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I have only a basic understanding of math, so I'll try my best to describe my problem.

I have an arbitrary number of items and I measure the numerical difference between them based on a set of criteria.

Example - the resulting difference between: A & B = 4 | A & C = 5 | B & C = 2

Is it possible to plot A,B,C onto a graph? Knowing the difference between each point, how can the position of a point be calculated relative to the other points it compares to?

If I were to add D, then it'll also have a difference score to A,B,C based on a criteria. Where will D be plotted relative to A,B,C?

I assume A would have the base coordinate of [0,0], then everything else is relative to that.

I hope thing makes sense.

Jyrki Lahtonen
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sebjwallace
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1 Answers1

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Let's imagine placing the points one at a time. You're right to define $A$ as some arbitrary fixed point.

The set of all points at a given distance from a central point is a circle. So with $A$ defined, you have a circle of points that $B$ can be on. You'll need to define $B$ as one of those points (arbitrarily).

Next, you'll need to place $C$. It has to be on a given circle around $A$ and a given circle around $B$, which will intersect at two points (or one, if the distances line up nicely). You'll need to pick one (again, arbitrarily).

Assuming $A, B,$ and $C$ don't all lie on a line, each subsequent point is defined uniquely by the distances. The distances to $A$ and $B$ determine two possible points, and the distance from $C$ tells you which side of line $AB$ the new point is on.

So the distances between points uniquely determine the positions of the points, assuming three arbitrary choices are made. Or you can think of those choices as ignoring translation (given by the position of $A$), rotation (given by the position of $B$), and reflection (given by the position of $C$).

  • "... which will intersect at two points (or one, if the distances line up nicely)." Or none, if you're particularly unlucky; the OP's distance criteria could fail to have a nice geometric visualization if they don't satisfy the triangle inequality. For example, they could end up with $\overline{AB} = 10$, $\overline{AC} = 1$, $\overline{BC} = 1$. – Michael Seifert Aug 25 '16 at 18:34
  • Yes, I'm assuming the system is consistent. It's certainly possible to have distances that don't work. –  Aug 25 '16 at 18:39