外轮廓在同一平面内

hbghlyj

球面、圆锥面、圆柱面在任何方向的外轮廓位于经过轴线的一个平面内。
正方体表面在从顶点到中心方向的外轮廓为六条棱，且不位于同一平面内。
圆环面在(1,0,1)方向的外轮廓不在同一平面内。
双曲面在任何方向的外轮廓在同一平面内吗？一定经过轴线吗？

kuing

第一句就有问题吧，圆柱面满足那啥一个平面内吗？

hbghlyj

kuing 发表于 2023-7-25 14:07
第一句就有问题吧，圆柱面满足那啥一个平面内吗？

满足啊，例如
圆柱面$x^2+y^2=1$从y方向看过去，外轮廓是两条直线(1,0,t)和(-1,0,t)，在同一平面内。

kuing

hbghlyj 发表于 2023-7-25 19:30
满足啊，例如
圆柱面$x^2+y^2=1$从y方向看过去，外轮廓是两条直线(1,0,t)和(-1,0,t)，在同一平面内。 ...

哦，我以为还包括上下底面圆……

hbghlyj

kuing 发表于 2023-7-25 21:28
哦，我以为还包括上下底面圆……

What is the difference between a closed manifold and a compact manifold?
A closed manifold is a compact manifold without a boundary.

Where we define a manifold with boundary as follows:
A paracompact space such that each of its points is homeomorphic to the upper half plane $H^n$
Some examples of closed and compact manifolds are :

• any closed interval of the real numbers is compact but has a boundary.
• a circle is compact and without boundary, hence closed

Examples of 2-manifolds: (a) without boundary, (b) with boundaries and (c) non-manifold
manifold in nLab
closed manifold in nLab
manifold with boundary in nLab
paracompact topological space in nLab