Finding a point on a circle that has a distance L (arc length) from another point

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Given the coordinates of a single point on a circle and a length of an arc $L$, how do I find the coordinates of another point?

Or, to put in another form: I have the radius $r$, the length of the arc $L$ and $(x_1,y_1)$ the coordinates. I need to express $(x_2,y_2)$ using only $r, L, x_1$, and $y_1$.

I'm at a dead end on this.

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Traveling along a fixed circle, is basically rotating a vector around a fixed point. For that we can use the well known rotation matrix $\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$. But to succesfully do this you need to know the center of you circle first, like ccorn mentioned