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.
 A: 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:
$p2.x = cp.x + r * cos (\alpha +\theta )$
$p2.y = cp.y + r * sin (\alpha +\theta )$

A: I see that it has been nearly five years since this problem was active, but perhaps the following answer will be useful to someone. A similar question asked more recently, and a longer and rather drawn out version of my answer that follows, were posted here: enter link description here.
What is requested is a relation between the coordinates of a point before and after a rotation in the plane by an angle $\theta$ about an axis through some center point $P_0(x_0,y_0)$. In this Figure I have tried to illustrate the geometry, labeling the initial point $P_1(a,b)$, and the final point $P_2(x, y)$. The distance $r$ from the axis of rotation to either point is unchanged by the rotation, i.e., the rotation traces a circular arc (of length $s = r\theta$) between the two points.

The rectangular coordinates of $P_2$ with respect to $P_0$ are $x-x_0$ and $y - y_0$, which are related to its polar coordinates $(r,\theta_2)$ by $$\begin{align}x -x_0 &= r\cos\theta_2,\\y-y_0 &= r\sin{\theta_2}.\end{align}$$ However, we see that $\theta_2 = \theta_1 + \theta$, so we have, equivalently, $$\begin{align}x -x_0 &= r\cos(\theta_1 + \theta) = r\,(\cos{\theta_1}\cos{\theta} - \sin{\theta_1}\sin{\theta}),\\y-y_0 &= r\sin{(\theta_1 + \theta)} = r\,(\sin{\theta_1}\cos{\theta} + \cos{\theta_1}\sin{\theta}),\end{align}$$ using trigonometric identities for the cosine and sine of the sum of two angles. But on the right-hand sides of the last pair of equations, we see that $r\cos{\theta_1} = a - x_0$ and $r\sin{\theta_1} = b - y_0$ are the coordinates of point $P_1$ with respect to $P_0$, which provides the relation requested between the coordinates of the two points: $$\begin{align}x - x_0 &= (a - x_0)\cos{\theta} - (b - y_0)\sin{\theta}, \\ y - y_0 &= (a - x_0)\sin{\theta} + (b - y_0)\cos{\theta}, \end{align}$$ hence the coordinates of the final point are $$\begin{align}x = x_0 + (a - x_0)\cos{\theta} - (b - y_0)\sin{\theta}, \\ y = y_0 + (a - x_0)\sin{\theta} + (b - y_0)\cos{\theta}, \end{align}$$ requiring only the coordinates of the center point and initial point, and the rotation angle $\theta$ (the radius $r$ of the circular arc is not necessary for the calculation).
