Use tikz (for example) to draw pictures in hyperbolic geometry

Question

I would like to draw pictures in the Poincaré disk model for hyperbolic geometry. Are there any built-in or add-on packages for tikz to do this? For example, it would be nice to have functions for drawing Saccheri quadrilaterals or Lambert quadrilaterals or asymptotic triangles.

(In my brief use so far, the tkz-euclide package looks very good for Euclidean geometry, and now I'm looking for a hyperbolic analogue of it.)

Answer 1

Here's another TikZ solution. Given two points on the Poincare plane or disc, it draws the geodesic between them. It does this by defining a (complicated!) to path which works out the circle and angles and then draws the correct arc. It has some basic error checking to see if it ought to draw a straight line, but it checks for the exact condition rather than to within the precision of TeX so if your points are almost one above the other (or on the same radius) so that to all intents and purposes the geodesic is straight then this code might not detect that in time and end up dividing by something smaller than it should (as it is, I had to be careful how I broke up the calculations as some methods gave Big Numbers at partial stages).

\documentclass{article}
\thispagestyle{empty}
\usepackage{tikz}

\makeatletter

\def\hyper@x#1,#2\relax{#1}
\def\hyper@y#1,#2\relax{#2}
\def\hyper@coords#1{#1}

\newif\ifhyper@vertical

\def\hyper@computer#1#2{%
  \edef\hyper@toscan{(#1)}
  \tikz@scan@one@point\hyper@coords\hyper@toscan
  \edef\hyper@sx{\the\pgf@x}
  \edef\hyper@sy{\the\pgf@y}
  \edef\hyper@toscan{(#2)}
  \tikz@scan@one@point\hyper@coords\hyper@toscan
  \edef\hyper@ex{\the\pgf@x}
  \edef\hyper@ey{\the\pgf@y}
  \pgfmathsetmacro{\hyper@mx}{(\hyper@ex + \hyper@sx)/2}
  \pgfmathsetmacro{\hyper@my}{(\hyper@ey + \hyper@sy)/2}
  \pgfmathsetmacro{\hyper@dx}{\hyper@ex - \hyper@sx}
  \pgfmathparse{\hyper@dx == 0 ? "\noexpand\hyper@verticaltrue" : "\noexpand\hyper@verticalfalse"}
  \pgfmathresult
  \ifhyper@vertical
  \edef\hyper@cmd{-- (\tikztotarget)}
  \else
  \pgfmathsetmacro{\hyper@dy}{\hyper@ey - \hyper@sy}
  \pgfmathsetmacro{\hyper@t}{\hyper@my/\hyper@dx}
  \pgfmathsetmacro{\hyper@cx}{\hyper@mx + \hyper@t * \hyper@dy}
  \pgfmathsetmacro{\hyper@radius}{veclen(\hyper@cx - \hyper@sx, \hyper@sy)}
  \pgfmathsetmacro{\hyper@sangle}{180 - atan2(\hyper@cx-\hyper@sx,\hyper@sy)}
  \pgfmathsetmacro{\hyper@eangle}{180 - atan2(\hyper@cx-\hyper@ex,\hyper@ey)}
  \edef\hyper@cmd{arc[radius=\hyper@radius pt, start angle=\hyper@sangle, end angle=\hyper@eangle]}
  \fi
}

\def\hyper@disc@computer#1#2{%
  \edef\hyper@toscan{(#1)}
  \tikz@scan@one@point\hyper@coords\hyper@toscan
  \edef\hyper@sx{\the\pgf@x}
  \edef\hyper@sy{\the\pgf@y}
  \edef\hyper@toscan{(#2)}
  \tikz@scan@one@point\hyper@coords\hyper@toscan
  \edef\hyper@ex{\the\pgf@x}
  \edef\hyper@ey{\the\pgf@y}
  \pgfmathsetmacro{\hyper@det}{\hyper@sx * \hyper@ey - \hyper@sy * \hyper@ex}
  \pgfmathparse{\hyper@det == 0 ? "\noexpand\hyper@verticaltrue" : "\noexpand\hyper@verticalfalse"}
  \pgfmathresult
  \ifhyper@vertical
  \edef\hyper@cmd{-- (\tikztotarget)}
  \else
  \pgfmathsetmacro{\hyper@mx}{(\hyper@ex + \hyper@sx)/2}
  \pgfmathsetmacro{\hyper@my}{(\hyper@ey + \hyper@sy)/2}
  \pgfmathsetmacro{\hyper@dx}{\hyper@ex - \hyper@sx}
  \pgfmathsetmacro{\hyper@dy}{\hyper@ey - \hyper@sy}
  \pgfmathsetmacro{\hyper@dradius}{\pgfkeysvalueof{/tikz/hyperbolic disc radius}}
  \pgfmathsetmacro{\hyper@t}{(\hyper@dradius^2 - \hyper@sx * \hyper@ex - \hyper@sy * \hyper@ey)/(2 * (\hyper@sx * \hyper@ey - \hyper@sy * \hyper@ex))}
  \pgfmathsetmacro{\hyper@radius}{sqrt(\hyper@t^2 + .25) * veclen(\hyper@dx,\hyper@dy)}
  \pgfmathsetmacro{\hyper@cx}{\hyper@mx + \hyper@t * \hyper@dy}
  \pgfmathsetmacro{\hyper@cy}{\hyper@my - \hyper@t * \hyper@dx}
  \pgfmathsetmacro{\hyper@sangle}{atan2(\hyper@sx-\hyper@cx,\hyper@sy - \hyper@cy)}
  \pgfmathsetmacro{\hyper@eangle}{atan2(\hyper@ex-\hyper@cx,\hyper@ey - \hyper@cy)}
  \pgfmathsetmacro{\hyper@eangle}{\hyper@eangle > \hyper@sangle + 180 ? \hyper@eangle - 360 : \hyper@eangle}
  \edef\hyper@cmd{arc[radius=\hyper@radius pt, start angle=\hyper@sangle, end angle=\hyper@eangle]}
\fi
}

\tikzset{%
  hyperbolic disc radius/.initial={1cm},
  hyperbolic plane/.style={
    to path={
      \pgfextra{\hyper@computer\tikztostart\tikztotarget}
      \hyper@cmd
    }
  },
  hyperbolic disc/.style={
    to path={
      \pgfextra{\hyper@disc@computer\tikztostart\tikztotarget}
      \hyper@cmd
    }
  }
}

\makeatother
\begin{document}
\begin{tikzpicture}[every to/.style={hyperbolic plane}]
\fill[blue] (0,1) \foreach \k in {0,...,7} { to ++(\k * 45:2)};
\coordinate (b) at (1,2);
\coordinate (a) at (3,4);
\fill (a) circle[radius=2pt];
\fill (b) circle[radius=2pt];
\draw (-2,0) -- (6,0);
\draw (a) to (b);
\end{tikzpicture}
\begin{tikzpicture}[hyperbolic disc radius=2cm,
every to/.style={hyperbolic disc}]
\draw (0,0) circle[radius=\pgfkeysvalueof{/tikz/hyperbolic disc radius}];
\pgfmathsetmacro{\hyperrad}{1/(2*sin(22.5))}
\fill[blue] (-112.5:\hyperrad) \foreach \k in {0,...,7} { to ++(\k * 45:1)};
\coordinate (b) at (1.2,.3);
\coordinate (a) at (.8,-.4);
\fill (a) circle[radius=2pt];
\fill (b) circle[radius=2pt];
\draw (a) to (b);
\end{tikzpicture}
\end{document}

Result:

One extension that would be nice would be to be able to draw an arc from a point starting in a particular direction and going on for a particular length. A hyperbolic version of the Euclidean \draw (1,0) -- ++(45:2);.

Answer 2

Here is very basic solution. It get (some of) the job done. The use of \hgline should be obvious from the examples. Feel free to suggest improvements.

\documentclass[]{minimal}
\usepackage{tikz}

\begin{document}

\newcommand{\hgline}[2]{
\pgfmathsetmacro{\thetaone}{#1}
\pgfmathsetmacro{\thetatwo}{#2}
\pgfmathsetmacro{\theta}{(\thetaone+\thetatwo)/2}
\pgfmathsetmacro{\phi}{abs(\thetaone-\thetatwo)/2}
\pgfmathsetmacro{\close}{less(abs(\phi-90),0.0001)}
\ifdim \close pt = 1pt
    \draw[blue] (\theta+180:1) -- (\theta:1);
\else
    \pgfmathsetmacro{\R}{tan(\phi)}
    \pgfmathsetmacro{\distance}{sqrt(1+\R^2)}
    \draw[blue] (\theta:\distance) circle (\R);
\fi
}

\begin{tikzpicture}
\draw (0,0) circle (1);
\clip (0,0) circle (1);
\hgline{30}{-30}
\hgline{180}{270}
\hgline{30}{120}
\hgline{0}{180}

\end{tikzpicture}

\end{document}

The result is

Answer 3

Here is an example of using tkz-euklide to draw a hyperbolic line through two given points:

\documentclass{article}
\usepackage{tkz-euclide}

\begin{document}
\begin{tikzpicture}[scale=3]
  \tkzDefPoint(0,0){O}
  \tkzDefPoint(1,0){A}
  \tkzDrawCircle(O,A)
  \tkzDefPoint(0.3,-0.25){z1}
  \tkzDefPoint(-0.5,-0.5){z2}
  \tkzClipCircle(O,A)
  \tkzDrawCircle[orthogonal through=z1 and z2](O,A)
  \tkzDrawPoints[color=black,fill=red,size=12](z1,z2)
  \tkzLabelPoints(z1,z2)
\end{tikzpicture}
\end{document}

The result looks like this:

Answer 4

Probably not. I googled "latex hyperbolic geometry" and the top hit was this page. There are plenty of papers about hyperbolic geometry written in latex but no packages that I found for hyperbolic geometry.

But I think these things can be done in TikZ, if one works hard enough.