Lua Hyperbolic Geometry

Version : 2026/03/16

luahyperbolic is a LaTeX package and a Lua library for performing operations and drawing pictures in hyperbolic geometry, intended for use with LuaLaTeX. The package provides complex number manipulation and hyperbolic geometric functions.

Installation

Manual Installation (for now)

If you wish to install the package manually, follow these steps:

• Put the file luahyperbolic.sty in your working directory
• Include the package in your document by adding:

  \usepackage{luahyperbolic}
  

Example Usage

A minimal working example is (see minimal_example.texin examples/) :

\documentclass[margin=.2cm,multi,tikz]{standalone}
\usepackage{luahyperbolic} %loads luacode package
\begin{document}
\begin{luacode*}
hyper.tikzBegin("scale=2.5")
local P = complex(0.5,-0.2)
local A = complex.exp_i(math.pi/10)
for k=1,5 do hyper.drawLine(P, A^k, "teal") end
hyper.labelPoint(P, "$P$", "left=.2cm")
hyper.drawLine(complex.J,-complex.I,"very thick, dashed, red")
hyper.tikzEnd("myfile.tikz")
\end{luacode*}
\end{document}

Compiling that file with lualatex produces the following output:

It also saves the TikZ picture to myfile.tikz, for later use. (Optional)

See the package documentation (pdf) for numerous examples.

More examples in examples/

License

This package is released under the Public Domain (CC0 1.0 Universal License). You may use, modify, and distribute it freely, without restriction.

For more information on the license, see the LICENSE file or visit CC0 1.0 Universal.

Todo

In luahyperbolic-core :

• function distancebetweengeodesics(z1, z2, w1, w2)
• function closestpointsbetweengeodesics(z1, z2, w1, w2)
• triangle intouch points, extouchpoints, excenters
• hide functions metricfactor, circletoeuclidean
• get rif of cosh, sinh, tanh
• IMPORTANT write function that computes triangle with given angles. Necessary for (p,q,r) tilings.
• change name fundamentalIdealTriangle if only one angle is zero
• power of a point, radical axis
• hyper.getType(phi) for automorphism
• hyper.getFixedPoints(phi) for automorphism
• symmetrySending (A to B)
• reflectionSending (A to B)
• expMap : transforme into function of type point -> (vector -> point) instead of (point, vector) -> point

In luahyperbolic-tikz :

• more constants : distances for angle drawing/labelling etc,
• function drawExcircle and variants
• more triangle geometry ? Gergonne, Nagel etc ?
• function markAngle(A, O, B, options)
• function labelSegment(A, B, label)
• function labelAngle(A, O, B, label)
• more tikz shapes if necessary
• draw external angle bisector ?
• replace old complex.isClose(z,w) etc with z:isNear(w) etc.
• fillTriangle, fillPolygon etc ?
• drawExcircle

In luahyperbolic-tilings

• faster tiling generation
• draw tiling step by step, triangle by triangle

In documentation

links to math articles on wikipedia for definitions ?

More examples

• ex circles

Contact

Do you really need to contact me ? Please don't contact me.