[Tex/LaTex] Dance of Venus (and variations) in TikZ/PGF

tikz-pgf

If we look down upon the Solar System in a geocentric model, the Sun and all the planets will be orbiting the Earth. In this model, the Earth is at the center of the universe. Let us focus on two celestial bodies other than Earth: Sun and Venus. If we assume the Earth is fixed, as the Sun orbits the Earth 8 times, Venus will do so 13 times. This will create a beautiful diagram known as the Pentagram of Venus (in reality, the ratio is 8:13.004 and the non-integer coefficient will create a wobble). Matthew Henderson posted this simplified image of the cosmic dance of Venus and Earth on his Tumblr webpage:

EDIT. The image above shows the dance of Venus from a heliocentric perspective, not a geocentric one as I explained. I missed it because I had several images of both heliocentric and geocentric Venus Pentagram open in my browser and I apparently copied the wrong one. Thanks to Keks Dose for mentioning that. Here is the link to an awesome video that shows the dance of Venus with Earth fixed at the center.

I was wondering if I can draw this diagram using TikZ/PGF. I have seen a similar topic on Mathematica Stack Exchange but I want to use LaTeX. I recently found the packages pst-solarsystem and tikz-planets and it would be cool to use their features if possible. I do not need the animation, any help to create the final static image would be appreciated.

Best Answer

Fascinating and lots of fun. This essentially one-liner:

\documentclass{article}

\usepackage[halfletter,margin=0.6in]{geometry}

\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \foreach \x in {0,5,...,2880}{
        \draw[help lines] (\x:5) -- ({(\x/8)*13}:4);%% <<<--- avoids arithmetic overflow
    }
\end{tikzpicture}

\end{document}

yields this beautiful graphic:

Update

Well, it had to happen. Here is a macro that will do the drawing, along with the ability to play with the parameters:

\documentclass[tikz,border=20pt]{standalone}

\usepackage{keyval}

\newlength{\outerlength}
\setlength{\outerlength}{4in}
\newlength{\innerlength}
\setlength{\innerlength}{3in}
\def\outerrate{8}
\def\innerrate{13}
\def\maxtimes{2880}
\def\increment{2}

\makeatletter
\define@key{venus}{outerlength}{\setlength{\outerlength}{#1}}
\define@key{venus}{innerlength}{\setlength{\innerlength}{#1}}
\define@key{venus}{outerrate}{\def\outerrate{#1}}
\define@key{venus}{innerrate}{\def\innerrate{#1}}
\define@key{venus}{maxtimes}{\def\maxtimes{#1}}
\define@key{venus}{increment}{\def\increment{#1}}

\newcommand{\makevenus}[1][]{%
    \setkeys{venus}{#1}
    \begin{tikzpicture}
        \foreach \x in {0,\increment,...,\maxtimes}{%
            \draw[help lines] (\x:\outerlength) --
                ({(\x/\outerrate)*\innerrate}:\innerlength);%% <<<--- avoids arithmetic overflow
        }%
    \end{tikzpicture}%
}

\begin{document}

\makevenus[innerrate=7,innerlength=2.25in,increment=3]

\end{document}

which produces:

while \makevenus[outerrate=5,innerrate=13,innerlength=3.75in,increment=1] produces (I love my new toy!):

Update 2

I was curious about what would happen if the centerpoint of each line drawn was shown. Interestingly, as the macro \newmakevenus shows, it traces the shapes formed by the intersections of the lines.

\RequirePackage[rgb]{xcolor}
\documentclass[tikz,border=20pt]{standalone}

\usepackage{xparse}
\usepackage{keyval}
\usepackage{fp}

\newlength{\outerlength}
\setlength{\outerlength}{4in}
\newlength{\innerlength}
\setlength{\innerlength}{2.8933in}
\def\outerrate{8}
\def\innerrate{13}
\def\maxtimes{2880}
\def\increment{2}
\colorlet{mycolor}{black}

\makeatletter
\define@key{venus}{outerlength}{\setlength{\outerlength}{#1}}
\define@key{venus}{innerlength}{\setlength{\innerlength}{#1}}
\define@key{venus}{outerrate}{\def\outerrate{#1}}
\define@key{venus}{innerrate}{\def\innerrate{#1}}
\define@key{venus}{maxtimes}{\def\maxtimes{#1}}
\define@key{venus}{increment}{\def\increment{#1}}
\define@key{venus}{venuscolor}{\colorlet{mycolor}{#1}}
\makeatother

\NewDocumentCommand{\makevenus}{sO{}}{%
    \setkeys{venus}{#2}
    \begin{tikzpicture}
        \foreach \x in {0,\increment,...,\maxtimes}{%
            \IfBooleanT{#1}{%
                \pgfmathsetmacro{\huenum}{\x/\maxtimes}
                \definecolor{mycolor}{hsb}{\huenum,0.5,1}
            }%
            \draw[help lines,mycolor] (\x:\outerlength) --
                ({(\x/\outerrate)*\innerrate}:\innerlength);%% <<<--- avoids arithmetic overflow
        }%
    \end{tikzpicture}%
}

\NewDocumentCommand{\newmakevenus}{sO{}}{%
    \setkeys{venus}{#2}
    \begin{tikzpicture}
        \foreach \x in {0,\increment,...,\maxtimes}{%
            \IfBooleanT{#1}{%
                \pgfmathsetmacro{\huenum}{\x/\maxtimes}
                \definecolor{mycolor}{hsb}{\huenum,0.5,1}
            }%
            \draw(\x:\outerlength) --node[circle,fill=orange,pos=.5]{}
                ({(\x/\outerrate)*\innerrate}:\innerlength);%% <<<--- avoids arithmetic overflow
        }%
    \end{tikzpicture}%
}

\begin{document}

%%\makevenus*[outerrate=8,innerrate=13,increment=2,maxtimes=2880,innerlength=2.9in]

\newmakevenus[outerrate=8,innerrate=13,increment=3,innerlength=2.9in,maxtimes=2880]

\end{document}

An optional * will colorize the graphic so that \makevenus*[outerrate=8,innerrate=13,increment=2,maxtimes=2880,innerlength=2.9in] will produce:

2880 is simply 8*360.

A festive holiday update

Colorizing the graphic is simple: using the hsb colorspace and changing the hue of each line as it is drawn (requires \RequirePackage[rgb]{xcolor} before \documentclass command). While interesting, I wondered what would happen if the number of color cycles (red-yellow->green->blue->purple->red) were to cycle as many times as the lobes of the pattern generated -- turns out this is simply abs(\innerrate-\outerrate) -- this works best if either/both \innerrate or \outerrate are prime numbers. I was stunned by just how effective this is:

\RequirePackage[rgb]{xcolor}
\documentclass[tikz,border=20pt]{standalone}

\usepackage{xparse}
\usepackage{keyval}

\newlength{\outerlength}
\setlength{\outerlength}{4in}
\newlength{\innerlength}
\setlength{\innerlength}{2.8933in}
\def\outerrate{8}
\def\innerrate{13}
\def\maxtimes{2880}
\def\increment{2}
\def\changephase{0}
\colorlet{mycolor}{black}

\makeatletter
\define@key{venus}{outerlength}{\setlength{\outerlength}{#1}}
\define@key{venus}{innerlength}{\setlength{\innerlength}{#1}}
\define@key{venus}{outerrate}{\def\outerrate{#1}}
\define@key{venus}{innerrate}{\def\innerrate{#1}}
\define@key{venus}{increment}{\def\increment{#1}}
\define@key{venus}{changephase}{\def\changephase{#1}}
\define@key{venus}{venuscolor}{\colorlet{mycolor}{#1}}
\makeatother

\NewDocumentCommand{\makevenus}{sO{}}{%
    \setkeys{venus}{#2}
    \begin{tikzpicture}
        \pgfmathsetmacro{\nmaxtimes}{360*\outerrate}
        \foreach \x in {0,\increment,...,\nmaxtimes}{%
            \IfBooleanT{#1}{%
                \pgfmathsetmacro{\huenum}{%
                    abs(sin((360/\nmaxtimes)*\x*abs(\innerrate-\outerrate)+\changephase))
                }%
                \definecolor{mycolor}{hsb}{\huenum,1,1}
            }%
            \draw[help lines,mycolor] (\x:\outerlength) --
                ({(\x/\outerrate)*\innerrate}:\innerlength);%% <<<--- avoids arithmetic overflow
        }%
    \end{tikzpicture}%
}


\begin{document}

\makevenus*[outerrate=8,innerrate=17,increment=2,innerlength=3.5in,changephase=60]

\end{document}

It is also possible to change the phase of the color cycling using changephase so that \makevenus*[outerrate=8,innerrate=13,increment=2,innerlength=3.5in,changephase=60] produces:

Hope you have fun with this -- I know I did.

A Festive New Year's Fireworks Update

Most importantly, a terrific vote of thanks to @Thruston on two counts: first for the elegant one-liner that does the drawing and, second, for the wonderful MP code that translates HSL to RGB.

The following code outputs three styles of graphics: 1: a monochrome version; 2: Thruston's original version of coloring; 3: phased coloring that repeats the spectrum (inner rate - outer rate) times. Note that I am not a MP expert by any manner of means, so that, no doubt, all sorts of improvements could be made to this code.

\documentclass[border=10pt]{standalone}
\usepackage{luamplib}
\usepackage{xparse}
\usepackage{graphicx}
\mplibnumbersystem{double}

%% TO DO: use keyval.sty
%% [#1 step delta]; #2 inner r; #3 outer r;
%% #4 [cycle offset]
%% #5 inner rate; #6 outer rate; 
%% #7[pen scale]; #8*=color (& #9* phased color)
%% #3 (outer r) is the radius of the finished graphic, assuming that it is
%% greater than #2 (inner r).
\NewDocumentCommand{\makevenusL}{O{1} m m O{0} m m O{.125} s s}{%
    \begin{mplibcode}
    vardef hsv_color_cyc(expr h,s,v) = % HSB->rgb following wikipedia article on "HSL and HSV"
          save chroma, hh, x, m;
          chroma = v*s;
          %% The following cycles through color spectrum (\innerrate-\outerrate) times
          %% a constant can be added to shift the cycle.
          hh = (((#6-#5)*h + #4) mod 360)/60;
          x  = chroma * (1-abs(hh mod 2 - 1));
          m  = v - chroma;
          if      hh < 1: (chroma,x,0)+(m,m,m)
          elseif hh < 2: (x,chroma,0)+(m,m,m)
          elseif hh < 3: (0,chroma,x)+(m,m,m)
          elseif hh < 4: (0,x,chroma)+(m,m,m)
          elseif hh < 5: (x,0,chroma)+(m,m,m)
          else:          (chroma,0,x)+(m,m,m)
          fi
    enddef;
    vardef hsv_color(expr h,s,v) = % HSB->rgb following wikipedia article on "HSL and HSV"
          save chroma, hh, x, m;
          chroma = v*s;
          %% The following cycles through color range once
          hh = h/60;
          x  = chroma * (1-abs(hh mod 2 - 1));
          m  = v - chroma;
          if      hh < 1: (chroma,x,0)+(m,m,m)
          elseif hh < 2: (x,chroma,0)+(m,m,m)
          elseif hh < 3: (0,chroma,x)+(m,m,m)
          elseif hh < 4: (0,x,chroma)+(m,m,m)
          elseif hh < 5: (x,0,chroma)+(m,m,m)
          else:          (chroma,0,x)+(m,m,m)
          fi
    enddef;
    \IfBooleanTF{#8}
    {%
        \IfBooleanTF{#9}
        {% cycle through spectrum multiple times
            beginfig(1);
                for t=0 step #1 until 360: 
                    draw #2 dir #5t -- #3 dir #6t
                        withpen pencircle scaled #7 withcolor hsv_color_cyc(t, 1, 1); 
                endfor;
                draw (288,0) .. (0,288) .. (-288,0) .. (0,-288) .. cycle;
            endfig;
        }%
        {% cycle through spectrum once
            beginfig(1);
                for t=0 step #1 until 360: 
                    draw #2 dir #5t -- #3 dir #6t
                        withpen pencircle scaled #7 withcolor hsv_color(t, 1, 1); 
                endfor;
                draw (288,0) .. (0,288) .. (-288,0) .. (0,-288) .. cycle;
            endfig;
        }%
    }%
    {% one color
        beginfig(1);
            for t=0 step #1 until 360: 
                 draw #2 dir #5t -- #3 dir #6t
                    %% << change (r,g,b) spec to suit
                    withpen pencircle scaled #7 withcolor (0.2,0.5,0.5); %% (0,0,0) for black
            endfor;
            draw (288,0) .. (0,288) .. (-288,0) .. (0,-288) .. cycle;
        endfig;
    }%
    \end{mplibcode}
}

\begin{document}

\makevenusL[.2]{260}{288}{8}{13}[0.125]
\makevenusL[.2]{260}{288}{8}{13}[0.125]*
\makevenusL[.2]{260}{288}[225]{8}{13}[0.125]**

\end{document}


%%% Local Variables: 
%%% coding: utf-8
%%% mode: latex
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%%% End:

Related Solutions

[Tex/LaTex] Vertical and horizontal lines in pgf-tikz

This answer extends Marc van Dongen's anwser.

The two styles are -|- and |-| with default value 0.5 to position the intermediate point (so, default is middle point).

enter image description here

\documentclass[margin=2mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\tikzset{
  -|-/.style={
    to path={
      (\tikztostart) -| ($(\tikztostart)!#1!(\tikztotarget)$) |- (\tikztotarget)
      \tikztonodes
    }
  },
  -|-/.default=0.5,
  |-|/.style={
    to path={
      (\tikztostart) |- ($(\tikztostart)!#1!(\tikztotarget)$) -| (\tikztotarget)
      \tikztonodes
    }
  },
  |-|/.default=0.5,
}
\begin{document}
\begin{tikzpicture}[thick]
  \draw[->,blue] (2.1,1.1) to[|-|] (1.1,0.1);
  \draw[->] (2,1) to[-|-] (1,0);

  \draw[->,blue] (3.1,0.1) to[|-|] (4.1,1.1);
  \draw[->] (3,0) to[-|-] (4,1);

  \begin{scope}[yshift=-1.5cm]
    \draw[->,red] (2.1,1.1) to[|-|=.2] (1.1,0.1);
    \draw[->,orange] (2,1) to[-|-=.2] (1,0);

    \draw[->,red] (3.1,0.1) to[|-|=.8] (4.1,1.1);
    \draw[->,orange] (3,0) to[-|-=.8] (4,1);
  \end{scope}
\end{tikzpicture}
\end{document}

[Tex/LaTex] Drawing of Sun–Earth–Moon system

A couple of months ago, I produced a beamer/tikz animation (available on texample.net) of the Earth's orbit around the Sun to illustrate the counterintuitive fact that Earth is farther from the Sun in Summer than it is in Winter. I used that example to demonstrate the power of inducing (and then resolving) cognitive dissonance in the classroom.

I've modified it to also show the Moon following an elliptical orbit around the Earth. (Of course, the Moon's orbit around the Earth is really only approximately elliptical, and does not lie in the Ecliptic Plane.) You can change the positions visited by the Earth and the Moon along their respective orbits (simply change how \Earthangle and \Moonangle are defined in the body of the \foreach, and/or modify the value of \N).

enter image description here

\documentclass{beamer}

\usepackage{lmodern}
\usepackage{tikz}

\setbeamertemplate{navigation symbols}{}

\begin{document}
\begin{frame}[fragile]
\frametitle{}
\begin{center}
    \begin{tikzpicture}[scale=2.5]
        \def\rS{0.3}                                % Sun radius

        \def\rE{0.1}                                % Earth radius
                                                    % Major radius of Earth's elliptical orbit = 1
        \def\eE{0.25}                               % Excentricity of Earth's elliptical orbit       
        \pgfmathsetmacro\bE{sqrt(1-\eE*\eE)}        % Minor radius of Earth's elliptical orbit    

        \pgfmathsetmacro\rM{.7*\rE}                 % Moon radius
        \pgfmathsetmacro\aM{2.5*\rE}                % Major radius of the Moon's elliptical orbit
        \def\eM{0.4}                                % Excentricity of Earth's elliptical orbit
        \pgfmathsetmacro\bM{\aM*sqrt(1-\eM*\eM)}    % Minor radius of the Moon's elliptical orbit 
        \def\offsetM{30}                            % angle offset between the major axes of Earth's and the Moon's orbits


        % This function computes the direction in which light hits the Earth.
        \pgfmathdeclarefunction{f}{1}{%
            \pgfmathparse{
                ((-\eE+cos(#1))<0) * ( 180 + atan( \bE*sin(#1)/(-\eE+cos(#1)) ) ) 
                +
                ((-\eE+cos(#1))>=0) * ( atan( \bE*sin(#1)/(-\eE+cos(#1)) ) ) 
            }
        }

        % This function computes the distance between Earth and the Sun,
        % which is used to calculate the varying radiation intensity on Earth.
        \pgfmathdeclarefunction{d}{1}{%
            \pgfmathparse{ sqrt((-\eE+cos(#1))*(-\eE+cos(#1))+\bE*sin(#1)*\bE*sin(#1)) }
        }

        % Draw the elliptical path of the Earth.
        \draw[thin,color=gray] (0,0) ellipse (1 and \bE);

        % Draw the Sun at the right-hand-side focus
        \shade[
            top color=yellow!70,
            bottom color=red!70,
            shading angle={45},
            ] ({sqrt(1-\bE*\bE)},0) circle (\rS);
         %\draw ({sqrt(1-\b*\b)},-\rS) node[below] {Sun};

        % Produces a series of frames showing one revolution
        % (the total number of frames is controlled by macro \N)
        \pgfmathtruncatemacro{\N}{12}
        \foreach \k in {0,1,...,\N}{
            \pgfmathsetmacro{\Earthangle}{360*\k/\N}
            \pgfmathsetmacro{\Moonangle}{3*360*\k/\N} % <--- change the multiplying factor to suit your needs
            % Draw the Earth at \Earthangle
            \pgfmathsetmacro{\radiation}{100*(1-\eE)/(d(\Earthangle)*d(\Earthangle))}
            \colorlet{Earthlight}{yellow!\radiation!blue}
            \pgfmathparse{int(\k+1)}
            \onslide<\pgfmathresult>{
                \shade[
                    top color=Earthlight,
                    bottom color=blue,
                    shading angle={90+f(\Earthangle)},
                    ] ({cos(\Earthangle)},{\bE*sin(\Earthangle)}) circle (\rE);
                 %\draw ({cos(\Earthangle)},{\bE*sin(\Earthangle)-\rE}) node[below] {Earth};  

                 % Draw the Moon's (circular) orbit and the Moon at \Moonangle
                 \draw[thin,color=gray,rotate around={{\offsetM}:({cos(\Earthangle)},{\bE*sin(\Earthangle)})}]
                    ({cos(\Earthangle)},{\bE*sin(\Earthangle)}) ellipse ({\aM} and {\bM});
                 \shade[
                    top color=black!70,
                    bottom color=black!30,
                    shading angle={45},
                    ]   ({cos(\Earthangle)+\aM*cos(\Moonangle)*cos(\offsetM)-\bM*sin(\Moonangle)*sin(\offsetM)},%
                         {\bE*sin(\Earthangle)+\aM*cos(\Moonangle)*sin(\offsetM)+\bM*sin(\Moonangle)*cos(\offsetM)}) circle (\rM);   
            }
        }
    \end{tikzpicture}
\end{center}
\end{frame}
\end{document}

Here is non-animated version in the article class.

enter image description here

\documentclass{article}

\usepackage{lmodern}
\usepackage{tikz}
\usepackage{kantlipsum}

\begin{document}
\section{Eclipses}
\kant[1]
\begin{figure}
\centering
    \begin{tikzpicture}[scale=2.5]
        \def\rS{0.3}                                % Sun radius

        \def\Earthangle{30}                         % angle wrt to horizontal        
        \def\rE{0.1}                                % Earth radius
                                                    % Major radius of Earth's elliptical orbit = 1
        \def\eE{0.25}                               % Excentricity of Earth's elliptical orbit       
        \pgfmathsetmacro\bE{sqrt(1-\eE*\eE)}        % Minor radius of Earth's elliptical orbit

        \def\Moonangle{-45}                         % angle wrt to horizontal           
        \pgfmathsetmacro\rM{.7*\rE}                 % Moon radius
        \pgfmathsetmacro\aM{2.5*\rE}                % Major radius of the Moon's elliptical orbit
        \def\eM{0.4}                                % Excentricity of Earth's elliptical orbit
        \pgfmathsetmacro\bM{\aM*sqrt(1-\eM*\eM)}    % Minor radius of the Moon's elliptical orbit 
        \def\offsetM{30}                            % angle offset between the major axes of Earth's and the Moon's orbits



        % This function computes the direction in which light hits the Earth.
        \pgfmathdeclarefunction{f}{1}{%
            \pgfmathparse{
                ((-\eE+cos(#1))<0) * ( 180 + atan( \bE*sin(#1)/(-\eE+cos(#1)) ) ) 
                +
                ((-\eE+cos(#1))>=0) * ( atan( \bE*sin(#1)/(-\eE+cos(#1)) ) ) 
            }
        }

        % This function computes the distance between Earth and the Sun,
        % which is used to calculate the varying radiation intensity on Earth.
        \pgfmathdeclarefunction{d}{1}{%
            \pgfmathparse{ sqrt((-\eE+cos(#1))*(-\eE+cos(#1))+\bE*sin(#1)*\bE*sin(#1)) }
        }

        % Draw the elliptical path of the Earth.
        \draw[thin,color=gray] (0,0) ellipse (1 and \bE);

        % Draw the Sun at the right-hand-side focus
        \shade[
            top color=yellow!70,
            bottom color=red!70,
            shading angle={45},
            ] ({sqrt(1-\bE*\bE)},0) circle (\rS);
         %\draw ({sqrt(1-\b*\b)},-\rS) node[below] {Sun};


        % Draw the Earth at \Earthangle
        \pgfmathsetmacro{\radiation}{100*(1-\eE)/(d(\Earthangle)*d(\Earthangle))}
        \colorlet{Earthlight}{yellow!\radiation!blue}
        \shade[%
            top color=Earthlight,%
            bottom color=blue,%
            shading angle={90+f(\Earthangle)},%
        ] ({cos(\Earthangle)},{\bE*sin(\Earthangle)}) circle (\rE);
        %\draw ({cos(\Earthangle)},{\bE*sin(\Earthangle)-\rE}) node[below] {Earth};  

        % Draw the Moon's (circular) orbit and the Moon at \Moonangle
        \draw[thin,color=gray,rotate around={{\offsetM}:({cos(\Earthangle)},{\bE*sin(\Earthangle)})}]
            ({cos(\Earthangle)},{\bE*sin(\Earthangle)}) ellipse ({\aM} and {\bM});
        \shade[
            top color=black!70,
            bottom color=black!30,
            shading angle={45},
        ]   ({cos(\Earthangle)+\aM*cos(\Moonangle)*cos(\offsetM)-\bM*sin(\Moonangle)*sin(\offsetM)},%
            {\bE*sin(\Earthangle)+\aM*cos(\Moonangle)*sin(\offsetM)+\bM*sin(\Moonangle)*cos(\offsetM)}) circle (\rM);   
    \end{tikzpicture}
    \caption{Sun, Earth and Moon}
\end{figure}
\end{document}

Best Answer

Update

Update 2

A festive holiday update

A Festive New Year's Fireworks Update

Related Solutions

[Tex/LaTex] Vertical and horizontal lines in pgf-tikz

[Tex/LaTex] Drawing of Sun–Earth–Moon system

Related Question