Here I have this loop, made of parts of two different circles with radiuses $r_1$ and $r_2$, joined with two lines intersecting at $90$ degrees and touching the circles only in one point, as shown in the picture below. How to calculate the length of a way from one point on the loop back to the same point?
Thanks in advance!
With a right angle in the middle, connecting the centers of the circles to the points of tangency produces two squares. Then you can see that the distance $s$ all around the figure eight is $3/4$ of each circle plus twice the sum of the radii: $$ s = \left(\frac{3 \pi}{2} + 2 \right)(r_1+r_2) = C(r_1+r_2). $$ If you know $s$ and $r_1$ then $$ r_2 = \frac{s - Cr_1}{Cr_2}. $$
Note how that requires $s > Cr_1$. When $s = Cr_1$ the second circle is a point and there's a right angle corner in the figure.