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Topic: New: Efficient use of Lines to make Points in 3D (Read 1145 times) |
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jmlyle
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New: Efficient use of Lines to make Points in 3D
« on: Jul 29th, 2002, 10:12am » |
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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
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Aleksi Liimatainen
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Re: New: Efficient use of Lines to make Points in
« Reply #1 on: Jul 30th, 2002, 1:38am » |
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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.
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Neil Sedaka
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Re: New: Efficient use of Lines to make Points in
« Reply #2 on: Jul 31st, 2002, 2:21am » |
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Quote:| Think of Points as valuable, and Lasers as expensive. |
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Ok.
Now...
Think of MIRRORS as CHEAP.
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