- Homework Statement
- Imagine you have five points on a flat plane. Name the points A, B, X, Y and Z. The position is arbitrary but fixed. The points X, Y and Z act as reference points. Points A and B are just some other points. Imagine now, that you know the following distances: The distances between the reference points X, Y and Z: XY, XZ, YZ. The distance of each reference point to points A and B: XA, YA, ZA, XB, YB, ZB.
Calculate the distance between points A and B.
- Relevant Equations
the following problem is not a home-work problem. I have come up with this question for myself. Nevertheless, I am stuck and need your help.
The question is: Can I calculate the distance between points A and B from this information? And if yes, how?
I think it should be possible. However, I do not know how to go about it.
These are my considerations so far:
The points X, Y, and Z form a kind of reference frame. Every point P on the plane is described by a unique combination of distances between the point and the three other points.
However, the difference is the following: In a coordinate system with two axes x and y, any combination of values for x and y will always return a valid point. In the case of the reference frame X, Y, and Z only very special combinations of three numbers/distances describe a point. Geometrically this can be shown as three circles around the points X, Y, and Z. Only those combinations of circles, that intersect in one point, define one specific point P. All other combinations do not describe a point. Inversely this can be drawn as three concentric circles around the point P where each of the circles goes through one of the fixed points.
In a cartesian coordinate system, I can also add this extra parameter in the form of the distance between the origin (0,0) (or any other arbitrary and the point P (x,y) that I want to describe. However, this additional distance adds no further information.
In a cartesian coordinate system, I can convert the two-dimensional coordinates (x,y) of a point P into three distances. A simple example would be:
- the distance between the P and the origin (0,0)
- the distance between P and the point (1,0)
- the distance between P and the point (0,1)
However, in my problem, the distances between the three points X, Y, and Z in the basic problem are not (necessarily) 1 and there are not (necessarily) any right angles.
It is also possible to write all those distances in the form of vectors and matrices, however, this gave me no further idea.
I also had the hunch to I think of the distances as radii of circles. If I do so, I can subtract the areas of those circles and calculate the radii of the resulting circles. However, by trying to do so I got lost and none of the results I got was meaningful. But I still think this is the way to go…
I think this problem cannot be new, but I have no idea where to look for a solution and I am stuck with my own ideas.