This is my first time posting so I hope my formatting is correct.
Consider this, I have two circles, one big one small with radius $r_1$ and $r_2$. The borders of both circles are touching. See image:
Correct me if I'm wrong, I believe the angle from the center of the big circle is $2\arcsin\left(\dfrac{r_2}{r_1-r_2}\right)$
What I am actually interested in is subtracting the smaller circle from the larger circle, making a small channel like this:
Is there an expression where I can find the radius of the bigger circle to any point of the arc of the channel? For the shortest distance is easy, basically just $r_1-2r_2$.
But what about all the other points? How do I go about calculating the distance to any point on the arc of the channel? I can approximate it from the middle and approximate triangles within small steps but if there is a mathematical expression for it, that would be great. The ideal expression would have $r_1,r_2,\theta$
Thank you
Best Answer
In reference to this image
and complementing other answers, the points on the green arc are represented by the equation $$ r = (r_1-r_2)\cos\theta-\sqrt{r_2^2-(r_1-r_2)^2\sin^2\theta},\qquad|\theta|\leq\arcsin\left(\frac{r_2}{r_1-r_2}\right), $$ while the points on the red arc are represented by the equation $$ r = (r_1-r_2)\cos\theta+\sqrt{r_2^2-(r_1-r_2)^2\sin^2\theta},\qquad|\theta|\leq\arcsin\left(\frac{r_2}{r_1-r_2}\right). $$ In particular, the points on the arc from $A$ to $B$ are represented by the second of previous equations with $$ \arctan\left(\frac{r_2}{r_1-r_2}\right)\leq\theta\leq\arcsin\left(\frac{r_2}{r_1-r_2}\right). $$