First a math question, then...

[SgtSpike]

woah, i just learned something...

it results in TWO points IF all 3 known points are in a straight line.

and it follows that an infinite number of known points will not reduce this to 1 known point as long as all the known points are in a straight line.

gotta have some kind of triangle there for triangulation

That's what I figured.

Allow me to be a little more specific.

You are holding a device that reads out that you are X miles away from a specific location. No GPS coordinates are given, with this fact being true for all as I continue with my example. You move to another location of which you know the direction and distance, then reread the device. It now states that you are Y miles from that same specific location. Either from the new location or the original one, you once again relocate to another point, also knowing how far and what direction you traveled. Looking at the device, you now see that you are Z distance from the specific location desired.

Given that, can you triangular where the desired specific location is by one result, or is there a second result that also must be checked?

Assume the distance is in a straight line as a crow flies.

Thanks to all who've answered so far.

~Bruno K~

What you are describing is called "Triangulation", it works to find a point in a plane if the three given points are not in a line. If you have three dimensions you would need a fourth point, as you will get two possible solutions from three points unless the given point is coplanar with the three.

You don't have to worry too much over short distances, but for longer distances you might need to know whether the distance follows the surface of the earth or goes through it?

Thank you for explaining it better than I could. 

[myrkul]

woah, i just learned something...

it results in TWO points IF all 3 known points are in a straight line.

and it follows that an infinite number of known points will not reduce this to 1 known point as long as all the known points are in a straight line.

gotta have some kind of triangle there for triangulation

That's what I figured.

Allow me to be a little more specific.

You are holding a device that reads out that you are X miles away from a specific location. No GPS coordinates are given, with this fact being true for all as I continue with my example. You move to another location of which you know the direction and distance, then reread the device. It now states that you are Y miles from that same specific location. Either from the new location or the original one, you once again relocate to another point, also knowing how far and what direction you traveled. Looking at the device, you now see that you are Z distance from the specific location desired.

Given that, can you triangular where the desired specific location is by one result, or is there a second result that also must be checked?

Assume the distance is in a straight line as a crow flies.

Thanks to all who've answered so far.

~Bruno K~

What you are describing is called "Triangulation", it works to find a point in a plane if the three given points are not in a line. If you have three dimensions you would need a fourth point, as you will get two possible solutions from three points unless the given point is coplanar with the three.

You don't have to worry too much over short distances, but for longer distances you might need to know whether the distance follows the surface of the earth or goes through it?

Thank you for explaining it better than I could. 

Still not an issue, though, since the other point is either up in the air, or within the earth, and is at the same x,y coordinates anyway.

[SgtSpike]

woah, i just learned something...

it results in TWO points IF all 3 known points are in a straight line.

and it follows that an infinite number of known points will not reduce this to 1 known point as long as all the known points are in a straight line.

gotta have some kind of triangle there for triangulation

That's what I figured.

Allow me to be a little more specific.

You are holding a device that reads out that you are X miles away from a specific location. No GPS coordinates are given, with this fact being true for all as I continue with my example. You move to another location of which you know the direction and distance, then reread the device. It now states that you are Y miles from that same specific location. Either from the new location or the original one, you once again relocate to another point, also knowing how far and what direction you traveled. Looking at the device, you now see that you are Z distance from the specific location desired.

Given that, can you triangular where the desired specific location is by one result, or is there a second result that also must be checked?

Assume the distance is in a straight line as a crow flies.

Thanks to all who've answered so far.

~Bruno K~

What you are describing is called "Triangulation", it works to find a point in a plane if the three given points are not in a line. If you have three dimensions you would need a fourth point, as you will get two possible solutions from three points unless the given point is coplanar with the three.

You don't have to worry too much over short distances, but for longer distances you might need to know whether the distance follows the surface of the earth or goes through it?

Thank you for explaining it better than I could. 

Still not an issue, though, since the other point is either up in the air, or within the earth, and is at the same x,y coordinates anyway.

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

[Peter Lambert]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

[SgtSpike]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

[myrkul]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

On the scale that GPS operates at, even the Himalayas are only +1 or 2 Z, and just about everything else is effectively 0.

[SgtSpike]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

On the scale that GPS operates at, even the Himalayas are only +1 or 2 Z, and just about everything else is effectively 0.

I don't see how this is relevant.  "Effectively 0" is not the same as 0, and will still have an effect on the calculation and subsequent answers received.

[myrkul]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

On the scale that GPS operates at, even the Himalayas are only +1 or 2 Z, and just about everything else is effectively 0.

I don't see how this is relevant.  "Effectively 0" is not the same as 0, and will still have an effect on the calculation and subsequent answers received.

But the effect is so minimal, that it can safely be ignored. At the scale you're operating on, everything is coplanar. We don't need millimeter resolution, here.

[SgtSpike]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

On the scale that GPS operates at, even the Himalayas are only +1 or 2 Z, and just about everything else is effectively 0.

I don't see how this is relevant.  "Effectively 0" is not the same as 0, and will still have an effect on the calculation and subsequent answers received.

But the effect is so minimal, that it can safely be ignored. At the scale you're operating on, everything is coplanar. We don't need millimeter resolution, here.

I disagree.  The scale of the GPS is small, sure, but not necessarily the scale of the three datapoints you have and the fourth one to be discovered.

Picture a pole that goes the same distance below ground as it does above ground, and is perfectly straight up and down relative to the earth's surface.  You could say that the top of the pole and the bottom of the pole have the same x,y coordinates on the surface of the earth.

Now, tilt that pole any direction.  The x,y has changed.  Even if the tilt is not large, it can have a significant impact on the x,y coordinates, depending on how long it is, what the point of rotation is, etc etc.

When you calculate the two potential answers, picture a pole between those two potential answers.  The only way the pole would be straight up and down is if the triangle is perfectly parallel to the earth (i.e., you took all three data points at the same elevation).  If the triangle rotates along an axis, the pole is then tilted to the same degree.  For instance, if the three data points are taken on a 45 degree incline, relative to the potential answer, then one answer would be at the correct x,y,z on the surface of the earth, and the other answer would be x/1.41, y/1.41, somewhere below the surface of the earth.

The tilt of the triangle of the three data points collected is what would determine the largeness of the difference between the two points, not the scale of the GPS triangulation.

[myrkul]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

On the scale that GPS operates at, even the Himalayas are only +1 or 2 Z, and just about everything else is effectively 0.

I don't see how this is relevant.  "Effectively 0" is not the same as 0, and will still have an effect on the calculation and subsequent answers received.

But the effect is so minimal, that it can safely be ignored. At the scale you're operating on, everything is coplanar. We don't need millimeter resolution, here.

I disagree.  The scale of the GPS is small, sure, but not necessarily the scale of the three datapoints you have and the fourth one to be discovered.

Not small. Large. The scale is extremely large. At large scales, small differences don't matter. For instance: Let's say I gave you a box tuned for some point in the continental US. To prove your point, you go to the highest point in Denver, 5,690 feet up, to take your first measurement, and for your second, you go to Death Valley, 282 feet below sea level. For your third measurement, you pick a spot out on the great plains.

Over a distance of about 700 miles, the difference in elevation is 5,972 feet. (1.131 miles) Not exactly what I would call "steep."

[Phinnaeus Gage]

Interesting!: http://en.wikipedia.org/wiki/Global_Positioning_System

The Global Positioning System (GPS) is a space-based satellite navigation system that provides location and time information in all weather conditions, anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites.

Then, why not three?

[myrkul]

Interesting!: http://en.wikipedia.org/wiki/Global_Positioning_System

The Global Positioning System (GPS) is a space-based satellite navigation system that provides location and time information in all weather conditions, anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites.

Then, why not three?

Because the reference points (the satellites) have significant elevation distance on the target point (your position). This needs to be accounted for, so that you're not marked as somewhere out in space.

[Phinnaeus Gage]

Interesting!: http://en.wikipedia.org/wiki/Global_Positioning_System

The Global Positioning System (GPS) is a space-based satellite navigation system that provides location and time information in all weather conditions, anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites.

Then, why not three?

Because the reference points (the satellites) have significant elevation distance on the target point (your position). This needs to be accounted for, so that you're not marked as somewhere out in space.

From the same source:

Although four satellites are required for normal operation, fewer apply in special cases. If one variable is already known, a receiver can determine its position using only three satellites. For example, a ship or aircraft may have known elevation. Some GPS receivers may use additional clues or assumptions such as reusing the last known altitude, dead reckoning, inertial navigation, or including information from the vehicle computer, to give a (possibly degraded) position when fewer than four satellites are visible.[39][40][41]

I'll assume that that's what you're eluding to, myrkul.

~Bruno K~

[myrkul]

Interesting!: http://en.wikipedia.org/wiki/Global_Positioning_System

The Global Positioning System (GPS) is a space-based satellite navigation system that provides location and time information in all weather conditions, anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites.

Then, why not three?

Because the reference points (the satellites) have significant elevation distance on the target point (your position). This needs to be accounted for, so that you're not marked as somewhere out in space.

From the same source:

Although four satellites are required for normal operation, fewer apply in special cases. If one variable is already known, a receiver can determine its position using only three satellites. For example, a ship or aircraft may have known elevation. Some GPS receivers may use additional clues or assumptions such as reusing the last known altitude, dead reckoning, inertial navigation, or including information from the vehicle computer, to give a (possibly degraded) position when fewer than four satellites are visible.[39][40][41]

I'll assume that that's what you're eluding to, myrkul.

~Bruno K~

Yup, a ship on the ocean, by definition, is at sea level. Thus, any result that does not put it reasonably near to 0 elevation can be thrown out.

But using 4 points is only necessary when you don't know the elevation of your target, and it's not coplanar with the reference points. When your reference points are all on the surface of the earth, you can treat them all as coplanar with the target, despite minor elevation differences, because the scales of the X and Y dimensions dwarf that of the Z in all but a very few cases. That's also why I said that the further apart your measurements are, the more accurate your targeting will be.

Edit:  Oh, and a minor grammatical quibble: elude = dodge; allude = refer

[SgtSpike]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

On the scale that GPS operates at, even the Himalayas are only +1 or 2 Z, and just about everything else is effectively 0.

I don't see how this is relevant.  "Effectively 0" is not the same as 0, and will still have an effect on the calculation and subsequent answers received.

But the effect is so minimal, that it can safely be ignored. At the scale you're operating on, everything is coplanar. We don't need millimeter resolution, here.

I disagree.  The scale of the GPS is small, sure, but not necessarily the scale of the three datapoints you have and the fourth one to be discovered.

Not small. Large. The scale is extremely large. At large scales, small differences don't matter. For instance: Let's say I gave you a box tuned for some point in the continental US. To prove your point, you go to the highest point in Denver, 5,690 feet up, to take your first measurement, and for your second, you go to Death Valley, 282 feet below sea level. For your third measurement, you pick a spot out on the great plains.

Over a distance of about 700 miles, the difference in elevation is 5,972 feet. (1.131 miles) Not exactly what I would call "steep."

I agree with you that large scales mean the difference doesn't matter, but that would only be because the triangle doesn't "tilt" relative to the surface of the earth very much.  The endpoint is irrelevant here - what matters is the distance between Denver, Death Valley, and the great plains, because that will determine the tilt of the triangle, thus how far off the x,y calculation will be for one of the calculated answers.

[myrkul]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

On the scale that GPS operates at, even the Himalayas are only +1 or 2 Z, and just about everything else is effectively 0.

I don't see how this is relevant.  "Effectively 0" is not the same as 0, and will still have an effect on the calculation and subsequent answers received.

But the effect is so minimal, that it can safely be ignored. At the scale you're operating on, everything is coplanar. We don't need millimeter resolution, here.

I disagree.  The scale of the GPS is small, sure, but not necessarily the scale of the three datapoints you have and the fourth one to be discovered.

Not small. Large. The scale is extremely large. At large scales, small differences don't matter. For instance: Let's say I gave you a box tuned for some point in the continental US. To prove your point, you go to the highest point in Denver, 5,690 feet up, to take your first measurement, and for your second, you go to Death Valley, 282 feet below sea level. For your third measurement, you pick a spot out on the great plains.

Over a distance of about 700 miles, the difference in elevation is 5,972 feet. (1.131 miles) Not exactly what I would call "steep."

I agree with you that large scales mean the difference doesn't matter, but that would only be because the triangle doesn't "tilt" relative to the surface of the earth very much.  The endpoint is irrelevant here - what matters is the distance between Denver, Death Valley, and the great plains, because that will determine the tilt of the triangle, thus how far off the x,y calculation will be for one of the calculated answers.

And have you considered that since it's all pretty much coplanar, that the elevation might be ignored in all the calculations? The distance from Denver to Death Valley might be measured simply by figuring the X/Y distance, and ignoring the Z, producing not a tilted triangle, but one that's perfectly flat along the "surface" and ignores terrain features? That would result in only one target location, wouldn't it? The math to determine distance would be simpler, too. So since it not only results in a more accurate result, but is actually easier to do, don't you think that's how it would be done?

[Peter Lambert]

And have you considered that since it's all pretty much coplanar, that the elevation might be ignored in all the calculations? The distance from Denver to Death Valley might be measured simply by figuring the X/Y distance, and ignoring the Z, producing not a tilted triangle, but one that's perfectly flat along the "surface" and ignores terrain features? That would result in only one target location, wouldn't it? The math to determine distance would be simpler, too. So since it not only results in a more accurate result, but is actually easier to do, don't you think that's how it would be done?

You could test this out - go to a very steep hill, get the distance to the top and then measure it yourself. If the GPS agrees with your measurement, then they take into account z-axis changes. This might depend on how nice of a GPS unit you have?

[SgtSpike]

It's not at the same x,y...

And yes, it would be up in the air or within the earth, but that must be kept in mind when determining which x,y is the proper one.  As long as you get the point that isn't in the air or in the ground, you're golden, but how do you determine that mathematically?  It must be kept in mind.

Actually, it would be the same x,y if all three points are in the x,y plane.

Exactly.  But my whole argument was based on the fact that they AREN'T on the x,y plane (i.e, the readings are taken from different real-world elevations).  Or, at least one point is not.  In that case, you would receive two points with different x and/or y.

On the scale that GPS operates at, even the Himalayas are only +1 or 2 Z, and just about everything else is effectively 0.

I don't see how this is relevant.  "Effectively 0" is not the same as 0, and will still have an effect on the calculation and subsequent answers received.

But the effect is so minimal, that it can safely be ignored. At the scale you're operating on, everything is coplanar. We don't need millimeter resolution, here.

I disagree.  The scale of the GPS is small, sure, but not necessarily the scale of the three datapoints you have and the fourth one to be discovered.

Not small. Large. The scale is extremely large. At large scales, small differences don't matter. For instance: Let's say I gave you a box tuned for some point in the continental US. To prove your point, you go to the highest point in Denver, 5,690 feet up, to take your first measurement, and for your second, you go to Death Valley, 282 feet below sea level. For your third measurement, you pick a spot out on the great plains.

Over a distance of about 700 miles, the difference in elevation is 5,972 feet. (1.131 miles) Not exactly what I would call "steep."

I agree with you that large scales mean the difference doesn't matter, but that would only be because the triangle doesn't "tilt" relative to the surface of the earth very much.  The endpoint is irrelevant here - what matters is the distance between Denver, Death Valley, and the great plains, because that will determine the tilt of the triangle, thus how far off the x,y calculation will be for one of the calculated answers.

And have you considered that since it's all pretty much coplanar, that the elevation might be ignored in all the calculations? The distance from Denver to Death Valley might be measured simply by figuring the X/Y distance, and ignoring the Z, producing not a tilted triangle, but one that's perfectly flat along the "surface" and ignores terrain features? That would result in only one target location, wouldn't it? The math to determine distance would be simpler, too. So since it not only results in a more accurate result, but is actually easier to do, don't you think that's how it would be done?

Good point, I hadn't thought of that...!

[myrkul]

Good point, I hadn't thought of that...!

Sometimes, we overthink things, looking for the complicated answer, when all we need is the simple one. I had a call once when I was doing tech support, that the previous techs had replaced several sound cards, and a number of sets of speakers. The problem? The sound was muted.

[SgtSpike]

Good point, I hadn't thought of that...!

Sometimes, we overthink things, looking for the complicated answer, when all we need is the simple one. I had a call once when I was doing tech support, that the previous techs had replaced several sound cards, and a number of sets of speakers. The problem? The sound was muted.

Absolutely.  I appreciate the discussion.