两个平面的立体投影

Question 1

正如 John Kormylo 所说，您可以使用tikz-3dplot。可见角度范围已经计算出来，例如这里，但约定不同。这个答案有纬度和经度弧的可见域的解析表达式，称为等。这允许您在球体上绘制网格。可以使用库中的键alpha1添加平面网格。canvas is xy plane at z=...3d

\documentclass[tikz]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{110}{40} 
\begin{tikzpicture}[tdplot_main_coords,declare function={R=3;
 alpha1(\th,\ph,\b)=\ph-asin(cot(\th)*tan(\b));%
 alpha2(\th,\ph,\b)=-180+\ph+asin(cot(\th)*tan(\b));%
 beta1(\th,\ph,\a)=90+atan(cot(\th)/sin(\a-\ph));%
 beta2(\th,\ph,\a)=270+atan(cot(\th)/sin(\a-\ph));%
}]
 \begin{scope}[canvas is xy plane at z=-R-1]
  \draw (-4,-4) grid (4,4);
 \end{scope}
 \draw[tdplot_screen_coords,ball color=gray!30] (0,0,0) circle[radius=R*1cm];
 \foreach \X in {60,90,...,210}
 {\draw plot[smooth,variable=\t,
     domain={beta1(\tdplotmaintheta,\tdplotmainphi,\X)}:{beta2(\tdplotmaintheta,\tdplotmainphi,\X)}] 
   (xyz spherical cs:radius=R,latitude=\t,longitude=\X);
 }
 \foreach \Y in {70,50,...,-70}
 {
  \draw plot[smooth,variable=\t,
     domain={alpha1(\tdplotmaintheta,\tdplotmainphi,\Y)}:{alpha2(\tdplotmaintheta,\tdplotmainphi,\Y)}] 
   (xyz spherical cs:radius=R,latitude=\Y,longitude=\t);
  }
 \begin{scope}[canvas is xy plane at z=R+1]
  \draw (-4,-4) grid (4,4);
 \end{scope}
\end{tikzpicture}
\end{document}

Answer

正如 John Kormylo 所说，您可以使用tikz-3dplot。可见角度范围已经计算出来，例如这里，但约定不同。这个答案有纬度和经度弧的可见域的解析表达式，称为等。这允许您在球体上绘制网格。可以使用库中的键alpha1添加平面网格。canvas is xy plane at z=...3d

\documentclass[tikz]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{110}{40} 
\begin{tikzpicture}[tdplot_main_coords,declare function={R=3;
 alpha1(\th,\ph,\b)=\ph-asin(cot(\th)*tan(\b));%
 alpha2(\th,\ph,\b)=-180+\ph+asin(cot(\th)*tan(\b));%
 beta1(\th,\ph,\a)=90+atan(cot(\th)/sin(\a-\ph));%
 beta2(\th,\ph,\a)=270+atan(cot(\th)/sin(\a-\ph));%
}]
 \begin{scope}[canvas is xy plane at z=-R-1]
  \draw (-4,-4) grid (4,4);
 \end{scope}
 \draw[tdplot_screen_coords,ball color=gray!30] (0,0,0) circle[radius=R*1cm];
 \foreach \X in {60,90,...,210}
 {\draw plot[smooth,variable=\t,
     domain={beta1(\tdplotmaintheta,\tdplotmainphi,\X)}:{beta2(\tdplotmaintheta,\tdplotmainphi,\X)}] 
   (xyz spherical cs:radius=R,latitude=\t,longitude=\X);
 }
 \foreach \Y in {70,50,...,-70}
 {
  \draw plot[smooth,variable=\t,
     domain={alpha1(\tdplotmaintheta,\tdplotmainphi,\Y)}:{alpha2(\tdplotmaintheta,\tdplotmainphi,\Y)}] 
   (xyz spherical cs:radius=R,latitude=\Y,longitude=\t);
  }
 \begin{scope}[canvas is xy plane at z=R+1]
  \draw (-4,-4) grid (4,4);
 \end{scope}
\end{tikzpicture}
\end{document}

Question 2

解决了。这是最终的 LaTeX 代码：

\documentclass[tikz, border=2mm]{standalone}

\usepackage{pgfplots}

\usepackage{amsmath,amssymb,amsfonts}

\usepackage{mathrsfs}

\pgfplotsset{compat=1.12}

\begin{document}

\begin{tikzpicture}[
  point/.style = {draw, circle, fill=black, inner sep=0.7pt},
]

\def\rad{2cm}

\coordinate (O) at (0,0); 
\coordinate (N) at (0,\rad); 
\coordinate (S) at (0,-\rad); 


\begin{scope}[xslant=0.65,yshift=-\rad,xshift=2]

\filldraw[fill=white,opacity=0.2]
  (-3,-1) -- (4,-1) -- (4,1) -- (-3,1) -- cycle;

\node[text=red] at (4.2,0.8) {$\mathscr{B}$};  

\draw[step=2mm, thick, black] (-3,-1) -- (4,-1) -- (4,1) -- (-3,1) -- cycle;

\draw[thin, gray, step=0.4cm] (-3,-1) grid (4,1);

\end{scope}
%
\filldraw[ball color=white] (O) circle [radius=\rad];

\draw[dashed,blue] 
  (\rad,0) arc [start angle=0,end angle=180,x radius=\rad,y radius=5mm];

\draw[blue]
  (\rad,0) arc [start angle=0,end angle=-180,x radius=\rad,y radius=5mm];
%
\begin{scope}[xslant=0.5,yshift=\rad,xshift=-2]

\filldraw[fill=gray!10,opacity=0.2]
  (-4,1) -- (3,1) -- (3,-1) -- (-4,-1) -- cycle;

\node[text=red] at (3.2,0.6) {$\mathscr{A}$};  

\draw[step=2mm, thick, black] (-4,1) -- (3,1) -- (3,-1) -- (-4,-1) -- cycle;

\draw[thin, gray, step=0.4cm] (-4,-1) grid (3,1);

\end{scope}
%

\draw[dashed]
  (N) node[above] {} -- (O) node[below] {};

 \draw[dashed]
  (O) node[above] {} -- (S) node[below] {}; 

\end{tikzpicture}

\end{document}

输出如下：

Answer

解决了。这是最终的 LaTeX 代码：

\documentclass[tikz, border=2mm]{standalone}

\usepackage{pgfplots}

\usepackage{amsmath,amssymb,amsfonts}

\usepackage{mathrsfs}

\pgfplotsset{compat=1.12}

\begin{document}

\begin{tikzpicture}[
  point/.style = {draw, circle, fill=black, inner sep=0.7pt},
]

\def\rad{2cm}

\coordinate (O) at (0,0); 
\coordinate (N) at (0,\rad); 
\coordinate (S) at (0,-\rad); 


\begin{scope}[xslant=0.65,yshift=-\rad,xshift=2]

\filldraw[fill=white,opacity=0.2]
  (-3,-1) -- (4,-1) -- (4,1) -- (-3,1) -- cycle;

\node[text=red] at (4.2,0.8) {$\mathscr{B}$};  

\draw[step=2mm, thick, black] (-3,-1) -- (4,-1) -- (4,1) -- (-3,1) -- cycle;

\draw[thin, gray, step=0.4cm] (-3,-1) grid (4,1);

\end{scope}
%
\filldraw[ball color=white] (O) circle [radius=\rad];

\draw[dashed,blue] 
  (\rad,0) arc [start angle=0,end angle=180,x radius=\rad,y radius=5mm];

\draw[blue]
  (\rad,0) arc [start angle=0,end angle=-180,x radius=\rad,y radius=5mm];
%
\begin{scope}[xslant=0.5,yshift=\rad,xshift=-2]

\filldraw[fill=gray!10,opacity=0.2]
  (-4,1) -- (3,1) -- (3,-1) -- (-4,-1) -- cycle;

\node[text=red] at (3.2,0.6) {$\mathscr{A}$};  

\draw[step=2mm, thick, black] (-4,1) -- (3,1) -- (3,-1) -- (-4,-1) -- cycle;

\draw[thin, gray, step=0.4cm] (-4,-1) grid (3,1);

\end{scope}
%

\draw[dashed]
  (N) node[above] {} -- (O) node[below] {};

 \draw[dashed]
  (O) node[above] {} -- (S) node[below] {}; 

\end{tikzpicture}

\end{document}

输出如下：

两个平面的立体投影

答案1

答案2

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