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In geometry, a fat object is an object in two or more dimensions, whose lengths in the different dimensions are similar. For example, a square is fat because its length and width are identical. A 2-by-1 rectangle is thinner than a square, but it is fat relative to a 10-by-1 rectangle. Similarly, a circle is fatter than a 1-by-10 ellipse and an equilateral triangle is fatter than a very obtuse triangle. Fat objects are especially important in computational geometry. Many algorithms in computational geometry can perform much better if their input consists of only fat objects. Some examples can be seen in the references below. == Global fatness == Given a constant ''R''≥1, an object ''o'' is called ''R''-fat if its "slimness factor" is at most ''R''. The "slimness factor" has different definitions in different papers. A common definition〔, 〕 is: : where ''o'' and the cubes are ''d''-dimensional. A 2-dimensional cube is a square, so the slimness factor of a square is 1 (since its smallest enclosing square is the same as its largest enclosed disk). The slimness factor of a 10-by-1 rectangle is 10. The slimness factor of a circle is √2. Hence, by this definition, a square is 1-fat but a disk and a 10×1 rectangle are not 1-fat. A square is also 2-fat (since its slimness factor is less than 2), 3-fat, etc. A disk is also 2-fat (and also 3-fat etc.), but a 10×1 rectangle is not 2-fat. Every shape is ∞-fat, since by definition the slimness factor is always at most ∞. The above definition can be termed two-cubes fatness since it is based on the ratio between the side-lengths of two cubes. Similarly, it is possible to define two-balls fatness, in which a d-dimensional ball is used instead. A 2-dimensional ball is a disk. According to this alternative definition, a disk is 1-fat but a square is not 1-fat, since its two-balls-slimness is √2. An alternative definition, that can be termed enclosing-ball fatness (also called "thickness") is based on the following slimness factor: : The exponent 1/''d'' makes this definition a ratio of two lengths, so that it is comparable to the two-balls-fatness. Here, too, a cube can be used instead of a ball. Similarly it is possible to define the enclosed-ball fatness based on the following slimness factor: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fat object」の詳細全文を読む スポンサード リンク
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