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        riddles >> general problem-solving / chatting / whatever >> New: Efficient use of Lines to make Points in 3D
        
(Message started by: jmlyle on Jul 29^th, 2002, 10:12am)

Title: New: Efficient use of Lines to make Points in 3D
Post by jmlyle on Jul 29^th, 2002, 10:12am

Sorry, I couldn't really find a better place to put this. I'm sure the esteemed willywu will move it if he deems appropriate.

Okay. I've been puzzling over this for a few years. I want to define multiple points in 3D space. I'm using lasers. The intersection of two Lasers (lines) in the general 3D work space is a Point.

Think of Points as valuable, and Lasers as expensive.

I want to determine how to most efficiently use the Lasers to define the Points. What kind of structure evolves as the number of Points increases?

2D space seems to be the same as 3D for the first 5 Points or so. After that, 3D space would begin to form crystaline structures, I would assume.


A Point is where 2 Lasers intersect

Points | Minimum # of Lasers
-------|--------------------
   1   |  2
   2   |  3
   3   |  3
   4   |  4
   5   |  4

How can I best use the Lasers to create Points? How many Lasers do I need for N Points?

And, just for fun, What would happen if we decided that we need a Point to be defined by the intersection of THREE lines, instead of two?

--jmlyle

Title: Re: New: Efficient use of Lines to make Points in
Post by Aleksi Liimatainen on Jul 30^th, 2002, 1:38am

It would seem to me that you would always get the most points with a 2d configuration: two straight lines can't intersect more than once, and two nonparallel lines on a plane will always intersect.

So you just need to place each successive laser so that it intersects all the already placed lasers. This should be trivial if no constraints are placed on the orientation and placement of the lasers. It also follows that each new laser N would add one new intersection for all N-1 lasers already in place, which would make the maximum number of points for N lasers a sum from 0 to N-1.

Title: Re: New: Efficient use of Lines to make Points in
Post by Neil Sedaka on Jul 31^st, 2002, 2:21am

Quote:

Think of Points as valuable, and Lasers as expensive.

Ok.

Now...

Think of MIRRORS as CHEAP.

;D