Knowing the distance from origin of two points, can you infer their distance from each other?

Question

This would be in a singular quadrant, so the coordinates of both points would be positive. I am wondering if, given two points and both of their distances from the origin, could you evaluate their distance from each other?

Say I have Point A, and Point B.

I know Point A has positive coordinates, and is X distance away from the origin.

Point B also has positive coordinates, and is distance Y from the origin.

I don't want to actually calculate the Euclidean distance between the points, but rather infer if Point A and Point B are within some variable Epsilon of each other. Given only this information, is it possible to find a solution?

Answer 1

Alternative perspective:

The Law of Cosines is controlling here.

Assume that points A and B are in $\Bbb{R^2}$, let O denote the origin, whose coordinates are $(0,0)$, and consider $\triangle$ ABO.

Let $a,b,c$ denote the lengths of the respective line segments $~\overline{AO}, ~\overline{BO}, ~$ and $~\overline{AB}.$

Let $\theta$ denote $\angle$ AOB.

Then:

• $a,b~$ are known.
• $c~$ and $~\theta~$ are unknown.
• $c^2 = a^2 + b^2 - 2ab[\cos(\theta)].$

Therefore, $~c~$ can not be determined unless $~(\theta)~$ is specified.

Answer 2

Hint

Geometrically, $A$ can be located anywhere in the first quadrant of a circle with center at zero and radius $X$, and $B$ - on a quarter circle centered at the origin with radius $Y$.

You need to find places where these quarter circles are closest and farthest from each other. Does this help?

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Speculation

Didn't prove this conclusively but it seems very geometrically motivated that the max distance is achieved when one lies on the $x$-axis and the other on the $y$-axis, which would cause the max distance to be from $(0,X)$ to $(Y,0)$, a total of $\sqrt{Y^2+X^2}$.

Answer 3

To picturize, consider the following image:

Here, If you are given that point $A$ is at a distance of $2$ units from the origin and point $B$ is at a distance of $4$ units from the origin, then from the figure, the point $A$ may lie anywhere on the arc $BC$, which is a part of a circle with radius $2$ units from the origin while the point $B$ may lie anywhere on the arc $DE$ (part of circle of radius $4$ units).
Assume point $A$ is at $F$ then if point $B$ is at $G$ then distance between $A$ and $B$ is $2$ units. If $B$ were at $H$ or $E$ then the distance would have been $2.32$ units and $2.96$ units respectively.

Thus, as pointed out by user2661923, unless we don't know the angle made by the line segment joining the $2$ points, we can't determine the distance between the $2$ points.