need three, if you have a single bearing and range that should get you pretty close by itself. Obtaining a fix with three lines of bearing only would seem to be more challenging.

quote:
Originally posted by bigcall:

Obtaining a fix with three lines of bearing only would seem to be more challenging.

You're right, it would be lot more challanging. But for myself, and other math geeks, it would make it an irresistable challange to find the actual cache.

I actually have a cache that is similar to this idea that has not been placed yet. I haven't placed it yet because the terrain of the area that I want to use is still snow and mud covered. I have to wait for it to dry out. Once placed I am not expecting a whole lot of cachers to try it because of the mathematical difficulty of it. For the ones that do attempt it and find it should be a fun challange.

I just wish somebody would post a cache like this for me to find.

-Gromit

quote:
Originally posted by bigcall:

Obtaining a fix with three lines of bearing only would seem to be more challenging.

You're right, it would be lot more challanging. But for myself, and other math geeks, it would make it an irresistable challange to find the actual cache.

I actually have a cache that is similar to this idea that has not been placed yet. I haven't placed it yet because the terrain of the area that I want to use is still snow and mud covered. I have to wait for it to dry out. Once placed I am not expecting a whole lot of cachers to try it because of the mathematical difficulty of it. For the ones that do attempt it and find it should be a fun challange.

I just wish somebody would post a cache like this for me to find.

-Gromit

Let me clarify ...

Three coordinates will be given. At each of the three locations, a microcache will contain a single number x.xxx miles (no bearings). Therefore, you will need to locate each of the three (two minimum) microcaches to get the distances.

In one of my caches, I did the reverse ... I gave the distances in the cache description but forced the finder to locate the coordinates (of the reference points). The real difference is that in the second example, the finder could use a pencil and compass to draw the arcs on a map. He only had to locate a bridge (at which the exact coordinates are found). However, in this case, he will need to calculate the location (or accept a fairly large search area).

quote:
I just wish somebody would post a cache like this for me to find.

My brother (Rodness) is currently building a twin cache in and around Boise, ID.

There are two ways to do this.

1) With a paper map. Requires the cacher to have a paper map of the appropriate projection.

2) With math. Keep in mind, unless the cache is near the equator, we are dealing with a NON-CARTESIAN coordinate system. This means that 1 degree longitude is less distance than 1 degree latitude. This would make doing some math difficult. On short, this is beyond geometry and into the realm of trigonometry.

I will provide UTM coordinates. The distances will be in the range of 1-10 miles. Since I'm providing the three reference points to 1 meter and the distances to 1/1000th of a mile, the finder should be able to use basic trigonometry to solve. As for paper ... if the distances are small, this will work but as the distances become larger, the scale of a map will make an accurate determination difficult.

I have a cache (HillsDevil's Triangle) that's similar to this, except I don't provide any distances or bearings or even waypoints except for the first one. Along the way I provide (and the seekers gather) just enough info for the seekers to determine distances, bearings, and waypoints on their own. Only two people have attempted it so far, it took theem both 2 trips, and both of them succeeded. L

quote:
AquaDyne wrote: Why not give three reference points with bearings? Or two with bearings and one with distances? (deleted)

There are many combinations or reference points, bearings, and distances that could be used. One could be given three arcs (two arcs would result in two possible solutions), two lines (only one solution), an arc and a line (two solutions but you could provide a N/S clue), etc.

I chose three arcs but calculating the location would require about the same effort for all of these.

Sorry for the lengthy post, but it all is pertinent to the question at hand.

I posed a question on the Geodashing forum (odd, I'm seeing some of the same names here, Gromit and Aquadyne ). We had been talking about the closest dashpoint to each other's homes and I said:

quote:

GD7-03926 was 8.23 south-southwest. That was my closest one, although GD6-03974 was 8.92 miles north-northeast. Each game they get closer and closer.

Someone with a good map and the right kind of info could now triangulate on my house.

Responses:

Other Geodasher:

So far, you've given enough info for two possible locations. Want to give a third?

My Response:

quote:
To Review:

GD7-***** was 8.23 miles from home

GD6-***** was 8.92 miles from home

and now...

GD5-***** was 12.79 miles from home

and for redundancy...

GD7-03961 was 17.17 miles from home

How many math geeks does it take to come up with a close approximation of my home coordinates?

Within a couple of hours...

AquaDyne came close, but no cigar.

geogromit posted a picture circling my neighbor's house.

Daniel B. Widdis even had my road name and a darn close approximation of the house number (within 500 on a 99999 system).

But geogromit explained it best:

quote:
Actually I only original two dashpoints that were closest to your house and I was able to triangulate 2 positions in your area. I did this by converting the coordinates of the dashpoints to UTM and the distances to meters. This made it fairly simple to calculate your position. The point that that I chose appeared to be in the middle of your street. The other appeared near a small lake. I guessed on the northside of your street. From the precision of your distances that you had given, I knew your house should have been within 50 feet of that point.

So - bottom line, it can be done to within the tolerance of a GPS. I suggest showing at least three decimal places (or better yet, show the distance in meters) and giving them four outlying locations.

Markwell

Non omnes vagi perditi sunt

Sorry for the lengthy post, but it all is pertinent to the question at hand.

I posed a question on the Geodashing forum (odd, I'm seeing some of the same names here, Gromit and Aquadyne ). We had been talking about the closest dashpoint to each other's homes and I said:

quote:

GD7-03926 was 8.23 south-southwest. That was my closest one, although GD6-03974 was 8.92 miles north-northeast. Each game they get closer and closer.

Someone with a good map and the right kind of info could now triangulate on my house.

Responses:

Other Geodasher:

So far, you've given enough info for two possible locations. Want to give a third?

My Response:

quote:
To Review:

GD7-***** was 8.23 miles from home

GD6-***** was 8.92 miles from home

and now...

GD5-***** was 12.79 miles from home

and for redundancy...

GD7-03961 was 17.17 miles from home

How many math geeks does it take to come up with a close approximation of my home coordinates?

Within a couple of hours...

AquaDyne came close, but no cigar.

geogromit posted a picture circling my neighbor's house.

Daniel B. Widdis even had my road name and a darn close approximation of the house number (within 500 on a 99999 system).

But geogromit explained it best:

quote:
Actually I only original two dashpoints that were closest to your house and I was able to triangulate 2 positions in your area. I did this by converting the coordinates of the dashpoints to UTM and the distances to meters. This made it fairly simple to calculate your position. The point that that I chose appeared to be in the middle of your street. The other appeared near a small lake. I guessed on the northside of your street. From the precision of your distances that you had given, I knew your house should have been within 50 feet of that point.

So - bottom line, it can be done to within the tolerance of a GPS. I suggest showing at least three decimal places (or better yet, show the distance in meters) and giving them four outlying locations.

Markwell

Non omnes vagi perditi sunt

Yes ... I plan to use "x.xxx miles". This will result in an accuracy of 1.6 meters. If solved mathematically, there should be no question. I chose miles so people will be forced to convert to meters.

After 16 votes, I'm guessing that only people with math backgounds are responding (surveyors, engineers, math majors or degres requiring tons of math). ... because frankly, it's not a simple calculation.

Yes ... I plan to use "x.xxx miles". This will result in an accuracy of 1.6 meters. If solved mathematically, there should be no question. I chose miles so people will be forced to convert to meters.

After 16 votes, I'm guessing that only people with math backgounds are responding (surveyors, engineers, math majors or degres requiring tons of math). ... because frankly, it's not a simple calculation.

quote:
Originally posted by harrkev:

... Keep in mind, unless the cache is near the equator, we are dealing with a NON-CARTESIAN coordinate system. This means that 1 degree longitude is less distance than 1 degree latitude.

[P]Well if a degree of latitude is not equal to a degree of longitude I don't see how this is really a problem. The real problem is that the degrees of longitude get smaller depending on the latitude. That's what really
quote:
would make doing some math difficult. On short, this is beyond geometry and into the realm of trigonometry.
[P] and that's where I get lost real fast. I had some trig in school, you know back in the 1800's. Remember precious little, and can't do a thing with it. Anybody have a good explanation? Relating it to geocaching might help... Thanks

King Pellinore

quote:
Originally posted by harrkev:

... Keep in mind, unless the cache is near the equator, we are dealing with a NON-CARTESIAN coordinate system. This means that 1 degree longitude is less distance than 1 degree latitude.

[P]Well if a degree of latitude is not equal to a degree of longitude I don't see how this is really a problem. The real problem is that the degrees of longitude get smaller depending on the latitude. That's what really
quote:
would make doing some math difficult. On short, this is beyond geometry and into the realm of trigonometry.
[P] and that's where I get lost real fast. I had some trig in school, you know back in the 1800's. Remember precious little, and can't do a thing with it. Anybody have a good explanation? Relating it to geocaching might help... Thanks

King Pellinore

Trogonometry is the math you learned that involved cosine, sine, tangent, etc. The intersection of arcs (i.e., the distances from the reference points to the cache) can be described by a series of triangles on a plane. Once you describe the triangles, you would simply use algebra to calculate the lengths of the sides of the triangles ... then you have the location of the cache. You can do it graphically with a compass and pencil (not a magnetic compass but a v-shaped tool used to draw circles) but this won't get you as close to the cache as the method described above.

quote:
Originally posted by DisQuoi:

It would help me to know if a typical finder can do this. This new poll option is perfect for this.

It looks like the typical polltaker can solve it, but I bet the typical geocacher can't.

Assuming 1 out of 10 geocachers can solve it, would that influence your decision to hide it?

quote:
Assuming 1 out of 10 geocachers can solve it, would that influence your decision to hide it?

I'm definately going ahead to place it ... All that's left is the placement of the micro-caches. However, I agree that in reality, less than 9 out of 10 typical people can do it. I may decide to offer an description of how to calculate this or even provide a spreadsheet wherein persons who are not interested in the challenge of the geometry can still follow the intended cache description. In my first cache I decided to provide solutions for anyone who wants to find the cache but not accept the challenge (so far 2/2 have located the cache without my help).

It would be interesting to solve this with 'high-tech' paper and compass. Using AutoCAD or other drafting software, the circles could be drawn and then you could get the coordinates of the intersection. I assume this would work, any other opinions out there?

Side note: Instead of three circles you could do two circles and either make the searchers look at two places or put one of the intersections in a 'low-probability area' like the ocean or middle of a lake. Not that this would be better, but would add a different wrinkle to it.

Yes, I agree that two circles are sufficient to almost solve this. I susspect that many will skip the third micro-cache and simply use their judgement of which of two solutions is correct. I will probably still place three. Maybe I'll make the third easier than the other two but provide only a discriminator (e.g., "northern intersection").

... and yes, if someone has access to AutoCAD or Microstation, they could do this graphically with high accuracy. Along that line, if you know any design civil engineers, they could solve it using coordinate geometry tools (or by hand I hope).

Since starting this geocaching pass-time in Jan 2002, I have posted 6 new caches and all of them require math calculations. They are:

DOUBLE CROSS GC364C

TRIANGLE GC3FD5

In The Square GC3E49

Walking In Circles GC3C43

Benches & Bearings GC391C

F'n Math GC3770

All can be solved using just your GPS during the search.I use a Garmin 12.

If anybody else out there has been making up challenging cache descriptions, I would be happy to hear from them.

binthair@rogers.com

Since starting this geocaching pass-time in Jan 2002, I have posted 6 new caches and all of them require math calculations. They are:

DOUBLE CROSS GC364C

TRIANGLE GC3FD5

In The Square GC3E49

Walking In Circles GC3C43

Benches & Bearings GC391C

F'n Math GC3770

All can be solved using just your GPS during the search.I use a Garmin 12.

If anybody else out there has been making up challenging cache descriptions, I would be happy to hear from them.

binthair@rogers.com

Here is a cache that is in a similiar vein. I think it is quite inventive. I have solved the location, but the weather and my schedule has been uncooperative, Oh well, maybe tomorrow I'' get to it.

null" TARGET=_blank>http://www.geocaching.com/seek/cache_details.asp?ID=15909[/url]

Here is a cache that is in a similiar vein. I think it is quite inventive. I have solved the location, but the weather and my schedule has been uncooperative, Oh well, maybe tomorrow I'' get to it.

null" TARGET=_blank>http://www.geocaching.com/seek/cache_details.asp?ID=15909[/url]

quote:
Originally posted by DisQuoi:

I'm preparing a cache that requires the finder to mathematically triangulate the location of the main cache from three reference points. It would help me to know if a typical finder can do this.

We have a very similar cache in my area: Flatland 3. If the activity on this cache is any indication, you should have no trouble finding geocachers able to successfully do the math and find the cache (although some have used methods other than mathematics to find the container).

Realizing I do not know half of what I need to less than half the time I should, I have always been good at finding places that will tell me what I need to know.

For math a good starting point is Mathworld

http://mathworld.wolfram.com/

----------------------------

TeamWSMF@wsmf.org

I just posted this cache last weekend:

http://www.geocaching.com/seek/cache_details.asp?ID=16197

It hasn't had any activity yet, but I'm hoping that's because it is -30C this week in Calgary.

I figured out how to solve this using math on my own, but since I hid the cache I'm not really sure I'm right.

I put a link to a page that describes how I think it could be solved. The math was found on the web (links provided).

I also provided a fairly descriptive hint so you can still find the cache by using circles on a map. Presumably.

A couple guys in my office are going to start geocaching just so they can figure this one out.

I just posted this cache last weekend:

http://www.geocaching.com/seek/cache_details.asp?ID=16197

It hasn't had any activity yet, but I'm hoping that's because it is -30C this week in Calgary.

I figured out how to solve this using math on my own, but since I hid the cache I'm not really sure I'm right.

I put a link to a page that describes how I think it could be solved. The math was found on the web (links provided).

I also provided a fairly descriptive hint so you can still find the cache by using circles on a map. Presumably.

A couple guys in my office are going to start geocaching just so they can figure this one out.

If you had 3 micro caches with a distance to the MAIN cache in x.xxx miles, it is easy to draw circles of x.xxx miles in Radius on Street Atlas USA, then find the intersection of the circles.

quote:
Originally posted by Geoffrey:

If you had 3 micro caches with a distance to the MAIN cache in x.xxx miles, it is easy to draw circles of x.xxx miles in Radius on _Street Atlas USA_, then find the intersection of the circles.

Exactly. And if you have only two points, you get two circles and two possible intersections. Try both or apply some clever guessing (like one is on the middle of a runway, it's probably the other one ).

Anders

quote:
Originally posted by Geoffrey:

If you had 3 micro caches with a distance to the MAIN cache in x.xxx miles, it is easy to draw circles of x.xxx miles in Radius on _Street Atlas USA_, then find the intersection of the circles.

Exactly. And if you have only two points, you get two circles and two possible intersections. Try both or apply some clever guessing (like one is on the middle of a runway, it's probably the other one ).

Anders

Excellent job DisQuoi!

Went out and tackled this one yesterday (satellites) and today (cache).

While I agree that this can be done graphically (whether w/ maps or computer software) with reasonable accuracy the fun is in exercising your brain. Unfortunately I don't get enough opportunities to clear out the cobwebs so these type of drills are very refreshing.

How long did it take you to set this up? I was fairly skeptical about finding the cache quickly since I thought there would be a sufficient number of minor deviations that would aggregate to at least a 90 ft search radius - I was wrong and now a believer.

Just finished decrypting the clues for fun as well. Thanks again. EVTAJCC

Being a true engineer I could solve it with math. But it would be quicker to draw it in AutoCAD and just ID the point!.

As to solving the math in the field...that would be another story. I don't do COGO very often at all and so would solve problems at home where I can grab whatever book I need. It's amazing how fast you forget.

quote:
Originally posted by bigcall:

How long did it take you to set this up? I was fairly skeptical about finding the cache quickly since I thought there would be a sufficient number of minor deviations that would aggregate to at least a 90 ft search radius - I was wrong and now a believer.

First, I placed the main cache. I spent an hour or so marking and remarking to ensure that the main cache was as accurate as possible. I also chose a hiding spot that would forgive a little error.

For each of the satellites, I spent about a half hour to get decent readings. But that error is eliminated since I calculated the distances (that I provide for you) using the same coordinates that I give in the cache description. However, if you used the coordinates of the satellites that YOU derived, your calculations would be off. I calculated the distances as I placed the satellites (much easier that calculating the intersections).

Of course, much more time was spent scouting satellite locations. Originally, I had marked a location along George Washington Parkway, Tyson's Corner, and even Downtown. In the end, I decided to make the driving minimal.

What does that little magic box made by Garmin or Magellan do? Simply put it does a three dimensional analysis of what DisQuoi wants to do in a two dimensional problem. (A purist may argue that the magic box does it in four dimensions by the way time is calculated.)

georgeandmary asks "when are we ever going to use this?" What do you are doing when you geocache?

DisQuoi wants to do a simplified GPS position calculation. GREAT IDEA. You are teaching a simplified understanding of the great toy that we have. (I am sure many will say that the calculations are still not very simple.)

In fact, the cache I posted is called "You are the GPS" and the reference points are referred to as "satellites". Also, as I discovered, some were expecting distance and bearing. This would be a very simple calculation. The method of using three "satellites" is very close to the method used by GPSRs. I agree that the satellites in orbit require a three-dimensional analysis but this just wasn't necessary when the satellites lie on the surface of the earth and are only a few miles apart. As for the question of "when will I ever need to know this?" ... as a civil engineer, I used the knowledge daily for years.

quote:
Originally posted by Moun10Bike:We have a very similar cache in my area: http://www.geocaching.com/seek/cache_details.asp?ID=13994. If the activity on this cache is any indication, you should have no trouble finding geocachers able to successfully do the math and find the cache (although some have used methods other than mathematics to find the container).

I think that this cache is a very clever use of geometry. It provides three coordinates and you have to derive the coordinates of the center of the circle that passes through each point. But in the hints it says,

There are actually TWO points on the earth's surface that are equidistant from the three given points. Fortunately, one of them lies in an ocean, and thus cannot be used by the flatlanders.

Is this true? I've always thought that there is only one circle for three given points and one center for a given circle.

quote:
Originally posted by DisQuoi:

There are actually TWO points on the earth's surface that are equidistant from the three given points.

Is this true? I've always thought that there is only one circle for three given points and one center for a given circle.

I think you are both correct. I think the point on the opposite side of the earth would be "equidistant from the three given points".

LOL ... is that what they meant? ... I was definately thinking "plane" geometry. There's still some flaws in that logic but it was definately clever if they wre trying to throw people off. Hats off.

quote:
Originally posted by DisQuoi:

I think that this cache is a very clever use of geometry. It provides three coordinates and you have to derive the coordinates of the center of the circle that passes through each point. But in the hints it says,

_There are actually TWO points on the earth's surface that are equidistant from the three given points. Fortunately, one of them lies in an ocean, and thus cannot be used by the flatlanders. _

Is this true? I've always thought that there is only one circle for three given points and one center for a given circle.

Yep, it's true that three points will determine a unique circle. The problem you're having is that you're thinking about things being planar (e.g. Euclidean Geometry) when you should be thinking about things being the surface of a sphere (well, actually, the earth is an oblate spheroid, but you get the same result).

So in the end, the result is that the three points determine a unique circle, and depending on what you decide is 'inside' and 'outside', there are actually two centers.

Don't believe it? Try this thought experiment. Pick a point on the surface of a sphere. Now, find the point that is farthest away from that point (e.g. on the other side of the sphere). Now, if you draw a circle with the first point as the center and a small radius, it's clear that the first point is the center. Now, gradually increase the radius. When you hit the point where the radius is 1/2 the circumference of the sphere, it will be an 'equator' with the two points as the 'poles'. Increasing the radius even more will make the circle 'closer' to the second point.

Hard to describe in words. Easy to demonstrate with a marking pen and a handy sphere (try a toy ball).

When I worked out the Flatland 3 cache, I expected people to do the math. To my great delight, people have come up with all sorts of clever ways to find it, including just going out into the field with all three points in their GPSR and wandering around until the distance to all three is equal. You try to come up with a hard puzzle, and clever people will come up with simple solutions!

-Paul

I had been planning on placing one like this and was wondering how much help to give. Then I found this thread, and it seems that people should be able to find it, although it may take a couple of trips.

My original plan was a three-stage multicache, each with the distance to the "real" cache. The third stage was to have a pad of graph paper, a compass, and a ruler/protractor.

I'm now thinking that I might just dispense with that altogether.

If people ask for a hint, perhaps I'll hide a fourth micro with detailed instructions on how to do the math.

Question: Am I right in assuming that when doing the math, you're best off working in UTM?

quote:
Question: Am I right in assuming that when doing the math, you're best off working in UTM?
If the reference points and final cache are all within one UTM zone, yes. Under these conditions, the curvature of the earth is negligble and you can solve the math using planar geometry. There are many ways to solve it but Warm Fuzzies/Fuzzy provides a great shortcut in this thread.

When I figure how to solve it myself I'm going to place a 5-5 geocache whose location is described as:

Cache's at a corner of the cube having two others at n-- --.--- w-- --.--- and n-- --.--- w-- --.---.

Of course a cube has 8 corners, not just 3. I will select the 2 given so the 5 wrong ones are not plausible.

The 2 corners I give will be diagonally across 2 of the 3 faces which come together at the cache.

These are 3 of the 4 corners of a tetrahedron (a 3-sided equilateral pyramid--with base also). Stating it as a cube instead makes it tougher because you can look up tetrahedron on the web and find a solution for the rectangular coordinates for the corners. These merely have to be transformed to polar. I've forgotten how to do it, but it's probably in any trig book. Working backwards from your cache's coordinates to those of the other corners will be challenging too. I havn't had time lately to persue it.

I think what's cute about this is how complex the solution is for such a short statement of the problem.

If I ever do it I don't expect many (any?) people to find it. That's why I'd like to somehow make it "locationless" in some way so the whole world could take a crack at at and not just the locals.

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