# Given one endpoint on an arc of a circle and the radius and arc angle, how to calculate the other endpoint of the arc?

I have a circle with an arc beginning at point $(x,y)$. The radius is $r$, the arc angle(w/ respect to center) is $\theta$. How do I calculate the end point of the arc $(a,b)$ ?

I know that the arc-length=radius*(arc angle)

I can't seem to find an easy way to solve this, I think the way to go is with parametric equations but I'm not sure.

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With the given information, all you can say is that $(a,b)$ is at a distance of $2r\sin(\theta/2)$ from $(x,y)$. What you need to pin the point down is some information about direction, for example: the slope of the tangent to the arc at $(x,y)$, or the slope of the line segment joining $(x,y)$ and $(a,b)$, or the angle with respect to the $x$-axis instead of the slope of either; that sort of thing. Do you see why? –  Rahul Narain Feb 6 at 5:41
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## 1 Answer

One way is to calculate the angle to the first point

$\alpha = \arctan \left ( \frac{p1.y-cp.y}{p1.x-cp.x} \right )$

Then you add your angle and calculate the new point:

$p2.x = cp.x + r * cos (\alpha +\theta )$

$p2.y = cp.y + r * sin (\alpha +\theta )$

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