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A stealth missile $M$ is launched from space station. You, at another space station far away, are trusted with the mission of intercepting $M$ using a single cruise missile $C$ at your disposal .

You know the target missile is traveling in straight line at constant speed $v_m$. You also know the precise location and time at which it was launched. $M$, built by state-of-the-art stealth technology however, is invisible (to you or your $C$). So you have no idea in which direction it is going. Your $C$ has a maximum speed $v_c>v_m$.

Can you control trajectory of $C$ so that it is guaranteed to intercept $M$ in finite time? Is this possible?

I can think of 3 apparent possibilities:

1. Precise interception is possible. (It is possible in two dimensions, by calibrating your missile's trajectory to the parameters of certain logarithm spiral).
2. Precise interception is impossible, but for any $\epsilon\gt 0$, paths can be designed so that $C$ can get as close to $M$ as $\epsilon$ in finite time.
3. There's no hope, and your chance of intercepting or getting close to $M$ diminishes as time goes by.

This question is inspire by a similar problem in two dimensions by Louis A. Graham in his book Ingenious Mathematical Problems and Methods.

A stealth missile $M$ is launched from space station. You, at another space station far away, are trusted with the mission of intercepting $M$ using a single cruise missile $C$ at your disposal .

You know the target missile is traveling in straight line at constant speed $v_m$. You also know the precise location and time at which it was launched. $M$, built by state-of-the-art stealth technology however, is invisible (to you or your $C$). So you have no idea in which direction it is going. Your $C$ has a maximum speed $v_c>v_m$.

Can you control trajectory of $C$ so that it is guaranteed to intercept $M$ in finite time? Is this possible?

I can think of 3 apparent possibilities:

1. Precise interception is possible. (It is possible in two dimensions, by calibrating your missile's trajectory to the parameters of certain logarithm spiral).
2. Precise interception is impossible, but for any $\epsilon\gt 0$, paths can be designed so that $C$ can get as close to $M$ as $\epsilon$ in finite time.
3. There's no hope, and your chance of intercepting or getting close to $M$ diminishes as time goes by.

This question is inspired by a similar problem in two dimensions by Louis A. Graham in his book Ingenious Mathematical Problems and Methods.