Given two points $p1$ and $p2$ on a circle of radius $R$ and knowing the following:
- Arc length $L$
- $p1$ (which is simply the origin at 0,0)
- The $y$ coordinate of $p2$ (but not the $x$ coordinate)
Is there an unambiguous way to calculate both ${\theta}$ and $R$?
This initially seemed like a trivial trigonometry problem but not knowing the $x$ coord of p2 has me stumped.
(The motivation for the problem is an algorithm for plotting a representation of a pipe of known length, which is effectively the arc length L, by a given deflection, which is effectively the y coord of $p2$.)
That problem has no solution.
Suppose that $R=1$, that the center of the circle is located at $(0,1)$, that $p_2=(0,2)$, that $\theta=\pi$, and that $L=\pi$.
And now suppose that $R=2$, that the center of the circle is located at $(0,2)$, that $p_2=(2,2)$, that $\theta=\frac\pi2$, and that $L=\pi$.
As you can see (see the picture below), you have the same $L$ in both cases, and you also have the same $y$-coordinate for $p_2$ in both cases. But you have distinct $\theta$'s and distinct $x$-coordinates of $p_2$.