需要绘制格子并构建环面的拓扑！

Question

正如标题所述，我需要在平面上绘制一个格子（不是以 (1,0), (0,1) 为基的标准格子），我想展示如何从它的拓扑正方形开始构建一个环面...如果有人能帮忙我将不胜感激！

Answer 1

作为一名拓扑学家，我可以证明经常绘制圆环的必要性，以及构建圆环的步骤。所以我认为这将是一个有用的资源。话虽如此，下面的实际圆环有点“老套”，只是经过调整看起来还可以，而不是完美的渲染。

这是代码。正如我所评论的，格子很简单。圆环的第一阶段也相当简单。唯一棘手的是让部分圆环和圆环看起来合理。它们可能可以改进。代码也可以通过使用箭头线的样式来稍微压缩一下。

（我刚刚发现在第二阶段，右边缘画错了方向，所以箭头指向了错误的方向。这很容易纠正。）

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,arrows}
\begin{document}
\begin{tikzpicture}
\draw[->,ultra thick,shorten >=-8pt] (-3,0) -- (3,0);
\draw[->,ultra thick,shorten >=-8pt] (0,-3) -- (0,3);
\begin{scope}
\clip (-3,-3) rectangle (3,3);
\begin{scope}[xslant=1]
\draw (-6,-6) grid (6,6);
\draw[ultra thick] (0,0) rectangle (1,1);
\end{scope}
\end{scope}
\end{tikzpicture}

\begin{tikzpicture}
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .145 with {\arrow{latex}},
    mark=at position .375 with {\arrow{latex}},
    mark=at position .395 with {\arrow{latex}},
    mark=at position .635 with {\arrowreversed{latex}},
    mark=at position .875 with {\arrowreversed{latex}},
    mark=at position .895 with {\arrowreversed{latex}}
    }
  ]
    (0,-1) -- +(2,0) -- +(2,2) -- +(0,2) -- cycle;

\begin{scope}[xshift=3cm]
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}}
    }
  ]
    (0,.25) -- ++(2,0);
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}}
    }
  ]
    (0,-.25) -- ++(2,0);
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    mark=at position .55 with {\arrow{latex}}
    }
  ] (0,-.25) to[out=-120,in=0] (-.35,-.75) to[out=180,in=180] (-.35,.75) to[out=0,in=120] (0,.25);
\draw (2,.25) to[out=120,in=0] (1.65,.75) -- (-.35,.75) (-.35,-.75) --  (1.65,-.75) to[out=0,in=-120] (2,-.25);
\begin{scope}
\clip (0,.25) rectangle (2,-.25);
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    mark=at position .58 with {\arrow{latex}}
    }
  ]  (1.65,.75) to[out=180,in=180] (1.65,-.75);
\end{scope}
\begin{scope}[xshift=3cm]
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    }
  ]  (0,0) -- (2,0);
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    mark=at position .55 with {\arrow{latex}},
    }
  ] (0,0) arc[start angle=0,delta angle=-360,x radius=.35,y radius=.75];
\draw (2,0) arc[start angle=0,delta angle=-90,x radius=.35,y radius=.75] -- ++(-2,0);
\draw (2,0) arc[start angle=0,delta angle=90,x radius=.35,y radius=.75] -- ++(-2,0);
\end{scope}
\end{scope}
\begin{scope}[yshift=-3cm]
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    }
  ]  (0,0) -- (3,0);
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    mark=at position .6 with {\arrow{latex}},
    }
  ] (0,0) arc[start angle=0,delta angle=-360,x radius=.15,y radius=.35];
\draw (3,0) arc[start angle=0,delta angle=-90,x radius=.15,y radius=.35] -- ++(-3,0);
\draw (3,0) arc[start angle=0,delta angle=90,x radius=.15,y radius=.35] -- ++(-3,0);
\begin{scope}[xshift=4cm]
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    mark=at position .6 with {\arrow{latex}},
    }
  ] (1,0) arc[start angle=0,delta angle=-360,x radius=.15,y radius=.35];
\draw (1,0) ++(-.15,-.35) .. controls +(170:1) and +(-90:.5) .. ++(-1.5,1) .. controls +(90:.5) and +(180:1) .. ++(2,1) .. controls +(0:1) and +(90:.5) .. ++(2,-1) .. controls +(-90:.5) and +(10:1) .. ++(-1.5,-1) coordinate (a);
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    mark=at position .6 with {\arrow{latex}},
    }
  ] (a) ++(-.15,.35) arc[start angle=0,delta angle=-360,x radius=-.15,y radius=.35];
\draw (1,0) ++(-.15,.35) .. controls +(170:.5) and +(-60:.25) .. ++(-.9,.5) coordinate (b);
\draw (a) ++(0,.7) .. controls +(10:.5) and +(240:.25) .. ++(.9,.5) coordinate (c);
\begin{scope}
\clip (1,0) ++(-.15,.35) .. controls +(170:.5) and +(-60:.25) .. ++(-.9,.5) -- ++(0,2) -| (c)  .. controls +(240:.25) and +(10:.5) .. ++(-.9,-.5);
\draw (1,0) ++(-.15,-.35) ++(0,-.7) .. controls +(170:1) and +(-90:.5) .. ++(-1.5,.8) .. controls +(90:.5) and +(180:1) .. ++(2,1.2) .. controls +(0:1) and +(90:.5) .. ++(2,-1.2) .. controls +(-90:.5) and +(10:1) .. ++(-1.5,-.8);
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    }
  ] (1,0) ++(-.15,-.35) ++(0,-.8) .. controls +(170:1) and +(-90:.5) .. ++(-1.5,.8) .. controls +(90:.5) and +(180:1.2) .. ++(2,1.5) .. controls +(0:1.2) and +(90:.5) .. ++(2,-1.5) .. controls +(-90:.5) and +(10:1) .. ++(-1.5,-.8);
\end{scope}
\begin{scope}
\clip  (a) ++(-.15,.35) arc[start angle=0,delta angle=-360,x radius=-.15,y radius=.35];
\draw (a) ++(-.15,.35) .. controls +(10:1) and +(-90:.5) .. ++(1.5,.8);
\end{scope}
\begin{scope}
\clip (1,0) arc[start angle=0,delta angle=-360,x radius=.15,y radius=.35];
\draw (1,0) .. controls +(170:1) and +(-90:.5) .. ++(-1.5,.8);
\end{scope}
\begin{scope}[xshift=5cm]
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    mark=at position .6 with {\arrow{latex}},
    }
  ] (1.5,.35) arc[start angle=90,end angle=-90,y radius=.35,x radius=.1];
\draw (1.5,-.35) .. controls +(180:1) and +(-90:.65) .. ++(-2,1) .. controls +(90:.65) and +(180:1) .. ++(2,1) .. controls +(0:1) and +(90:.65) .. ++(2,-1) .. controls +(-90:.65) and +(0:1) .. ++(-2,-1); \draw (1.5,.35) .. controls +(180:.5) and +(-50:.25) .. ++(-1.3,.35) coordinate (b);
\draw (1.5,.35) .. controls +(0:.5) and +(230:.25) .. ++(1.3,.35) coordinate (c);
\begin{scope}
\clip (1.5,.35) .. controls +(180:.5) and +(-50:.25) .. ++(-1.3,.35) -- ++(0,2) -| (c)  .. controls +(230:.25) and +(0:.5) .. ++(-1.3,-.35);
\draw (1.5,-.35) ++(0,-.7) .. controls +(180:1) and +(-90:.65) .. ++(-1.5,1) .. controls +(90:.65) and +(180:1) .. ++(1.5,1) .. controls +(0:1) and +(90:.65) .. ++(1.5,-1) .. controls +(-90:.65) and +(0:1) .. ++(-1.5,-1);
\draw[postaction={decorate},decoration={
    markings,
    mark=at position .5 with {\arrow{latex}},
    }
  ] (1.5,-.35) ++(0,-.6) .. controls +(180:1) and +(-90:.65) .. ++(-1.5,1) .. controls +(90:.65) and +(180:1) .. ++(1.5,1) .. controls +(0:1) and +(90:.65) .. ++(1.5,-1) .. controls +(-90:.65) and +(0:1) .. ++(-1.5,-1);
\end{scope}
\end{scope}
\end{scope}
\end{scope}
\end{tikzpicture}
\end{document}