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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.

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

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