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如何在 TikZ 中绘制环形网格图

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如何在 TikZ 中绘制环形网格图

我试图通过将顶部边缘与底部、左侧边缘与右侧边缘进行识别，将平面网格图转换为 3D 环形图，即：

在此处输入图片描述

因此，左下方的网格将转变为环面上的右手三叶结： 在此处输入图片描述 在此处输入图片描述

我想画一个看起来像这样的圆环，但带有网格结构（包括网格本身​​、o、x 和红线）。请帮我一下。提前谢谢！

编辑：我用 A4 创建了一个“圆环”用于说明 - 希望将 x 和 o 放置在某些单位网格中，并且红线像原始网格图中一样是直线： 在此处输入图片描述

答案1

更新：象限问题已解决，现在可以只绘制可见（或隐藏）的延伸部分。您需要做的就是定义圆环坐标u和的函数v，然后可以使用 pgfplots 仅绘制可见部分。

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\tikzset{declare function={torusx(\u,\v,\R,\r)=cos(\u)*(\R + \r*cos(\v)); 
torusy(\u,\v,\R,\r)=(\R + \r*cos(\v))*sin(\u);
torusz(\u,\v,\R,\r)=\r*sin(\v);
vcrit1(\u,\th)=atan(tan(\th)*sin(\u));% first critical v value
vcrit2(\u,\th)=180+atan(tan(\th)*sin(\u));% second critical v value
vtest(\u,\v,\az,\el)=sin(-vcrit1(\u-\az,\el)+\v);
disc(\th,\R,\r)=((pow(\r,2)-pow(\R,2))*pow(cot(\th),2)+% 
pow(\r,2)*(2+pow(tan(\th),2)))/pow(\R,2);% discriminant
umax(\th,\R,\r)=ifthenelse(disc(\th,\R,\r)>0,asin(sqrt(abs(disc(\th,\R,\r)))),0);
}}
\pgfplotsset{%
visible stretch/.style={restrict expr to domain={vtest(atan2(rawy,rawx),%
ifthenelse(veclen(rawx,rawy)>\R,asin(rawz/\r),180-asin(rawz/\r)),\pgfkeysvalueof{/pgfplots/view/az},\pgfkeysvalueof{/pgfplots/view/el})}{-0.05:1.1}},
hidden stretch/.style={restrict expr to domain={vtest(atan2(rawy,rawx),%
ifthenelse(veclen(rawx,rawy)>\R,asin(rawz/\r),180-asin(rawz/\r)),\pgfkeysvalueof{/pgfplots/view/az},\pgfkeysvalueof{/pgfplots/view/el})}{-1.1:0.05}}}
\begin{document}
\begin{tikzpicture}
\pgfmathsetmacro{\R}{4}
\pgfmathsetmacro{\r}{1}
\begin{axis}[colormap/blackwhite,
   view={40}{60},axis lines=none]
%\typeout{el=\pgfkeysvalueof{/pgfplots/view/el},az=\pgfkeysvalueof{/pgfplots/view/az}}     
\tikzset{declare function={%
myu(\t)=ifthenelse(\t<108,36,ifthenelse(\t<324,\t-72,ifthenelse(\t<432,252,\t-180)));
myv(\t)=ifthenelse(\t<108,\t,ifthenelse(\t<324,108,ifthenelse(\t<432,\t-216,216)));}}
%   \addplot3[very thick,red,samples y=0,domain=0:576,smooth,samples=46,hidden stretch
%   ]  
%         ({torusx(myu(x),myv(x),\R,\r)}, 
%         {torusy(myu(x),myv(x),\R,\r)}, 
%         {torusz(myu(x),myv(x),\R,\r)});

       \addplot3[surf,shader=interp,
       samples=61, point meta=z+sin(2*y),
       %surf,shader=flat,
       domain=0:360,y domain=0:360,
       z buffer=sort]
       ({torusx(x,y,\R,\r)}, 
        {torusy(x,y,\R,\r)}, 
        {torusz(x,y,\R,\r)});
    \pgfplotsinvokeforeach{0,30,...,330}  
    {\addplot3[samples y=0,domain=0:360,smooth,samples=71,visible stretch]  
        ({torusx(x,#1,\R,\r)}, 
        {torusy(x,#1,\R,\r)}, 
        {torusz(x,#1,\R,\r)});}

    \pgfplotsinvokeforeach{0,30,...,330}  
    {\addplot3[samples y=0,domain=0:360,smooth,samples=71,visible stretch]  
        ({torusx(#1,x,\R,\r)}, 
        {torusy(#1,x,\R,\r)}, 
        {torusz(#1,x,\R,\r)});}

    \addplot3[very thick,red,samples y=0,domain=0:576,smooth,samples=46,visible stretch]  
        ({torusx(myu(x),myv(x),\R,\r)}, 
        {torusy(myu(x),myv(x),\R,\r)}, 
        {torusz(myu(x),myv(x),\R,\r)});
\end{axis}
\end{tikzpicture}
\end{document}

在此处输入图片描述

或者

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\tikzset{declare function={torusx(\u,\v,\R,\r)=cos(\u)*(\R + \r*cos(\v)); 
torusy(\u,\v,\R,\r)=(\R + \r*cos(\v))*sin(\u);
torusz(\u,\v,\R,\r)=\r*sin(\v);
vcrit1(\u,\th)=atan(tan(\th)*sin(\u));% first critical v value
vcrit2(\u,\th)=180+atan(tan(\th)*sin(\u));% second critical v value
vtest(\u,\v,\az,\el)=sin(-vcrit1(\u-\az,\el)+\v);
disc(\th,\R,\r)=((pow(\r,2)-pow(\R,2))*pow(cot(\th),2)+% 
pow(\r,2)*(2+pow(tan(\th),2)))/pow(\R,2);% discriminant
umax(\th,\R,\r)=ifthenelse(disc(\th,\R,\r)>0,asin(sqrt(abs(disc(\th,\R,\r)))),0);
}}
\pgfplotsset{%
visible stretch/.style={restrict expr to domain={vtest(atan2(rawy,rawx),%
ifthenelse(veclen(rawx,rawy)>\R,asin(rawz/\r),180-asin(rawz/\r)),\pgfkeysvalueof{/pgfplots/view/az},\pgfkeysvalueof{/pgfplots/view/el})}{-0.05:1.1}},
hidden stretch/.style={restrict expr to domain={vtest(atan2(rawy,rawx),%
ifthenelse(veclen(rawx,rawy)>\R,asin(rawz/\r),180-asin(rawz/\r)),\pgfkeysvalueof{/pgfplots/view/az},\pgfkeysvalueof{/pgfplots/view/el})}{-1.1:0.05}}}
\begin{document}
\begin{tikzpicture}
\pgfmathsetmacro{\R}{4}
\pgfmathsetmacro{\r}{1}
\begin{axis}[colormap/blackwhite,
   view={40}{60},axis lines=none]
%\typeout{el=\pgfkeysvalueof{/pgfplots/view/el},az=\pgfkeysvalueof{/pgfplots/view/az}}     
\tikzset{declare function={%
myu(\t)=\t;
myv(\t)=3*\t;}}
    \addplot3[very thick,gray,samples y=0,domain=0:576,smooth,samples=101]  
        ({torusx(myu(x),myv(x),\R,\r)}, 
        {torusy(myu(x),myv(x),\R,\r)}, 
        {torusz(myu(x),myv(x),\R,\r)});

       \addplot3[surf,shader=interp,opacity=0.8,
       samples=61, point meta=z+sin(2*y),
       %surf,shader=flat,
       domain=0:360,y domain=0:360,
       z buffer=sort]
       ({torusx(x,y,\R,\r)}, 
        {torusy(x,y,\R,\r)}, 
        {torusz(x,y,\R,\r)});
    \pgfplotsinvokeforeach{0,30,...,330}  
    {\addplot3[samples y=0,domain=0:360,smooth,samples=71,visible stretch]  
        ({torusx(x,#1,\R,\r)}, 
        {torusy(x,#1,\R,\r)}, 
        {torusz(x,#1,\R,\r)});}

    \pgfplotsinvokeforeach{0,30,...,330}  
    {\addplot3[samples y=0,domain=0:360,smooth,samples=71,visible stretch]  
        ({torusx(#1,x,\R,\r)}, 
        {torusy(#1,x,\R,\r)}, 
        {torusz(#1,x,\R,\r)});}

    \addplot3[very thick,red,samples y=0,domain=0:576,smooth,samples=101,visible stretch]  
        ({torusx(myu(x),myv(x),\R,\r)}, 
        {torusy(myu(x),myv(x),\R,\r)}, 
        {torusz(myu(x),myv(x),\R,\r)});
\end{axis}
\end{tikzpicture}
\end{document}

在此处输入图片描述

原来的。

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\tikzset{declare function={%
torusx(\u,\v,\R,\r)=cos(\u)*(\R + \r*cos(\v)); 
torusy(\u,\v,\R,\r)=(\R + \r*cos(\v))*sin(\u);
torusz(\u,\v,\R,\r)=\r*sin(\v);
vcrit1(\u,\th)=atan(tan(\th)*sin(\u));% first critical v value
vcrit2(\u,\th)=180+atan(tan(\th)*sin(\u));% second critical v value
thetacritA(\R,\r)=atan(sqrt(\R/\r-1));
thetacritB(\R,\r)=acos(\r/\R);
ucritA(\R,\r,\th)=180+(90/pi)*sqrt(abs(-(\R^2*pow(cot(\th),2))+4*pow(\r,2)/pow(sin(2*\th),2)))/\R; 
ucritB(\R,\r,\th)=540-ucritA(\R,\r,\th);
umaxA(\R,\r,\th)=asin(sqrt(abs(-pow(cot(\th),2)+4*pow(\r,2)/(pow((sin(2*\th)*\R),2)))));
umaxB(\R,\r,\th)=180-umaxA(\R,\r,\th);}}
\begin{document} 
\tdplotsetmaincoords{65}{0}
\begin{tikzpicture}[tdplot_main_coords]
     \pgfmathsetmacro{\RadiusA}{3}
     \pgfmathsetmacro{\RadiusB}{1}
     \pgfmathsetmacro{\rprime}{1.25}
     % all v curves
     \foreach \X in {0,10,...,350}
     {\draw
        plot[variable=\x,domain=0:360,smooth]
    ({torusx(\X,\x,\RadiusA,\RadiusB)},{torusy(\X,\x,\RadiusA,\RadiusB)},{torusz(\X,\x,\RadiusA,\RadiusB)});
     }
     % all u curves
     \foreach \X in {0,30,...,330}
     {\draw plot[variable=\x,domain=0:360,smooth]
        ({torusx(\x,\X,\RadiusA,\RadiusB)},{torusy(\x,\X,\RadiusA,\RadiusB)},{torusz(\x,\X,\RadiusA,\RadiusB)});
     }
  \end{tikzpicture}
\end{document}

在此处输入图片描述

它们可以用来辨别隐藏在圆环上的物体和可见的物体，如图所示这个答案其中解释了各个函数。如果您发现将各个函数拼凑在一起太麻烦，那么您可能需要考虑切换到渐近线。

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\tikzset{declare function={%
torusx(\u,\v,\R,\r)=cos(\u)*(\R + \r*cos(\v)); 
torusy(\u,\v,\R,\r)=(\R + \r*cos(\v))*sin(\u);
torusz(\u,\v,\R,\r)=\r*sin(\v);
vcrit1(\u,\th)=atan(tan(\th)*sin(\u));% first critical v value
vcrit2(\u,\th)=180+atan(tan(\th)*sin(\u));% second critical v value
thetacritA(\R,\r)=atan(sqrt(\R/\r-1));
thetacritB(\R,\r)=acos(\r/\R);
ucritA(\R,\r,\th)=180+(90/pi)*sqrt(abs(-(\R^2*pow(cot(\th),2))+4*pow(\r,2)/pow(sin(2*\th),2)))/\R; 
ucritB(\R,\r,\th)=540-ucritA(\R,\r,\th);
umaxA(\R,\r,\th)=asin(sqrt(abs(-pow(cot(\th),2)+4*pow(\r,2)/(pow((sin(2*\th)*\R),2)))));
umaxB(\R,\r,\th)=180-umaxA(\R,\r,\th);}}
\tikzset{3d torus/.style n
args={2}{/utils/exec=\pgfmathsetmacro{\DDA}{int(sign(sin(thetacritA(#1,#2))-sin(\tdplotmaintheta)))}
  \pgfmathsetmacro{\DDB}{int(sign(sin(thetacritB(#1,#2))-sin(\tdplotmaintheta)))},
  insert path={
  plot[variable=\x,domain=1:359,smooth cycle,samples=71]
  ({torusx(\x,vcrit1(\x,\tdplotmaintheta),#1,#2)},
 {torusy(\x,vcrit1(\x,\tdplotmaintheta),#1,#2)},
 {torusz(\x,vcrit1(\x,\tdplotmaintheta),#1,#2)}) 
   \ifnum\DDA=1
    plot[variable=\x,domain=0:360,smooth cycle,samples=71]
    ({torusx(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)},
    {torusy(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)},
    {torusz(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)})    
   \else
   \ifnum\DDB=1 
    plot[variable=\x,domain={umaxA(#1,#2,\tdplotmaintheta)}:{umaxB(#1,#2,\tdplotmaintheta)},smooth,samples=71]
    ({torusx(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)},
    {torusy(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)},
    {torusz(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)})    --
    plot[variable=\x,domain={180+umaxA(#1,#2,\tdplotmaintheta)}:{180+umaxB(#1,#2,\tdplotmaintheta)},smooth,samples=71]
    ({torusx(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)},
    {torusy(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)},
    {torusz(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)})  -- cycle  
    \fi 
   \fi
  }},3d torus stretch/.style n args={2}{/utils/exec=\pgfmathsetmacro{\DDA}{int(sign(thetacritA(#1,#2)-\tdplotmaintheta))},
  insert path={\ifnum\DDA=-1
   plot[variable=\x,domain={ucritA(#1,#2,\tdplotmaintheta)}:{ucritB(#1,#2,\tdplotmaintheta)},smooth,samples=71]
    ({torusx(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)},
    {torusy(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)},
    {torusz(\x,vcrit2(\x,\tdplotmaintheta),#1,#2)}) 
  \fi 
}}}
\begin{document} 
\tdplotsetmaincoords{65}{0}
\begin{tikzpicture}[tdplot_main_coords]
     \pgfmathsetmacro{\RadiusA}{3}
     \pgfmathsetmacro{\RadiusB}{1}
     \pgfmathsetmacro{\rprime}{1.25}
     \foreach \X/\Y in {105/195,245/335}
     {\draw[line width=2mm,blue] plot[variable=\x,domain=\X:\Y,smooth]
    ({torusx(\x,2*\x,\RadiusA,\rprime)},{torusy(\x,2*\x,\RadiusA,\rprime)},{torusz(\x,2*\x,\RadiusA,\rprime)});}
     \draw[thick,samples=71,fill=gray,fill opacity=0.7,even odd
    rule,3d torus={\RadiusA}{\RadiusB}] ;
    \draw[thick,samples=71,3d torus stretch={\RadiusA}{\RadiusB}];
     \foreach \X/\Y in {-27/107,193/247}
     {\draw[line width=2mm,blue] plot[variable=\x,domain=\X:\Y,smooth]
    ({torusx(\x,2*\x,\RadiusA,\rprime)},{torusy(\x,2*\x,\RadiusA,\rprime)},{torusz(\x,2*\x,\RadiusA,\rprime)});}
  \end{tikzpicture}
\end{document}

在此处输入图片描述

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