Elliptic anomaly in TikZ

ellipse.tex

% vim:nolist lbr tw=78 expandtab autoindent nocindent
\documentclass[a4paper]{minimal}
\usepackage[landscape,top=1cm,bottom=1cm,left=1cm,right=1cm]{geometry}

\usepackage{tikz}
%\usetikzlibrary{decorations.pathreplacing}

\everymath{\displaystyle} % set math in display style, not text style (ie. larger)

\begin{document}

\begin{tikzpicture}[scale=1.0]
    \pgfmathsetmacro{\maxr}{7}

    \pgfmathsetmacro{\p}{2}
    \pgfmathsetmacro{\e}{0.7}
    \pgfmathsetmacro{\a}{\p / (1 - \e^2)}
    \pgfmathsetmacro{\b}{\p / sqrt(1 - \e^2)}
    \pgfmathsetmacro{\c}{sqrt(\a^2 - \b^2)} % XXX: ellipse only!
    %\pgfmathsetmacro{\c}{\a*\e}
    \pgfmathsetmacro{\q}{\p / (1 + \e)}
    \pgfmathsetmacro{\directrixx}{\p / \e}

    \pgfmathsetmacro{\f}{115} % true anomaly in degrees
    % atan2() argument order ?!
    \pgfmathsetmacro{\E}{atan2(sqrt(1 - \e^2) * sin(\f), cos(\f) + \e)}

    \pgfmathsetmacro{\Pr}{\p / (1 + \e * cos(\f))}
    \pgfmathsetmacro{\Px}{\Pr * cos(\f)}
    \pgfmathsetmacro{\Py}{\Pr * sin(\f)}

    \pgfmathsetmacro{\Qx}{-\c + \a*cos(\E)}
    \pgfmathsetmacro{\Qy}{\a * sin(\E)}

    \coordinate (F) at (0,0);
    \coordinate (C) at (-\c,0);
    \coordinate (Ap) at (-\c - \a,0);
    \coordinate (Pe) at (\q,0);
    \coordinate (Mi) at (-\c,-\b);
    \coordinate (Re) at (0,-\p);

    \coordinate (P) at (\Px, \Py);
    \coordinate (Q) at (\Qx, \Qy);
    \coordinate (N) at (\directrixx, \Py);

    % grid and axes
    %\draw[gray,thin] (-\maxr,-\maxr) grid (\maxr,\maxr);
    \draw[->, thin] (-\maxr, 0) -- (\maxr+0.2, 0)
        node[right]{$x$};
    \draw[->, thin] (0, -\maxr) -- (0, \maxr+0.2)
        node[above]{$y$};

    % conic section
    \fill[color=gray, opacity=0.5, domain=0:rad(\f), samples=100, smooth]
        (F) --
        plot (xy polar cs:angle=\x r, radius={\p / (1 + \e * cos(\x r))});

    \draw[thick, domain=-pi:pi, samples=200, smooth]
        plot (xy polar cs:angle=\x r, radius={\p / (1 + \e * cos(\x r))});

    % directrix
    \draw[thick] (\directrixx,-\a) -- (\directrixx,\a)
        node[above, color=black]{Directrix};
    \draw[<->, thick] (0,-\p) -- (\directrixx,-\p)
        node[below, midway]{$\frac{p}{e}$};

    % conic section measurements (major axis, minor axis, latus rectum)
    \draw[<->,thick] (C) -- (Ap)
        node[below, midway]{$a$};
    \draw[<->,thick] (C) -- (F)
        node[below, midway]{$c$};
    \draw[<->,thick] (C) -- (Mi)
        node[left, midway]{$b$};
    \draw[<->,thick] (F) -- (Re)
        node[left, midway]{$p$};
    \draw[<->,thick] (F) -- (Pe)
        node[below, midway]{$q$};

    % auxiliary circle
    \draw[thin, dashed] (C) circle[radius={\a}];

    % arrows from focus and center to orbit and aux circle
    \draw[->, thick] (F) -- (P)
        node[midway, anchor=east]{$r$}
        node[anchor=south west]{$P$};
    \draw[->, thick] (C) -- (Q)
        node[midway, anchor=east]{$a$}
        node[anchor=south west]{$Q$};

    % arrow to directrix
    \draw[<->, thick] (P) -- (N)
        node[above, midway]{$\frac{r}{e}$};

    % arcs for true and eccentric anomaly
    \draw[->, thick]
        (F) node[anchor=south west]{$f$}
        ++(0.7,0) arc [start angle=0, end angle={\f}, radius=0.7];
    \draw[->, thick]
        (C) node[anchor=south west]{$E$}
        ++(0.7,0) arc [start angle=0, end angle={\E}, radius=0.7];
\end{tikzpicture}

\begin{tikzpicture}[scale=1.0]

    \pgfmathsetmacro{\maxr}{7}

    \pgfmathsetmacro{\p}{2.3}
    \pgfmathsetmacro{\e}{1.2}
    \pgfmathsetmacro{\a}{\p / (1 - \e^2)}
    \pgfmathsetmacro{\b}{\p / sqrt(\e^2 - 1)}
    %\pgfmathsetmacro{\c}{sqrt(\a^2 - \b^2)} % XXX: ellipse only!
    \pgfmathsetmacro{\c}{-\a*\e}
    \pgfmathsetmacro{\q}{\p / (1 + \e)}
    \pgfmathsetmacro{\directrixx}{\p / \e}
    \pgfmathsetmacro{\maxf}{rad(acos((\p / \maxr - 1) / \e))}

    \pgfmathsetmacro{\f}{110} % true anomaly in degrees
    \pgfmathsetmacro{\tanhF}{sqrt(\e^2 - 1) * sin(\f) / (\e + cos(\f))} % tanh(F)
    % note: atanh(x) = ln((1+x)/(1-x))/2
    \pgfmathsetmacro{\F}{ln((1+\tanhF)/(1-\tanhF))/2}

    \pgfmathsetmacro{\Pr}{\p / (1 + \e * cos(\f))}
    \pgfmathsetmacro{\Px}{\Pr * cos(\f)}
    \pgfmathsetmacro{\Py}{\Pr * sin(\f)}

    \pgfmathsetmacro{\Qx}{\a*(cosh(\F) - \e)}
    \pgfmathsetmacro{\Qy}{-\a * sinh(\F)}

    \coordinate (F) at (0,0);
    \coordinate (C) at (\c,0);
    \coordinate (Pe) at (\q,0);
    \coordinate (Mi) at (-\c,-\b);
    \coordinate (Re) at (0,-\p);

    \coordinate (P) at (\Px, \Py);
    \coordinate (Q) at (\Qx, \Qy);
    \coordinate (N) at (\directrixx, \Py);

    % grid and axes
    %\draw[gray,thin] (-\maxr,-\maxr) grid (\maxr,\maxr);
    \draw[->, thin] (-\maxr, 0) -- (\maxr+0.2, 0)
        node[right]{$x$};
    \draw[->, thin] (0, -\maxr) -- (0, \maxr+0.2)
        node[above]{$y$};

    % conic section
    \draw[thick, domain=-\maxf:\maxf, samples=200, smooth]
        plot (xy polar cs:angle=\x r, radius={\p / (1 + \e * cos(\x r))});

    % arrow to point on conic
    \draw[->, thick] (F) -- (P)
        node[midway, anchor=east]{$r$}
        node[anchor=south west]{$P$};
    \draw[->, thick] (C) -- (Q)
        node[anchor=south west]{$Q$};

    \draw[thin, dashed] (C) -- ({\a * (cosh(\F) - \e)}, {\b * sinh(\F)});

    % directrix
    \draw[thick] (\directrixx,-\maxr) -- (\directrixx,\maxr)
        node[above, color=black]{Directrix};
    \draw[thick] (\c,-\maxr) -- (\c,\maxr);
        %node[above, color=black]{Center};
    %\draw[thick] (2*\c-\directrixx,-\maxr) -- (2*\c-\directrixx,\maxr)
        %node[above, color=black]{Directrix};

    % asymptotes
    \draw[thin, dashed]
        ({\maxr * \a / \b + \c}, \maxr) --
        (\maxr, {(\b / \a) * (\maxr - \c)});
    \draw[thin, dashed]
        ({\maxr * \a / \b + \c}, -\maxr) --
        (\maxr, {-(\b / \a) * (\maxr - \c)});

    % major and minor axes
    \draw[thin, dashed]
        (\c, \b) -- (\q, \b) -- (\q, -\b) -- (\c, -\b);
    \draw[thin, dashed] (C) ++(0,\c) arc [radius={\c}, start angle=90, end angle=270];


    % auxiliary hyperbola with b = a (concentric)
    \draw[thin, smooth, domain=-\maxr:\maxr]
        plot ({\c - sqrt((\a)^2 * (1 + (\x)^2 / (\a)^2))}, {\x});

    % fill area swept by hyperbolic anomaly
    \fill[color=gray, opacity=0.5, domain=0:\F, smooth]
        (C) --
        plot ({\c + \a * cosh(\x)}, {-\a * sinh(\x)});

    % fill area swept by satellite
    \fill[color=gray, opacity=0.5, domain=0:rad(\f), samples=100, smooth]
        (F) --
        plot (xy polar cs:angle=\x r, radius={\p / (1 + \e * cos(\x r))});

    % arcs for true and hyperbolic anomaly
    \draw[->, thick]
        (F) node[anchor=south west]{$f$}
        ++(0.7,0) arc [start angle=0, end angle={\f}, radius=0.7];
    \draw[thick]
        (C) node[anchor=north east]{$\frac{F}{2}$}
        ++(-0.7,0) arc [start angle=180, end angle={180-deg(\F / 2)}, radius=0.7];

    % XXX: is the damn auxiliary hyperbola concentric or confocal???
    % XXX: confocal auxiliary parabola
    \pgfmathsetmacro{\paux}{\q * (1 + sqrt(2))}
    \pgfmathsetmacro{\eaux}{sqrt(2)}
    \pgfmathsetmacro{\maxfaux}{rad(acos((\paux / \maxr - 1) / \eaux))}
    \pgfmathsetmacro{\aaux}{\paux / (1 - \eaux^2)}
    \pgfmathsetmacro{\baux}{\paux / sqrt(\eaux^2 - 1)}
    \pgfmathsetmacro{\caux}{-\aaux*\eaux}

    \coordinate (Caux) at (\caux,0);

    %\draw[thin, domain=-\maxfaux:\maxfaux, smooth]
        %plot (xy polar cs:angle=\x r, radius={\paux / (1 + \eaux * cos(\x r))});

    %\draw[thin, dashed]
        %({\Px}, 0) -- ({\Px}, {\maxr});

    % \draw[thin] ({\caux}, 0) -- (P);

    %\draw[->, thick]
    %   (Caux) node[anchor=north east]{$\frac{F}{2}$}
    %   ++(-0.7,0) arc [start angle=180, end angle={180-deg(\F / 2)}, radius=0.7];

\end{tikzpicture}

\end{document}