We have a circle with the known radius $r$ and the circumference $2\pi r$, and we know a point $P_1$ somewhere on it's circumference. Now, we want to get the coordinates $[x_{P_2},y_{P_2}]$ of the point $P_2$. We know the arc between $P_1$ and $P_2$ as $d = \frac{2\pi r}{x}$ where $x$ is known and $\geq 1$.
As a matter of fact, by knowing $d$ we know the angle from the center between $P_1$ and $P_2$, but I am unable to find a formula to get me the correct coordinates of $P_2$ for any combination of known $P_1$, $r$ and $d$.
According to the law of cosines, $\cos (\theta )={{r^2+r^2-d^2}\over {2rr}}=1- {{d^2}\over {2r^2}}$.
Best Answer
Let the coordinates of $P_1$ be $(x_{P_1},y_{P_1})$. Let the angle between the points $P_1$ and $P_2$ be $\theta$. Then from the arc $d$ you get, $\theta=d/r=\frac{2\pi}{x}$. So, now, we have $$x_{P_2}=x_{P_1}+r\sin{\theta};\ \\ y_{P_2}=y_{P_1}-r(1-\cos{\theta})$$.