Area of intersection of two general overlapping circles

Question

My algebra is letting me down here, I can't figure out how to arrange this equation - anyone prepared to give me a hand?

The area of the intersection of two circles can be defined as $$A = r^2 \cos^{-1}\left(\frac{d}{r}\right) - d \sqrt{r^2 - d^2} + R^2 \cos^{-1}\left(\frac{d}{R}\right) - d \sqrt{R^2 - d^2}$$ Where:

$r$ is the radius of circle one
$R$ is the radius of circle two
$A$ is the area of their intersection
$d$ is the distance between the two circles' centres.

But, if I have numerical values for all of these variables except $d$, how can I reform the equation to find it?

Something is wrong with your formula. For $d=r=R$ it makes the area $0$. — Robert Israel, Mar 25 '13 at 09:12
The actual formula is equation $(14)$ or maybe $(10)$ of the question’s link — Тyma Gaidash, Jun 17 '23 at 13:10
If someone find this question, then equation $(10)$ requires inverting $x-\cos(x)\sin(x)$ which is possible with inverse of $\sin(x)+x$ — Тyma Gaidash, Jun 17 '23 at 13:28

score 2 · Accepted Answer · answered Mar 25 '13 at 07:50

2

Don't try to do this analitically. You can find $d$ only by a numerical method (http://en.wikipedia.org/wiki/Root-finding_algorithm).

answered Mar 25 '13 at 07:50

Boris Novikov

17,470

Ah ha! That'd explain it. Care to expand with recommendations as to which method(s) I should look at? – JP. Mar 25 '13 at 09:10
1

Sorry, I'm not a specialist in numerical methods. I think you will find experts in this site. But first, I would advise to transform your question in such a way: There is an equation $$B = F(\frac{d}{r})+ F(\frac{d}{R})$$ where $$F(x)=(x^2 \cos x)^{-1}-\sqrt{x^{-2} - 1}$$ (and besides $B=Ad^{-2}$). Which is an appropriate numerical method to find $r$? – Boris Novikov Mar 25 '13 at 10:54

Area of intersection of two general overlapping circles

1 Answers1

Linked