Hi Ken,

Suppose the ends of the arc are $P_1 = \left(x_1< y_1\right)$ and $P_2 = \left(x_2< y_2\right)$ and the radius is $r.$

I know how to do this geometrically. Draw a circle with center $P_1$ and radius $r$ and a second circle with center $P_2$ and radius $r.$ These circle intersect at two points and these are the points you are seeking. Let's try to describe this process algebraically.

The two circles are

\begin{eqnarray*}
\left(x - x_1\right)^2 + \left(y - y_1\right)^2 &=& r^2\\
\left(x - x_2\right)^2 + \left(y - y_1\right)^2 &=& r^2.
\end{eqnarray*}

Subtract the two equations to obtain

\[ \left[ \left(x - x_1\right)^2- \left(x - x_2\right)^2 \right] + \left[ \left(y - y_1\right)^2- \left(y - y_2\right)^2 \right]= 0\]

Expand each expression inside square brackets as a difference of squares and simplify. This will result in a linear equation in $x$ and $y.$ Solve for $y$ and substitute into the equation for the first circle. The result is a quadratic equation in $x$ which you can solve to yield the $x$-coordinates of the two points you are seeking. Substitute these values into the linear equation relating $x$ and $y$ to obtain the $y$ values.

Harley