# Equation of an arbitrary circular arc

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.

• "Semicircles and quarter-circles are easy to derive": what do you mean ? – Yves Daoust Jul 8 at 15:24

Using your notation of $$(h,k)$$ being the center and $$r$$ being the radius, you can represent the circle as $$x=h+r\cos \theta\\y=k+r\sin \theta$$ Now allow $$\theta$$ to range as desired.