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克莱因瓶 .
浸入 三维空间中的克莱因瓶在数学 领域中,克莱因瓶 (德語:Kleinsche Flasche )是指一种无定向性的平面 ,比如二维平面 ,就没有“内部”和“外部”之分。克莱因瓶最初的概念提出是由德国 数学家 费利克斯·克莱因 提出的。克莱因瓶和莫比乌斯带 非常相像。
要想像克萊因瓶的結構,可先試想一個底部鏤空的紅酒瓶。現在延長其頸部,向外扭曲後伸進瓶子的內部,再與底部的洞相連接。
和我们平时用来喝水的杯子不一样,这个物体没有“边”,它的表面不会终结。它也不类似于气球,一只苍蝇可以从瓶子的内部直接飞到外部而不用穿过表面(所以说它没有内外部之分)。
其名稱可能源自德語中的「Kleinsche Fläche 」(克萊因平面),後來被誤解為「Kleinsche Flasche 」(克萊因瓶)。德語最終也沿用了「克萊因瓶」這種稱呼。[1]
性质
从拓扑学 角度上看,克莱因瓶可以定义为[0,1] × [0,1]的矩阵,边定义为(0,y ) ~ (1,y ),其中0 ≤ y ≤ 1;和(x ,0) ~ (1-x ,1),其中0 ≤ x ≤ 1。
可以用图表示为
就像莫比乌斯带 一样,克莱因瓶是不可定向 的。但是与之不同的是,克莱因瓶是一个闭合的曲面,也就是说它没有边界。莫比乌斯带可以嵌入到三维 或更高维的欧几里得空间 ,克莱因瓶只能嵌入到于四维或更高维空间。
参数方程模型
克莱因瓶的参数十分复杂:
x
(
u
,
v
)
=
−
2
15
cos
u
(
3
cos
v
−
30
sin
u
+
90
cos
4
u
sin
u
−
60
cos
6
u
sin
u
+
5
cos
u
cos
v
sin
u
)
y
(
u
,
v
)
=
−
1
15
sin
u
(
3
cos
v
−
3
cos
2
u
cos
v
−
48
cos
4
u
cos
v
+
48
cos
6
u
cos
v
−
60
sin
u
+
5
cos
u
cos
v
sin
u
−
5
cos
3
u
cos
v
sin
u
−
80
cos
5
u
cos
v
sin
u
+
80
cos
7
u
cos
v
sin
u
)
z
(
u
,
v
)
=
2
15
(
3
+
5
cos
u
sin
u
)
sin
v
(
0
≤
u
<
π
,
0
≤
v
<
2
π
)
{\displaystyle {\begin{aligned}&x(u,v)=-{\frac {2}{15))\cos u(3\cos {v}-30\sin {u}+90\cos ^{4}{u}\sin {u}-60\cos ^{6}{u}\sin {u}+5\cos {u}\cos {v}\sin {u})\\&y(u,v)=-{\frac {1}{15))\sin u(3\cos {v}-3\cos ^{2}{u}\cos {v}-48\cos ^{4}{u}\cos {v}+48\cos ^{6}{u}\cos {v}-60\sin {u}+5\cos {u}\cos {v}\sin {u}\\&\quad \quad \quad \quad -5\cos ^{3}{u}\cos {v}\sin {u}-80\cos ^{5}{u}\cos {v}\sin {u}+80\cos ^{7}{u}\cos {v}\sin {u})\\&z(u,v)={\frac {2}{15))(3+5\cos {u}\sin {u})\sin {v}\\&(0\leq u<\pi ,0\leq v<2\pi )\end{aligned))}
还有一个较简单的
x
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=
cos
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cos
u
2
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2
+
cos
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+
sin
u
2
sin
v
cos
v
)
y
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u
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v
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=
sin
u
(
cos
u
2
(
2
+
cos
v
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+
sin
u
2
sin
v
cos
v
)
z
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u
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v
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=
−
sin
u
2
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2
+
cos
v
)
+
cos
u
2
sin
v
cos
v
{\displaystyle {\begin{aligned}&x(u,v)=\cos u(\cos {\frac {u}{2))({\sqrt {2))+\cos v)+\sin {\frac {u}{2))\sin v\cos v)\\&y(u,v)=\sin u(\cos {\frac {u}{2))({\sqrt {2))+\cos v)+\sin {\frac {u}{2))\sin v\cos v)\\&z(u,v)=-\sin {\frac {u}{2))({\sqrt {2))+\cos v)+\cos {\frac {u}{2))\sin v\cos v\end{aligned))}
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