10-10-2020, 04:27 AM
(This post was last modified: 10-10-2020, 04:48 AM by Ottia Tuota.)
(10-09-2020, 10:51 PM)Krikor Wrote: The plugin worked very well here. Fast and easy to use. Thx a lot!Great!
(10-09-2020, 08:55 PM)Ottia Tuota Wrote: Imagine a path. Imagine its bounding box. Then imagine that we put there a large number of slicing lines, criss-crossing the bounding box. They slice the bounding box into a large number of small regions, and the path becomes sliced accordingly into small slices. The number of the slices grows fast with the number of the slicing lines. I don't know how fast it grows, but it could be quadratic(??). I seem to remember vaguely that I have seen a theorem about this years ago, but I don't remember the exact result.
I figured it out. No need for any old theorems. Consider a rectangular box. Cut it with n straight lines. The box becomes divided into small regions; denote their number by f(n). Then
1+n <= f(n) <= 1+n(n+1)/2.
I can write and post the proof if somebody wants it. What f(n) exactly is, depends on how the lines are arranged. By choosing the arrangement suitably, either of the cases 1+n and 1+n(n+1)/2 can be realized. So, the worst case does indeed grow quadratically. I suppose this explains it.