I have two circles which intersect in such a way that the overhang of the smaller circle is a percentage of that smaller circle's radius.
How do I define the function of the radius of the smaller circle, X, in terms of the angle from the center of the larger circle whose rays run tangent to it, Θ. Picture of the geometry attached.
I assume $R$ is the radius of the big circle. We have a right triangle where one of the acute angles is $\theta$ and leg opposite to $\theta$ is $x$. All we need to do is to find the hypotenuse $h$. We can see that $h=R+kx-x=R+(k-1)x$ which gives us the function for $$\sin \theta=\frac{x}{R+(k-1)x}$$ or $$\theta=\arcsin \frac{x}{R+(k-1)x}$$