# term for a rectangle with a bounded aspect ratio

I am writing a peper about dividing a shape into rectangles, where the main issue is to make sure that the rectangles have a limited aspect ratio. I am looking for a clear, unambiguous term for such rectangles.

Initially I used the term "r-balanced rectangle", for a rectangle whose width/height ratio is between $r$ and $1/r$. However, I noticed that this term is used with different meanings in different papers, so I look for another term.

Recently I found the term "fat rectangle" in two papers: Agarwal et al., "Binary space partitions for fat rectangles": "the aspect ratio of each rectangle in S is at most $\alpha$ , for some constant $\alpha \ge 1$", and similarly in Csaba D. Tóth, "Binary Space Partitions for Axis-Aligned Fat Rectangles". In my paper I study different bounds on the aspect ratio, so I must add the ratio bound (r) to the term, so I thought of using the term "r-fat rectangle", however, when r is larger, the rectangle is thinner, so maybe the term "r-thin rectangle" is better.

On the other hand, the term "fat rectangle" is used in other papers with a slightly different meaning, for example, Joseph O'Rourke and Geetika Tewari, "Partitioning Orthogonal Polygons into Fat Rectangles in Polynomial Time": "the shortest rectangle side is maximized over all rectangles", i.e., fatness relates to the absolute dimension of the shortest side, and not to the aspect ratio.

What, in your opinion, is the best term to use? "r-balanced rectangle"? "r-fat rectangle"? "r-thin rectangle"? Some other name?

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One standard pair of terms is "thick" versus "thin" for bounded versus unbounded aspect ratios. – Sam Nead Jul 13 '13 at 19:20
Clearly this area needs consistent terminology! What I have done in a few analogous situations is: Decide on what should be the terminology, and then edit Wikipedia in the hope that your preference spreads. :-) – Joseph O'Rourke Jul 13 '13 at 19:37
@ErelSegal-haLevi I suspect that your question will be better received if you ask something like "what, if any, terms have been used in the literature for referring to rectangles with bounded aspect ratio?" instead of soliciting opinions on "the best" term. But since you ask, I am partial to "$r$-conditioned rectangles". – Vidit Nanda Jul 13 '13 at 20:20
I was writing very quickly before - here are more details. In Teichmuller theory (of which rectangles are a first, important, but easy case - cf the Grötzsch's problem) we call a surface "thin" if contains a long skinny annulus and "thick" otherwise. This leads to the "thick-thin" decomposition of hyperbolic surfaces, for example. Another equivalent way to talk about such annuli is to discuss their "modulus". Of course, you get an annulus by gluing two opposite sides of a rectangle - all of these things are tightly connected. – Sam Nead Jul 13 '13 at 21:19
I'm not posting this as an answer since you already have these references, and I'm not sure what your real question is, but in computational geometry "fat" generally means having a bounded aspect ratio or very closely related concepts — that is, the Agarwal meaning is much more standard than the O'Rourke one. The notion of an $r$-fat shape, having fatness (bounded by some measure) at most $r$, is also very standard. So that is what I would use in this case. – David Eppstein Jul 13 '13 at 23:43