<p>I think that "<a href="http://en.wikipedia.org/wiki/Trilateration" rel="nofollow">Trilateration</a>" should solve your problem.</p>

<p>In geometry, <em>trilateration</em> is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and navigation, including global positioning systems (GPS). In contrast to triangulation it does not involve the measurement of angles. ...</p>

CC BY-SA 3.0
2012-02-08T15:39:05.737Z
Interesting. Thank you for putting a name for the process. I will investigate this term further for application toward my problem.
Mat Kelly
CC BY-SA 3.0
2012-02-08T11:02:44.260Z
2012-02-08T11:02:44.260Z
Marzio De Biasi
<p>I am attempting to resolve a problem of inferring Cartesian position from distance. I have a set of nodes arbitrarily but statically positioned on a 2-D plane. Every node is aware of its position relative to the other $n-1$ nodes. From this set of $n$ nodes, is it possible to infer each node's position ($x$,$y$) by choosing any node and using it as a basis?</p>

<p>For example, with two nodes, this is not possible, as the distance alone leaves open the possibility of the node not chosen as the basis for being located anywhere in the radial distance from the basis node.</p>

<p>With three nodes, it also seems impossible because the orientation can be flipped and is thus ambiguous based solely on distance. How many nodes are needed to unambiguously infer the position of the elements on a Cartesian plane?</p>

CC BY-SA 3.0
2012-02-07T16:39:11.553Z
I am not very clear about your problem. You have $n$ nodes, one of them is fixed. You can rotate the $n$ nodes around the fixed node, so no matter how many nodes you have, it is impossible.
Peng Zhang
CC BY-SA 3.0
2012-02-07T17:21:24.793Z
I can rotate them, yes, but I hope to obtain a coordinate system derived from these distances. I can rotate them and have the coordinates be at the new location but each inter-coordinate distance will remain the same. Is there some way to determine the layout of these coordinate unambiguously? If so, how many nodes are needed to accomplish this?
Mat Kelly
CC BY-SA 3.0
2012-02-07T19:23:54.957Z
You can't ever do this *completely* unambiguously. If all you have is distances, you can always reflect across some axis or rotate the set of points or translate them and the distances won't change. -- If you want uniqueness mod rigid transformations like this, then all you need is three points in general position. Then I think you can triangulate all the others.
Josephine Moeller
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2012-02-07T19:25:59.353Z
@Mat Oh, by the way, what's your research interest in this question? It would be nice if you would edit your question to reflect that.
Josephine Moeller
CC BY-SA 3.0
2012-02-08T07:03:05.813Z
Adding to what @JohnMoeller said - given 3 points, you can easily deduce that they are in general position from the distances: the triangle inequality will be sharp.
Shir
CC BY-SA 3.0
2012-02-08T22:10:46.943Z
It would still be nice if you edited your question to reflect your research interest in the problem.
Josephine Moeller
CC BY-SA 3.0
2012-02-13T13:41:10.787Z
@JohnMoeller, I am more than willing to do so but do not think that it adds anything useful to the question. I tried to keep it relatively abstract so it might be useful to others without causing one to get distracted by the application of the above question.
Mat Kelly
<ds.algorithms><graph-theory>
Inferring Cartesian position from a set of nodes where only distance is known
Mat Kelly
# Inferring Cartesian position from a set of nodes where only distance is known

(Asked by: username_0 on 2012-02-07)

I am attempting to resolve a problem of inferring Cartesian position from distance. I have a set of nodes arbitrarily but statically positioned on a 2-D plane. Every node is aware of its position relative to the other $n-1$ nodes. From this set of $n$ nodes, is it possible to infer each node's position ($x$,$y$) by choosing any node and using it as a basis?

For example, with two nodes, this is not possible, as the distance alone leaves open the possibility of the node not chosen as the basis for being located anywhere in the radial distance from the basis node.

With three nodes, it also seems impossible because the orientation can be flipped and is thus ambiguous based solely on distance. How many nodes are needed to unambiguously infer the position of the elements on a Cartesian plane?

**Tags:** ds.algorithms, graph-theory
> **Comments:**
> **username_1**: I am not very clear about your problem. You have $n$ nodes, one of them is fixed. You can rotate the $n$ nodes around the fixed node, so no matter how many nodes you have, it is impossible.  
> **username_0**: I can rotate them, yes, but I hope to obtain a coordinate system derived from these distances. I can rotate them and have the coordinates be at the new location but each inter-coordinate distance will remain the same. Is there some way to determine the layout of these coordinate unambiguously? If so, how many nodes are needed to accomplish this?  
> **username_2**: You can't ever do this *completely* unambiguously. If all you have is distances, you can always reflect across some axis or rotate the set of points or translate them and the distances won't change. -- If you want uniqueness mod rigid transformations like this, then all you need is three points in general position. Then I think you can triangulate all the others.  
> **username_2**: @Mat Oh, by the way, what's your research interest in this question? It would be nice if you would edit your question to reflect that.  
> **username_3**: Adding to what @JohnMoeller said - given 3 points, you can easily deduce that they are in general position from the distances: the triangle inequality will be sharp.  
> **username_2**: It would still be nice if you edited your question to reflect your research interest in the problem.  
> **username_0**: @JohnMoeller, I am more than willing to do so but do not think that it adds anything useful to the question. I tried to keep it relatively abstract so it might be useful to others without causing one to get distracted by the application of the above question.  

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## 1 answers

### Answer by username_4 on 2012-02-08. Score: 2 | Accepted answer

I think that "[Trilateration](http://en.wikipedia.org/wiki/Trilateration)" should solve your problem.

In geometry, *trilateration* is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and navigation, including global positioning systems (GPS). In contrast to triangulation it does not involve the measurement of angles. ...

> **Comments:**
> **username_0**: Interesting. Thank you for putting a name for the process. I will investigate this term further for application toward my problem.  

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