Trying to establish a connection between the 2 rotating quadrants

[bugatti79]

TL;DR Summary

establish connection between the 2 rotating quadrants

Hello All,

I struggled to find a closed form solution connecting the 2 quadrants via the angles alpha and beta. I filled the schematic with right angle triangles hoping to find a connection but to no avail.
In the end, I decided to create a polynomial function of beta in terms of alpha as I know the coordinates of the vertices. Could someone confirm that a closed form solution is not possible? Is there any way I could use a custom analytic function of beta in terms of alpha? Thanks!

 

[TSny]

I find the following relation between $\alpha$ and $\beta$: $$\sin \beta = \frac 1 {K_2} \left( 1 - \cos \alpha + \sin \alpha \right).$$ See if you can confirm this using the method sketched below.

In the figure, $L$ and $L_2$ are the lengths of the chords of the two quadrants: $L = \sqrt 2 R$ and $L_2 = \sqrt 2 K_2 R$.

$\theta$ and $\phi$ denote the angles that the two chords make with respect to the horizontal. You can show that $\theta = \pi/4 - \alpha$ and $\phi = \pi/4 - \beta$.

Thus, $\sin \theta = (\cos \alpha – \sin\alpha)/\sqrt 2$ and $\sin \phi = (\cos \beta – \sin\beta)/\sqrt 2$.

$A, B, C, D,$ and $E$ label various points. Note that $$AC = AB + BC = AB + BD +DE -CE$$ where

$AC = L \sin \theta$

$AB = 2K_1R$

$BD = K_2R(1-\cos \beta)$

$DE = L_2 \sin \phi$

$CE = R – K_2 R$

Use these relations along with $K_1 + K_2 = 1$.