$$\begin{align} x^2+(z-c)^2 &= a^2\\y^2+(z+c)^2 &= b^2\end{align}$$

$$\begin{align} x &= acos(t)\\y &= \pm\sqrt{b^2-(2c+asin(t))^2}\\z &= c+asint(t)\end{align}$$