finding points on the line connecting two 3D points

Question

I have two points in 3D space. Say these are P = (x, y, z) and Q = (u, v, w). Now, I would like to get line segments from P to Q of length epsilon each. So, for that, I need all the points on the line from P to Q that are apart by length epsilon. I can do this for a line in 2D, but I am confused with how to do this for a line in 3D?

Here is what I tried.

Set current point to be P and the next point to be current point + epsilon * (Q-P)/|Q-P|.

Z = {current point, next point}

Then, iteratively, went through adding to the set Z, next point + epsilon * (Q-P)/|Q-P|.

However, I can not figure out how to stop this iterative process.

What is the correct way of thinking about this in 3D?

Thank you in advance!

Answer 1

The number of line segments ($n$) of length $\epsilon$ starting from point $P$ and ending at or before point $Q$ must be,

$ \displaystyle n = \left \lfloor \frac{|\textbf{Q}-\textbf{P}|}{\epsilon} \right \rfloor$

At the start of the iteration, $ \textbf{P}_{1} = \textbf{P}$

Iterate for $1 \leq m \leq n$,

$ \displaystyle \small \textbf{Q}_m = \textbf {P}_m + \frac{\textbf{Q}-\textbf{P}}{|\textbf{Q}-\textbf{P}|} \epsilon ~$

and $\textbf{P}_{m+1} = \textbf{Q}_{m}$