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使用 Tikz 在球体上绘制椭圆莫比乌斯地图

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使用 Tikz 在球体上绘制椭圆莫比乌斯地图

我正在努力适应此示例来自 texample制作一个带有箭头的球体，表示椭圆莫比乌斯映射的效果。

事实上，我正在尝试重新绘制下面的球体，但有一段时间我只处理其中两个球体。

在此处输入图片描述

使用下面的文件我得到了这个：

在此处输入图片描述

但我不知道如何减少箭头的长度以及如何将其移动到其他圆圈。我尝试改变角度但没有成功。欢迎提供帮助。

以下是代码

% Stereographic and cylindrical map projections
% Author: Tomasz M. Trzeciak
% Source: LaTeX-Community.org 
%         <http://www.latex-community.org/viewtopic.php?f=4&t=2111>
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}
\usepackage{verbatim}

%% helper macros

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,dashed}}
   % 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\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,line width=.7pt}}
  \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[red,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{document}

\begin{tikzpicture}[scale=1.4] % "THE GLOBE" showcase

\def\R{2.5} % sphere radius
\def\angEl{35} % elevation angle
\def\angAz{-105} % azimuth angle
\def\angPhi{-40} % longitude of point P
\def\angBeta{0} % latitude of point P
%
\filldraw[ball color=white] (0,0) circle (\R);
\foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} }
\foreach \t in {-5,-35,...,-175} { \DrawLongitudeCircle[\R]{\t} }
%
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate[mark coordinate] (N) at (0,\H);
\draw (N)  node[inner sep=1pt,below=0pt] {$\infty$}; %+(0.3ex,0.6ex)
%
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhi}
\LatitudePlane[equator]{\angEl}{0}
\draw[red,equator,->,thick] (\angAz:\R) to[bend right=30] (\angPhi:\R);
\end{tikzpicture}

\end{document}

答案1

改变箭头的大小意味着改变箭头尖端和尾部之间的角度间隔，因为它们以极坐标形式出现。减小角度间隔后，还必须减小箭头的“弯曲”，使其在赤道处模糊：

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}
\usepackage{verbatim}

%% helper macros

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,dashed}}
   % 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\DrawLatitudeCircle[2][2]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,line width=.7pt}}
  \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[red,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{document}

\begin{tikzpicture}[scale=1.4] % "THE GLOBE" showcase

\def\R{2.5} % sphere radius
\def\angEl{35} % elevation angle
\def\angAz{-105} % azimuth angle
\def\angPhi{-40} % longitude of point P
\def\angBeta{0} % latitude of point P
%
\filldraw[ball color=white] (0,0) circle (\R);
\foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} }
\foreach \t in {-5,-35,...,-175} { \DrawLongitudeCircle[\R]{\t} }
%
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate[mark coordinate] (N) at (0,\H);
\draw (N)  node[inner sep=1pt,below=0pt] {$\infty$}; %+(0.3ex,0.6ex)
%
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhi}
\LatitudePlane[equator]{\angEl}{0}
\draw[red,equator,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\end{tikzpicture}

\end{document}

在此处输入图片描述

为了在其他纬度上绘制矢量场，可以在其他纬度定义另一个平面，就像原始代码定义“赤道”一样。

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}
\usepackage{verbatim}

%% helper macros

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][4]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,dashed}}
   % 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\DrawLatitudeCircle[2][5]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,line width=.7pt}}
  \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[gray,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{document}

\begin{tikzpicture}[scale=1.4] % "THE GLOBE" showcase

\def\R{2.5} % sphere radius
\def\angEl{35} % elevation angle
\def\angAz{-105} % azimuth angle
\def\angPhi{-40} % longitude of point P
\def\angBeta{0} % latitude of point P
%
\filldraw[ball color=white] (0,0) circle (\R);
\foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} }
\foreach \t in {-5,-35,...,-175} { \DrawLongitudeCircle[\R]{\t} }
%
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate[mark coordinate] (N) at (0,\H);
\draw (N)  node[inner sep=1pt,below=0pt] {$\infty$}; %+(0.3ex,0.6ex)
%
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhi}
\LatitudePlane[equator]{\angEl}{0}
\LatitudePlane[capr]{\angEl}{37}
\LatitudePlane[sag]{\angEl}{68}
\draw[red,equator,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\draw[green,sag,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\draw[blue,capr,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\end{tikzpicture}

\end{document}

结果如下（您可以更改颜色，我只是为了使箭头更易于追踪而显示它们）：在此处输入图片描述

要在同一个圆圈上添加更多箭头，可以使用此代码

\draw[red,equator,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\draw[red,equator,->,thick] (-135:\R) to[bend right=11] (-110:\R);
\draw[red,equator,->,thick] (-75:\R) to[bend right=11] (-50:\R);
\draw[red,equator,->,thick] (-40:\R) to[bend right=8] (-25:\R);
\draw[blue,capr,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\draw[blue,capr,->,thick] (-135:2.6) to[bend right=11] (-110:\R);
\draw[blue,capr,->,thick] (-75:\R) to[bend right=11] (-50:2.6);
\draw[blue,capr,->,thick] (-40:2.65) to[bend right=8] (-25:2.7);

在此处输入图片描述

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