I'm working on a gear outline that uses involute of a circle. I want to find the intersection (I) between an involute (D) of radius r1 and a circle (C) of radius r2 (where u,v in ]0;pi/2[) :
D : xi = r1 * (cos(u) + u*sin(u))
yi = r1 * (sin(u) - u*cos(u))
C : xi = r2 * cos(v)
yi = r2 * sin(v)
Thus the equation to calculate I coordinates is:
r1 * (cos(u) + u*sin(u)) = r2 * cos(v)
r1 * (sin(u) - u*cos(u)) = r2 * sin(v)
Help is wanted to resolve this equation.
Ok for the parametric equations of the involute:
But for the circle, don't use parametric equations ; express plainly that you are looking for the (unique) point of the evolute that is at distance $r_2$ from $0$, i.e., constraint $x_i^2+y_i^2=r_2^2$
$$r_1^2 [(\cos(u) + u*\sin(u))^2+ (\sin(u) - u*\cos(u))^2]=r_2^2$$
Dividing by $r_1^2$, setting $r:=r_2/r_1$ and expanding, one obtains the following result (using $\sin^2u+\cos^2u=1$...):
$$1+u^2=r^2 \ \ \implies \ \ u=\sqrt{r^2 - 1} \ \ \ \ \ ( \text{we assume} \ 1 \leq \ r \ \ \iff \ \ r_1 \leq r_2 )$$
(because $u >0$). It suffices now to plug this value of $u$ into the equations of the involute to get the intersection point, if any (because the other condition $u < \pi/2$ has to be fulfilled).