I have a circle with radius $r$ and several rectangular objects. All rectangles have the same sides $a$ and $b$ (although the side $a$ may be different from side $b$). I would like to place the rectangles centered on the circumference, all the way around, so they don't overlap. Suppose I place the first rectangle (blue) at angel 0.
I want to place the next rectangle (green) side by side without overlapping, then the gray and so on... How do I calculate the angle $\alpha$ for each rectangle?
I don't have any issues with placing the rectangle centered on the circumference, it is the angel $\alpha$ for each rectangle that is my problem to calculate.
If things get more simple we can assume quadrants ($a$=$b$) instead of rectangles and even use the worst case as the common angle for all rectangles...
I'm aware of that there might be a gap in the end (as shown in the picture between blue and gray) and I intend to spread this gap evenly between all rectangles.
EDIT
Okay I sort of came up with one dirty solution where I use the same angle for all quadrants.
If I tilt the quadrat 45 degree (worst case), the diagonal will form a chord on the circumference. I know the radius so from this I can calculate the occupied arc angle.
$H=\sqrt{b^2*2}$
$\alpha = 2*\arcsin{(\frac{H}{2*r}})$
But I'm still looking forward to a "real" solution as the first picture shows...
Call your initial rectangle number $0$, the next number $1$ and so on. Sooner or later you are going to hit the top of the circle and not be able to place another rectangle on top of the previous one, but until that happens, the centre of rectangle $k$ will have $y$ coordinate $kb$, distance from the origin $r$, so the angle above the horizontal will be $$\sin^{-1}\frac{kb}{r}\ .$$