☰

绘制 3D 交换图

tag-icon • tag-icon 3d asymptote tikz-cd commutative-diagrams
绘制 3D 交换图

当我在绘制交换图时，我突然想到要绘制它的四面体版本（这有点可疑）。我使用了 Asymptote，希望同时获得好看的缩略图和漂亮的嵌入式 3D 模型。这是我目前得到的结果：

左侧是平面交换图。右侧是四面体的缩略图。

激活图表并自由旋转后，我意识到我的 3D 模型看起来不太美观：

左侧为之前的平面交换图。右侧为激活的嵌入四面体 3D 模型。

我想做出以下改进：

  1. 箭头的开始和结束位置会自动计算，这样tikz-cd我就不需要使用Fill标签或使用例程convexcomb了。
  2. 允许前面的箭头穿过后面的箭头，类似于tikz-cd选项crossing over。
  3. 无需激活模型，箭头看起来也不错。
  4. 随着相机旋转，箭头在 3D 中看起来不错。

我真的不知道如何实现 1 和 2。此外，3 和 4 似乎是相互矛盾的目标。我的现实目标应该是什么，我应该做出哪些改进？来源：

settings.render = 0;
import solids;
import three;
size(240,120);
defaultpen(basealign);

real side = 1;
triple origin = (0,0,0);
triple voa = (0,0,sqrt(6)/3*side);
triple vob = (sqrt(3)/4*side,-.5*side,0);
triple voc = (sqrt(3)/4*side,.5*side,0);
triple vod = (-sqrt(3)/3*side,0,0);

triple convexcomb(triple a, triple b, real x)
{
  return (1-x) * a + x * b;
}

// A: U
// B: V1x...xVn
// C: W
// D: V

draw(convexcomb(vob,voa,.15)--convexcomb(vob,voa,.85),L=Label("$\varphi$",align=NW)         ,Arrow3(TeXHead2(normal=O)));
draw(convexcomb(vob,voc,.15)--convexcomb(vob,voc,.85),L=Label("$\psi$"   ,align=unit(4*S+W)),Arrow3(TeXHead2(normal=X)));
draw(convexcomb(vob,vod,.15)--convexcomb(vob,vod,.85),L=Label("$\phi$"   ,align=unit(4*N+E)),Arrow3(TeXHead2(normal=O)));
draw(convexcomb(voa,voc,.15)--convexcomb(voa,voc,.85),L=Label("$T_{0}$"  ,align=E)          ,Arrow3(TeXHead2(normal=O)));
draw(convexcomb(voa,vod,.15)--convexcomb(voa,vod,.85),L=Label("$\Pi$"    ,align=unit(E+NE)) ,Arrow3(TeXHead2(normal=O)));
draw(convexcomb(vod,voc,.15)--convexcomb(vod,voc,.85),L=Label("$T$"      ,align=unit(E+SE)) ,Arrow3(TeXHead2(normal=O)));

label(Label("$U$",Fill(white)),voa);
label(Label("$V_{1}\times\cdots\times V_{n}$",Fill(white)),vob);
label(Label("$W$",Fill(white)),voc);
label(Label("$V$",Fill(white)),vod);

答案1

这是关于伪 3D tikz-cd 图的建议。

\documentclass[border=2pt]{standalone}
\usepackage{amssymb,amsmath}
\usepackage{tikz-cd}
\usetikzlibrary{arrows}

\usepackage{tikz}

\begin{document}
\tikzset{zshift/.style={xshift={-0.3*#1},yshift={-0.9*#1}}}

\begin{tikzcd}[row sep=2cm,column sep=2cm,inner sep=1ex]
& U \arrow{d}{T_0} \arrow{dr}{\Pi} & \\
V_1\times\cdots V_n \arrow{ur}{\varphi} 
\arrow{r}{\phi} \arrow{rr}{\psi}&
|[zshift=-1.5cm]| V \arrow{r}{T}& |[zshift=1cm]|W
\end{tikzcd}

\end{document}

在此处输入图片描述

答案2

以下是从另一个背景中截取的类似形状的图表；也许它们有帮助：

\begin{tikzcd}[row sep={7.2em,between origins}, column sep={9.0em,between origins}, background color=backgroundColor, ampersand replacement=\&,execute at end picture={
  \foreach \Number in  {A,B,...,D}
      {\coordinate (\Number) at (\Number.center);}
  \begin{pgfonlayer}{background}
      \fill[left color=grey!98!black, right color=grey!80!white, shading angle=60] (A) -- (C) -- (B) -- cycle;
      \fill[left color=grey!85!black, right color=grey!85!white, shading angle=-60] (D) -- (C) -- (B) -- cycle;
      \fill[left color=grey!80!black, right color=grey!98!white, shading angle=180] (A) -- (B) -- (D) -- cycle;
      \draw[dashed, dash pattern=on 3.0pt off 3.0pt] (B) -- (A);
      \draw[dashed, dash pattern=on 3.0pt off 3.0pt] (B) -- (C);
      \draw[dashed, dash pattern=on 3.0pt off 3.0pt] (B) -- (D);
  \end{pgfonlayer}
  \filldraw[black!30!white] (A) circle (6pt) node[black,align=center] {$A$};
  \filldraw[black!30!white] (B) circle (6pt) node[black,align=center] {$B$};
  \filldraw[black!30!white] (C) circle (6pt) node[black,align=center] {$C$};
  \filldraw[black!30!white] (D) circle (6pt) node[black,align=center] {$D$};
}]
    \&
    |[alias=C]|C
    \arrow[rdd, "h"]
    \&\\[4.5em]\&
    |[alias=B]|B
    \arrow[u, dashed, "\textcolor{white}{\phantom{g}}"{description,background color=grey,opacity=0.75}, dash pattern=on 3.0pt off 3.0pt]
    \arrow[u, phantom, "\textcolor{white}{g}"]
    \arrow[dr, dashed, "\textcolor{white}{\phantom{h\circ g}}"{description,background color=grey,opacity=0.5}, dash pattern=on 3.0pt off 3.0pt]
    \arrow[dr, dashed, "\textcolor{white}{h\circ g}", phantom]
    \&\\[-1.85em]
    |[alias=A]|A
    \arrow[ruu, "g\circ f"]
    \arrow[rr, "h\circ g\circ f"']
    \arrow[ru, dashed, "\textcolor{white}{\phantom{f}}"{description,background color=grey,opacity=0.75}, dash pattern=on 3.0pt off 3.0pt]
    \arrow[ru, dashed, "\textcolor{white}{f}", phantom]
    \&\&
    |[alias=D]|D\mathrlap{.}
\end{tikzcd}

  1. （忽略 2-态射）：

    \begin{tikzcd}[row sep={7.2em,between origins}, column sep={9.0em,between origins}, background color=backgroundColor, ampersand replacement=\&]
    \&
    A_{2}
    \arrow[rdd, "f_{23}", bend left=20]
    \&\\\&
    A_{1}
    \arrow[u, "f_{12}"{description}, dash pattern=on 4.0pt off 4.0pt]
    \arrow[rd, "f_{13}"'{description,name=f13}, dash pattern=on 4.0pt off 4.0pt]
    \&\\
    A_{0}
    \arrow[ruu, "f_{02}"{name=f02}, bend left=20]
    \arrow[rr, "f_{03}"'{name=f03}, bend right=20]
    \arrow[ru, "f_{01}"{description}, dash pattern=on 4.0pt off 4.0pt]
    \&\&
    A_{3}
    % 2-Arrows
    \arrow[from=2-2, to=f02, Rightarrow, shorten=1.0em, "\theta_{012}"description, pos=0.475]
    \arrow[from=1-2, to=f13, Rightarrow, yshift=-0.25em, xshift=0.75em, shorten=3.5em, "\theta_{123}"description, pos=0.475]
    \arrow[from=2-2, to=f03, Rightarrow, shorten=2.5em, "\theta_{013}"description, pos=0.475]
    \arrow[from=1-2, to=f03, bend left=20, yshift=+0.0em, xshift=0.0em, crossing over, Rightarrow, shorten=2.0em, "\hspace{+0.625em}\theta_{023}"description, pos=0.425, crossing over clearance=1.5ex]
\end{tikzcd}

相关内容