Math CentralQuandaries & Queries


Find the Cartesian coordinates of the center of an arc with the given location of the beginning and end points and radius length. Not the midpoint of the circumference but the actual point that the arc
is drawn around.

I know their are two answers depending on the direction of the arc. Unless we assume that all arcs are drawn counter clock wise.


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

\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.

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.


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