I need help checking my work. I'm working on a representation of multiple regression effect sizes using a Venn Diagram for educational policy research. I see that the area of a geometric lens has a closed form solution. Given a Venn diagram made from circles ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$with centers ${\displaystyle X}$, ${\displaystyle Y}$, and ${\displaystyle Z}$, that ${\displaystyle r=R={\sqrt {1/\pi }}}$, and ${\displaystyle A_{A\cap B}=0.23}$, then ${\displaystyle d_{XY}\approx 0.429}$. Can someone validate this? The Venn diagram is represented at this link here.Schyler (exquirito veritatem bonumque) 04:07, 21 December 2022 (UTC)[reply]

Circle ${\displaystyle C}$has no role. I cannot replicate these numbers. When each circle passes through the centre of the other circle, so the circle centres are ${\displaystyle r\approx 0.56419}$apart, a rhombus of two equilateral triangles fits within the lens. The area of this rhombus is ${\displaystyle {\tfrac {\sqrt {3}}{2\pi }}\approx 0.27566,}$so when ${\displaystyle d_{XY}=0.429<r}$, ${\displaystyle A_{A\cap B}>0.27566,}$contradicting ${\displaystyle A_{A\cap B}\approx 0.23\,.}$My calculations give me that distance ${\displaystyle d_{XY}=0.429}$gets you ${\displaystyle A_{A\cap B}\approx 0.52785\,.}$Conversely, to get lens area ${\displaystyle A_{A\cap B}=0.23\,,}$I find we need ${\displaystyle d_{XY}\approx 0.73938\,.}$ --Lambiam 07:38, 21 December 2022 (UTC)[reply]

Yes, circle ${\displaystyle C}$has no role. Let ${\displaystyle \theta }$be the angle at X (or Y) between the lines to the junctions of ABD and of CEFG. The lens has area ${\displaystyle A=R^{2}\left(\theta -\sin \theta \right)}$, where ${\displaystyle R={\sqrt {1/\pi }}}$, and the diagram says ${\displaystyle A_{A\cap B}}$(D+G) has area 0.03, so ${\displaystyle \theta -\sin \theta =0.03\pi }$. That gives ${\displaystyle \theta \approx 0.8235}$(in radians). The length from X (or Y) to the midpoint of XY is ${\displaystyle R\cos(\theta /2)}$, and twice this gives ${\displaystyle d_{XY}\approx 1.0341}$. My trig is rusty and I may have made some errors, but I think the principle and the order of magnitude are right. If ${\displaystyle A_{A\cap B}}$were 0.23, we'd get ${\displaystyle \theta \approx 1.1377}$and ${\displaystyle d_{XY}\approx 0.9507}$. Certes (talk) 13:31, 21 December 2022 (UTC)[reply]

For ${\displaystyle \theta =0.8235,}$we have ${\displaystyle (\theta -\sin \theta )/\pi \approx 0.02864,}$not quite ${\displaystyle 0.03}$but at least in the ballpark. But in the second case, you missed a factor ${\displaystyle \pi }$: for ${\displaystyle \theta =1.1377,}$we have ${\displaystyle \theta -\sin \theta \approx 0.23}$while ${\displaystyle (\theta -\sin \theta )/\pi \approx 0.07322.}$ --Lambiam 15:52, 21 December 2022 (UTC)[reply]

Here are the calculations for lens area ${\displaystyle 0.23,}$step by step:

${\displaystyle {\begin{array}{cccc}\theta &=&1.71254\\\sin \theta &=&\sin 1.71254&\approx &0.98997\\\theta -\sin \theta &\approx &1.71254-0.98997&\approx &0.72257\\(\theta -\sin \theta )/\pi &\approx &0.72257/3.14159&\approx &0.23000\\\cos(\theta /2)&\approx &\cos 0.85627&\approx &0.65526\\2R\cos(\theta /2)&\approx &2\times 0.56419\times 0.65526&\approx &0.73938\\\end{array}}}$

--Lambiam 16:03, 21 December 2022 (UTC)[reply]

Oops. Thanks, that looks more credible. Certes (talk) 18:40, 21 December 2022 (UTC)[reply]

Okay, well this is interesting. Yes, in the example, circle C has no role here. Someone said "When each circle passes through the centre of the other circle," but I do not think that is probable. Here were my steps:

Circles ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$have centers ${\displaystyle X}$, ${\displaystyle Y}$, and ${\displaystyle Z}$and radii ${\displaystyle r_{A}=r_{B}=r_{C}={\sqrt {1/\pi }}}$. The centers form ${\displaystyle \triangle XYZ}$and intersect such that ${\displaystyle Area_{A\cap B}=0.03}$, ${\displaystyle Area_{B\cap C}=0.11}$, ${\displaystyle Area_{A\cap C}=0.23}$

${\displaystyle d_{1}={\overline {\rm {XY}}};d_{2}={\overline {\rm {XZ}}};d_{3}={\overline {\rm {YZ}}}}$

${\displaystyle Area_{A\cap C}=r_{A}^{2}\cos ^{-1}({\frac {d_{1}^{2}+r_{A}^{2}-r_{C}^{2}}{2d_{1}r_{A}}})+r_{C}^{2}\cos ^{-1}({\frac {d_{1}^{2}+r_{C}^{2}-r_{A}^{2}}{2d_{1}r_{C}}})-2\Delta }$

${\displaystyle \Delta ={\frac {1}{4}}{\sqrt {(-d_{1}+r_{A}+r_{B})(d_{1}-r_{A}+r_{B})(d_{1}+r_{A}-r_{B})(d_{1}+r_{A}+r_{B})}}}$

Simplifying:

${\displaystyle {\cancel {0.23=2({\frac {1}{\pi }}\cos ^{-1}({\frac {d_{1}^{2}}{2d_{1}{\sqrt {\frac {1}{\pi }}}}}))-{\frac {1}{2}}{\sqrt {(-d_{1}+2{\frac {1}{\pi }})(d_{1}+2{\frac {1}{\pi }})}}}}}$

oh I see this was my problem, I think... I distributed the 2 onto ${\displaystyle {\frac {1}{4}}\Delta }$

${\displaystyle {\cancel {0.23=2({\frac {1}{\pi }}\cos ^{-1}({\frac {d_{1}^{2}}{2d_{1}{\sqrt {\frac {1}{\pi }}}}}))-{\frac {1}{4}}{\sqrt {(-d_{1}+2{\frac {1}{\pi }})(d_{1}+2{\frac {1}{\pi }})}}}}}$

Update: well, here i am checking my own work again. i found another error. i deleted some values of d within ${\displaystyle \Delta }$. I should know better than to ask for help right away... i can do this... but the doubt is strong in this one

${\displaystyle {\cancel {0.23=2({\frac {1}{\pi }}\cos ^{-1}({\frac {d_{1}^{2}}{2d_{1}{\sqrt {\frac {1}{\pi }}}}}))-{\frac {1}{4}}{\sqrt {(-d_{1}+2{\frac {1}{\pi }})(d_{1}+2{\frac {1}{\pi }})}}}}}$

no that's wrong too, ${\displaystyle r={\sqrt {\frac {1}{\pi }}}}$... i got it mixed up in ${\displaystyle \Delta }$again...

You can shift two equal-sized circles such that each passes through the other's centre; I used this special case merely to easily establish bounds that were violated by a purported solution.

The function ${\displaystyle f(\theta )=(\theta -\sin \theta )/\pi }$is transcendental, and one cannot hope to solve the equation ${\displaystyle f(\theta )=x}$by a combination of algebraic and trigonometric manipulations for rational values of ${\displaystyle x,}$except when ${\displaystyle x}$is an integer, in which case the equation is solved by ${\displaystyle \theta =x\pi .}$ --Lambiam 09:01, 22 December 2022 (UTC)[reply]