Timeline for Intersection/Union of Rectangles as a Group (or Monoid or...?)
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 2, 2010 at 0:05 | vote | accept | tony | ||
| Mar 1, 2010 at 22:01 | comment | added | Zev Chonoles | I've edited my answer to summarize what the set of rectangles becomes under each of the main set-theoretic operaitons, since that seems to be what you're looking for. However, you may still have luck - both with getting a stronger structure, and with implementing the concept of rectangles with negative width or height - if you first experiment with different operations on the set $\mathbb{R}^4=${$(x,y,w,h):x,y,w,h\in\mathbb{R}$}, say using +, *, min, max, and so on, and then seeing how it is interpretable as being about rectangles. | |
| Mar 1, 2010 at 20:30 | comment | added | tony | @Zev: yeah, I was once a pure mathematician, but that was 20 years ago :-( I started out thinking about negative rectangles and what that means for intersection, but I would in fact need to make 'as most sense as possible' of a number of operations. If I could convince myself that operationA makes sense because it now builds a Monoid or whatever instead of just a 'dumb set', I'd be happy. Maybe I should just consider each operation, and see whether they can be sensibly made to be associative or reflexive, etc. And see how it goes from there. | |
| Mar 1, 2010 at 20:18 | comment | added | Zev Chonoles | I apologize if you were already aware of this - it is hard to tell from what you've said - but a group has just one operation. | |
| Mar 1, 2010 at 20:14 | history | edited | tony | CC BY-SA 2.5 |
added 1435 characters in body
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| Mar 1, 2010 at 20:11 | comment | added | tony | I'll update the quesiton with a few more thoughts... | |
| Mar 1, 2010 at 20:06 | comment | added | Mariano Suárez-Álvarez | You are not going to get a group (independently of what you decide to do with negative widths and heights), because you already have problems with cancellation: the union of {x:0, y:0, w:1, h:1} with {1, 1, 1, 1} is the same as the union of {0,0,2,1} with {1, 1, 1, 1}, namely {0, 0, 2, 2}. A similar thing happens with intersections. | |
| Mar 1, 2010 at 20:04 | answer | added | Zev Chonoles | timeline score: 1 | |
| Mar 1, 2010 at 20:02 | comment | added | José Figueroa-O'Farrill | I'd say that what the union/intersection allows you to do is to define a lattice (i.e., a poset with finite meets and joins) in the set of rectangles. I haven't checked but you might get a monoid by considering the endomorphisms of the lattice, which you could then turn into a group using the Grothendieck construction. I don't see, though, how this is related to rectangles with negative dimensions. | |
| Mar 1, 2010 at 20:01 | comment | added | Qiaochu Yuan | Any definition that you could want to make depends on what you want the definition to do for you. What purpose do you want this to serve? | |
| Mar 1, 2010 at 19:43 | history | asked | tony | CC BY-SA 2.5 |