Is there a pure algebraic way to calculate intersection of two disks (extended to spheres, ellipses)?
I think what you mean is a parametric representation. Suppose the circles $C_1$ and $C_2$ intersect at points $P_1$ and $P_2$. Let $f$ be inversion around $P_1$. This maps the circles to straight lines $L_1 = f(C_1)$ and $L_2 = f(C_2)$, and the intersection of the disks to one of the four regions into which these lines divide the plane. This can be parametrized by $f(P_2) + r (\cos(\theta),\sin(\theta))$, $0 \le r < \infty$, $\theta_1 \le \theta \le \theta_2$. Invert back and you get the intersection of the disks.