I am attempting to calculate the intersection of a spiral, defined by the parametric formula $x=\left(t\cdot\cos\left(t\right)\right),\ y=\left(t\cdot\sin\left(t\right)\right)$, and a circle defined as $(x+x_0)^{2}+(y+y_0)^{2}=r^{2}$, where $x_0$ and $y_0$ are origin points of the circle, lying on the spiral itself.
The issue is, it needs to be the point, which would have the lowest value of $t$, since I'm looking for generating a set of points on the spiral, which iteratively intersects the next point (see attached picture for more detail). My question is, how would I calculate such intersection point?
Technically, if I were able to calculate all of the points, I could go through them and see which one matches the lowest parameter, but since I'm looking for an efficient algorithm, I would like to do this as efficiently as possible.
Visual interpretation of what I'm looking for.
Image description: Circle, with origin in A, intersects the spiral in point B. Another circle, with origin in B, intersects the spiral in point C, ...
I can't see the difficulty hast put x(t1), y(t1) in the circle and find t2 with $t_1^2+t_2^2-2t_1t_2 cos(t_2-t_1)=r^2$ and of cause the smaller $t_2$ is the solution you can see in the general solution?