I was recently introduced to the concept of inversion in the plane, in which a circle of inversion is chosen so that all points $A$ that can be described relative to the center of inversion $O$ have dance partners $A’$, whose length from the center of inversion is related to they length of the original point from the center by $OA\cdot OA’ = R^2$.
I was wondering whether there is a means by which circular inversion may be used to derive the hyperbolic trigonometric functions, because the equation above is a rotated hyperbola. I am new to the subject so I would appreciate any thoughts you might have on this.