I'm attempting to solve a problem similar to trilateration but with slightly different data.
I want to find the center of a circle (in my case, its coordinate is the origin of a sound wave). Let's call the center of this circle $c$.
Imagine then I have a set of three points $p1...p3$ which are equidistant from each other, but all different distances from the origin of the sound, $c$.
Imagine a circle travelling outward from $c$ (this is the sound wave propagating). It will "hit" one of these points first - let's call this point $p_1$. This point then, by definition, intersects with the circle propagating from $c$ at this point in time.
Imagine for the sake of this explanation that this circle then stops propagating outwards, and instead, circles begin propagating from each of the remaining points $p_2$ and $p_3$ until these circles lie tangentially on the circle centered on $c$ at the moment that it intersects with $p_1$.
I'm hoping that this can be visually intuited with the graph below (The green point is $c$, the point in the bottom right is $p_1$, and the other two points whose circles lie tangentially to the circle centered on $c$ are naturally $p_1...p_2$):
In actuality though, I only have a subset of the data depicted above known to me. That subset is:
- The coordinates of $p_1$, $p_2$, and $p_3$.
- The radii of the circles centered on $p_2$ and $p_3$.
My question: with only these known quantities, is it even possible for me to deduce the center of $c$? If so, how?
I've tried following trilateration tutorials, done a fair bit of article reading, but this seems like a weird sub-field / nuanced problem that seems to be preventing the information from applying completely.
Any help would be much appreciated. Thank you!