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如何使用 TikZ 在球体表面绘制带有褪色的挤压椭圆？

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如何使用 TikZ 在球体表面绘制带有褪色的挤压椭圆？

理想情况下，我想绘制类似下图 (c) 部分的内容： 椭圆球
（来源：nbi.ku.dk)
。在这里，我将问题缩小到如何在球体表面上用渐变颜色绘制椭圆，如图 (b) 部分所示。（注：也许我可以跟进这个例子在球体表面画箭头，根据（b）完成（c）。

  1. 第一次尝试基本上是遵循一个简单的例子绘制球体，发现它不适用于当前版本的 TikZ。现在我卡在这一步。这是我第一次尝试完成这项工作的一部分的最少代码：

    
\documentclass[border=2pt]{standalone} %\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{calc,decorations.pathmorphing,shapes,arrows,math}
\begin{document}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}
\newcommand\pgfmathsinandcos3{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane3[current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane3[current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle2{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,dashed] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}
\newcommand\DrawLatitudeCircle2{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}
    


    %% document-wide tikz options and styles
    


    \tikzset{%
    


    
  
    =latex, % option for nice arrows
    inner sep=0pt,%
    outer sep=2pt,%
    mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
      fill=black,circle}%
  }
    

    


    \begin{figure}[ht]
\begin{center}
\begin{tikzpicture} % MERC
    


    % Modified from Stereographic and cylindrical map projections
% Author: Tomasz M. Trzeciak
% Source: LaTeX-Community.org 
%         http://www.latex-community.org/viewtopic.php?f=4&t=2111
    


    %% some definitions
    


    \def\R{3} % sphere radius
\def\angEl{25} % elevation angle
\def\angAz{-100} % azimuth angle
\def\angPhiOne{-50} % longitude of point P
\def\angPhiTwo{-35} % longitude of point Q
\def\angBeta{33} % latitude of point P and Q
    


    %% working planes
    


    \pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhiOne}
\LongitudePlane[qzplane]{\angEl}{\angPhiTwo}
\LatitudePlane[equator]{\angEl}{0}
    


    %% draw background sphere
    


    \fill[ball color=white] (0,0) circle (\R); % 3D lighting effect
%\fill[white] (0,0) circle (\R); % just a white circle
\draw (0,0) circle (\R);
    


    %% characteristic points
    


    \coordinate (O) at (0,0);
\coordinate[mark coordinate] (N) at (0,\H);
\coordinate[mark coordinate] (S) at (0,-\H);
\path[xzplane] (\R,0) coordinate (XE);
\path[pzplane] (\angBeta:\R) coordinate (P);
\path[pzplane] (\R,0) coordinate (PE);
\path[qzplane] (\angBeta:\R) coordinate (Q);
\path[qzplane] (\R,0) coordinate (QE);
    


    %% meridians and latitude circles
    


    % \DrawLongitudeCircle[\R]{\angAz} % xzplane
% \DrawLongitudeCircle[\R]{\angAz+90} % yzplane
\DrawLongitudeCircle[\R]{\angPhiOne} % pzplane
%\DrawLongitudeCircle[\R]{\angPhiTwo} % qzplane
\DrawLatitudeCircle[\R]{\angBeta}
\DrawLatitudeCircle[\R]{0} % equator
    


    % shifted equator in node with nested call to tikz 
% (I didn't know it's possible)
%\node at (0,1.6*\R) { \tikz{\DrawLatitudeCircle[\R]{0}} };
    


    %% draw lines and put labels
    


    %\draw (-\R,-\H) -- (-\R,2*\R) (\R,-\H) -- (\R,2*\R);
%\draw[->] (O) -- +(0,1.5*\R) node[above] {$|0\rangle$};
\node[above=8pt] at (N) {$|0\rangle$};%{$\mathbf{N}$};
\node[below=8pt] at (S) {$|1\rangle$};%{$\mathbf{S}$};
\node[right=8pt] at (P) {$|\Phi \rangle$};
\draw[->] (O) -- (P);
\draw[dashed] (O) -- (N);
\draw[dashed] (XE) -- (O) -- (PE);
%\draw[dashed] (O) -- (QE);
%\draw[pzplane,->,thin] (0:0.5*\R) to[bend right=15]
%    node[midway,right] {$\beta$} (\angBeta:0.5*\R);
\path[pzplane] (0.5*\angBeta:\R) ;%node[right] {$\hat{1}$};
\path[qzplane] (0.5*\angBeta:\R) ;%node[right] {$\hat{2}$};
\draw[equator,->,thin] (\angAz:0.5*\R) to[bend right=30]
    node[pos=0.4,above] {$\phi$} (\angPhiOne:0.5*\R);
\draw[pzplane,->,thin] (90:0.5*\R) to[bend left=30]
    node[midway,right] {$\theta$} (\angBeta:0.5*\R);
\draw[equator,->] (-90:\R) arc (-90:-70:\R) ;%node[below=0.3ex] {$x = a\phi$};
\path[xzplane] (0:\R) node[below] {$\theta=0$};
\path[xzplane] (\angBeta:\R);% node[below left] {$\beta=\beta_0$};
    


    \end{tikzpicture}
\end{center}
\caption{Ellipse on a Sphere}
\label{fig:sphere}
\end{figure}
\end{document}

预览：在此处输入图片描述 如你所见，与原始示例（左下图），赤道上的两条虚线无法在我的修改版本中显示。即使我编译原始示例的代码，也会出现此错误。注意到我已经替换了风格到风格。最重要的是，我接下来该怎么做才能在任意位置制作球体上的椭圆和渐变颜色？再次感谢。

  1. 第二次尝试是使用3dplot来完成球体，代码非常简单（有缺陷）。以下是代码： 
    
\documentclass[11pt]{standalone}
%\usepackage[utf8]{inputenc}
\usepackage{tikz,tikz-3dplot}
\usetikzlibrary{arrows}
\begin{document}
\tdplotsetmaincoords{70}{135}
\begin{tikzpicture}
\begin{scope}[tdplot_main_coords, fill opacity=.7,>=latex,shape=circle]
\pgfsetlinewidth{.1pt}]
%draw sphere
\tdplotsphericalsurfaceplot{36}{12}{1}{black!85!white}{blue!40!white}
{\draw[color=black,thick,->]  (-1.5,0,0) --  (1.5,0,0)  node[anchor=north  east]{$x$};}
{\draw[color=black,thick,->]  (0,-1.5,0) --  (0,1.5,0)  node[anchor=north  west]{$y$};}
{\draw[color=black,thick,->]  (0,0,-1.5) --  (0,0,1.5)  node[anchor=south]{$z$};}
\end{scope}
\end{tikzpicture}
\end{document}
    预览：球体三维绘图

我不知道接下来该如何画椭圆。您有什么想法吗？

答案1

这是获取所需图像的替代方法。我只是对地球上的路径进行了近似（没有数学正确的绘制）。将其视为简单的操作方法。

\documentclass[tikz, border=6mm]{standalone}

\usetikzlibrary{backgrounds}

\begin{document}
  \begin{tikzpicture}[x={(215:.75cm)},y={(-10:1cm)},z={(90:1cm)}]

    % Axes
    \begin{scope}[black!50, >=latex, ->, font=\scriptsize]
        \draw (0,0,0) -- ++(1.5,0,0) coordinate [label={0:$x$}];
        \draw (0,0,0) -- ++(0,1.5,0) coordinate [label={35:$y$}];
        \draw (0,0,0) -- ++(0,0,1.5) coordinate [label={90:$z$}];
    \end{scope}

    % Globe
    \begin{scope}[blue!50!white, samples=250, domain=0:2*pi]
        \draw plot ({cos(\x r)}, {sin(\x r)}, 0); % xy
        \draw plot ({cos(\x r)}, 0, {sin(\x r)}); % xz
        \draw plot (0, {cos(\x r)}, {sin(\x r)}); % yz
        \fill [opacity=.5] (0,0) circle (.85cm);
    \end{scope}

    % Plot around globe - just approximation!
    \begin{scope}[smooth, red, >=latex, ->, scale=.85]
        \draw [domain=0:.875] plot ({cos(\x r)},{sin(\x r)},0);
        \draw [domain=.19:pi/3.5] plot ({cos(\x r)},{cos(\x r)},{sin(\x r)}) coordinate (c-x);
    \end{scope}

    \begin{scope}[on background layer, smooth]
        % projection area with plot
        \fill [blue!50!white] (-4,-1.5,-1.5) -- (-4,1.5,-1.5) -- (-4,1.5,1.5) -- (-4,-1.5,1.5);
        \draw [black!25, very thin] (-4,0,-1.5) -- (-4,0,1.5);
        \draw [black!25, very thin] (-4,-1.5,0) -- (-4,1.5,0);
        \draw [black!75, domain=-1.5:1.5] plot (-4,{\x},{sin((2*\x) r)});
        \draw [black!25, domain=-1.5:1.5] plot (-4,{\x},{0.8*sin((2*\x) r)});

        % grey filled circle
        \fill [black!15, opacity=.5] (0,0,0) circle (1.155cm);

        % projection of globe-coordinate
        \draw [dashed] (c-x) -- +(-4,0,0);
    \end{scope}     
  \end{tikzpicture}
\end{document}

渲染图像

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