Sorry if this is phrased weird, I don't have background in math whatsoever.
Is there a way to find the function $F$ that gives you Cartesian coordinates $(x, y, z)$ with just the knowledge of $F(t) - G(t) = \text{given range}$ i.e. $\sqrt{(x_f - x_g)^2 + (y_f - y_g)^2 + (z_f - z_g)^2} = R$; and $F'(t) - G'(t) = 0$ i.e. at the minimum range? They are both ellipses if that helps. I have the initial $x, y, z, \dot{x}, \dot{y}$, and $\dot{z}$ of $F(t)$ at $t_0$ and all of the data on $G$ including $\dot{x}$, $\dot{y}$, and $\dot{z}$. The $\dot{x}$, $\dot{y}$, and $\dot{z}$ are going to change to form a new ellipse that has the minimum distance equal to the given range. I am realizing that it might be a family of functions that will get that answer or a sphere of possible $x, y, z$ coordinates at the minimum time. I wasn't sure if it could be calculated so I can optimize that sphere down to a point based on the minimum amount of change from the $\dot{x}$, $\dot{y}$, and $\dot{z}$ of $F(t)$. If the 3-Dimensional solution doesn't work but a 2-Dimensional one would then I'll take that too.