在扭曲的正方形/轴上绘制均匀的线条/网格？

Question 1

以下是我的尝试。最初，我开始寻找类似于 John Kormylo 在他的评论，最后使用了该decorations.markings库。我希望这不是一个大问题 :-) ！在每种情况下，网格都是自适应的，正如您从以下图片中看到的那样：

我的代码是：

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\def\NrLines{8}
\pgfmathsetmacro{\UnitSegment}{1/(\NrLines+1)}

\begin{document}
\begin{tikzpicture}[draw=red]

% Nodes definition
\node[inner sep=0pt,minimum size=0pt,label=below:1] (a) at (2,5) {};
\node[inner sep=0pt,minimum size=0pt,label=below:2] (b) at (5,1) {};
\node[inner sep=0pt,minimum size=0pt,label=above:3] (c) at (6,2) {};
\node[inner sep=0pt,minimum size=0pt,label=above:4] (d) at (5,3) {};

% Paths between nodes
\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-1-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (a) -- (b);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-2-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (b) -- (c);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-3-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (c) -- (d);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-4-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (d) -- (a);

% Grid
\draw (mark-1-2) -- (mark-3-9);
\draw (mark-1-3) -- (mark-3-8);
\draw (mark-1-4) -- (mark-3-7);
\draw (mark-1-5) -- (mark-3-6);
\draw (mark-1-6) -- (mark-3-5);
\draw (mark-1-7) -- (mark-3-4);
\draw (mark-1-8) -- (mark-3-3);
\draw (mark-1-9) -- (mark-3-2);

\draw (mark-2-2) -- (mark-4-9);
\draw (mark-2-3) -- (mark-4-8);
\draw (mark-2-4) -- (mark-4-7);
\draw (mark-2-5) -- (mark-4-6);
\draw (mark-2-6) -- (mark-4-5);
\draw (mark-2-7) -- (mark-4-4);
\draw (mark-2-8) -- (mark-4-3);
\draw (mark-2-9) -- (mark-4-2);

% Nodes circles
\foreach \a in {a,...,d}{%
    \fill[black] (\a) circle (2pt);
}

\end{tikzpicture}
\end{document}

在此代码中\NrLines，定义为内部线的数量，这些内部线之间的距离等于\UnitSegment。

不幸的是，我找不到自动绘制网格的方法；我尝试过：

\pgfmathsetmacro{\End}{\NrLines+1}

\foreach \x in {1,2}{%
    \pgfmathsetmacro{\y}{\x+2}
    \foreach \i/\j in {2/\End,...,\End/2}{%
        \draw (mark-\x-\i) -- (mark-\y-\j);
    }
}

\pgfsetmacro{}但在节点名称中使用结果时会出现问题。也许比我更专业的人可能会建议使用此\foreach循环的解决方案。

编辑：最后，我改进了代码，使其能够自动绘制网格！有两个问题：

节点名称内部的使用\pgfmathsetmacro，感谢 Stefan Kottwitz 的回答“指向具有计算名称的节点，为什么我到达东边，而不是中心？“：而不是pgfmathsetmacro，应该用来\pgfmathtruncatemacro删除结果中的小数点；
内部\foreach循环无法使用，\foreach \i/\j in {2/\End,...,\End/2}因为不清楚（对于解算器 :-) ）\i和如何\j变化。因此，\j在每一步中，作为的函数进行评估\i，它是循环的唯一计数器。

新的改进代码为：

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\def\NrLines{8}
\pgfmathsetmacro{\UnitSegment}{1/(\NrLines+1)}
\pgfmathsetmacro{\End}{\NrLines+1}

\begin{document}
\begin{tikzpicture}[draw=red]

% Nodes definition
\node[inner sep=0pt,minimum size=0pt,label=below:1] (a) at (2,0) {};
\node[inner sep=0pt,minimum size=0pt,label=below:2] (b) at (5,0) {};
\node[inner sep=0pt,minimum size=0pt,label=above:3] (c) at (4,3) {};
\node[inner sep=0pt,minimum size=0pt,label=above:4] (d) at (3,3) {};

% Paths between nodes
\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-1-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (a) -- (b);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-2-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (b) -- (c);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-3-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (c) -- (d);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-4-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (d) -- (a);

% Grid
\foreach \x in {1,2}{%
    \pgfmathtruncatemacro{\y}{\x+2}
    \foreach \i in {2,...,\End}{%
        \pgfmathtruncatemacro{\j}{\End-\i+2}
        \draw (mark-\x-\i) -- (mark-\y-\j);
    }
}

% Nodes circles
\foreach \a in {a,...,d}{%
    \fill[black] (\a) circle (2pt);
}

\end{tikzpicture}
\end{document}

现在您可以轻松修改网格设置内部线的数量\NrLines，并且代码将毫无问题地运行！

Answer