Do an axis-aligned rectangle and a circle overlap? Given a circle of radius $r$ located at $(x_c, y_c)$ and a rectangle defined by the points $(x_l, y_l), (x_l+w, y_l+h)$ is there a way to determine whether the the two overlap? The square's edges are parallel to the $x$ and $y$ axes.
I am thinking that overlap will occur if one of the rectangle's corners is contained in the circle or one of the circle's circumference points at ($n\frac{\pi}{2}, n=\{0,1,2,3\}$ radians) is contained in the rectangle. Is this true?
EDIT: One answer has pointed out a case not covered by the above which is resolved by also checking whether the center of the circle is contained.
Is there a method which doesn't involve checking all points on the circle's circumference?

No. Imagine a square and enlarge its incircle a bit. They will overlap, but wouldn't satisfy neither of your requirement.
Unfortunately, you have to check all points of the circle. Or, rather, solve the arising inequalities (I assume you are talking about filled idoms):
$$\begin{align}  (x-x_c)^2+(y-y_c)^2 & \le r \\
x\in [x_l,x_l+w]\  &\quad y\in [y_l,y_l+h]
\end{align}$$
Or.. Perhaps it is enough to add to your requirements, that the center of the circle is contained in the rectangle.
