0
$\begingroup$

enter image description here

If you have two intersecting circles with different length radii as shown here (and adapted/borrowed from another question), is the ratio of the two intersected arc's sagitta (FE and GE) equivalent to the ratio of their radii (AG and CF)?

If not, what is the best way to determine the lengths of the individual sagitta if only the total amount of overlap (FG) and the circles' radii are known?

2 Answers 2

2

No, the ratio is not, in general, the same. If $L$ is the length of $BE$, and $R_1$ is the length of $AB$ then $$ EG = R_1\left(1 - \sqrt{1-\left(\frac{L}{R_1}\right)^2}\right) $$ while $$ EF = R_2\left(1 - \sqrt{1- \left(\frac{L}{R_2}\right)^2}\right) $$ so that the ratio is \begin{align} \rho &= \frac{R_1}{R_2} \frac{1 - \sqrt{1-\left(\frac{L}{R_1}\right)^2}} {1 - \sqrt{1-\left(\frac{L}{R_2}\right)^2}} \end{align} which differs from the ratio of the radii by a factor of \begin{align} s &= \frac{1 - \sqrt{1-\left(\frac{L}{R_1}\right)^2}} {1 - \sqrt{1-\left(\frac{L}{R_2}\right)^2}}\\ \end{align}

I got the formula for the sagitta from wikipedia.

  • 0
    Thanks, that's a great direct answer to my question. It still doesn't look like I can calculate the sagitta lengths with what you provided because _L_ cannot be calculated until I know the length of _EF_ or _EG_. Am I reading that right? – 2017-02-24
  • 0
    Nope -- you've mis-read. $L$ is half of the chord-length, and you can compute EG (which is the first sagitta) knowing only $L$ and $R_1$, and similarly for $EF$. Check out that wikipedia article for more details and more ways to compute the sagitta. – 2017-02-24
  • 0
    @John Hughes--Using Wikipedia's equation for the sagitta, shouldn't you have L^2/R^2 under the radical in your expression, since Wikipedia's c^2/4 = L^2 for you? But besides, as OP comments, L is not given as Wikipedia assumes it is. We don't know the chord length. Can it be calculated from FG and the radii? – 2017-02-27
  • 0
    @JohnHughes Thanks for the update. I spent quite a bit of time with that Wikipedia page before and after writing my question. For my particular application, I only know $FG$ and of course the two radii. I don't know the position of $E$ on $FG$. In any case your answer was VERY helpful in getting going down the right road (and keeping me from going down the wrong one). Thanks so much for your help. – 2017-02-27
  • 0
    @EdwardPorcella -- thanks; fixed the missing squares problem. You're right -- $L$ is not given. But there is some value $L$ that *is* the length of the chord, and using it lets me show that the ratio isn't what OP thought it might be. – 2017-02-28
  • 0
    @D.Woods: glad to have been of some use, and sorry my answer had to involve the unknown length $L$; I have a feeling that actually locating $E$, given what you know, is likely to be pretty tough... – 2017-02-28
  • 0
    @John Hughes--I agree that although we can't calculate your L, we don't need to in order to show that the sagittae are not proportional to the radii. In the OP's setting out of the problem, however, I was uncertain whether he was contemplating a direct or an inverse proportion between the sagittae and respective radii. So I've been thinking how to rule out both. – 2017-02-28
0

In unequal circles with a common chord, the circle with the lesser radius has the greater sagitta, since the "bow" is more bent. Therefore the sagittae are not directly proportional to the radii. Suppose they are inversely proportional. In the posted figure, extending GA and FC to meet the circles at H and K, and joining BH, BF, BG, BK, then since $$\frac{EG}{EB}=\frac{EB}{EH}$$ we have $$EG = \frac{(BE)^2}{EH}$$ Similarly we get $$EF = \frac{(BE)^2}{EK}$$ Therefore $$\frac{EG}{EF}=\frac{EK}{EH}=\frac{R_2}{R_1}$$ But suppose the overlap is such that BD is the diameter of the lesser circle. Then since $$EK=R_2$$ it follows that $$EH=R_1$$ which is false, since BD is less than the diameter of the greater circle. Therefore the sagittae are not inversely proportional to the radii.