A circle is defined by $(x-h)^2+(y-k)^2=r^2$ for radius $r$ and centre $(h,k)$. Semicircles and quarter-circles are easy to derive.
Is there a more general formula that will graph any portion of an arbitrary circle's circumference, e.g. the arc between $\pi/2$ and $2\pi/3$, or a three-quarters circle starting at $\pi/6$ and ending $-\pi/3$? If this question is too general, are there some parameters one can add to solve for a non-trivial special case?
I know so far that for a function $y=f(x)$, rotating by $\theta$ degrees about the origin results in the relation: $$y\cos\theta-x\sin\theta=f(y\sin\theta+x\cos\theta)$$
I'm not sure how useful this. However, it has helped me rotate semicircles.