tkz-elements — for euclidean geometry

Release 4.10c 2025/06/10

Description

tkz-elements v.4.10c is the new release and a major update of a library written in lua, allowing to make all the necessary calculations to define the objects of a Euclidean geometry figure. You need to compile with LuaLaTeX. With tkz-elements, the definitions and calculations are only done with Lua.

The main possibility of programmation proposed is oriented object programming with object classes like point, line, triangle, circle and now, conic. For the moment, once the calculations are done, it is tkz-euclide or TikZ which allows the drawings. You can use the option mini with tkz-euclide to load only the modules required for tracing.

Licence

This package may be modified and distributed under the terms and conditions of the LaTeX Project Public License, version 1.3 or greater.

Requirements

The package compiles with utf8 and lualatex. You need actually to load:

• tkz-euclide
• or tikz

Installation

The package tkz-elements is present in TeXLive and MiKTeX, use the package manager to install.

You can experiment with the tkz-elements package by placing all of the distribution files in the directory containing your current tex file.

The different files must be moved into the different directories in your installation TDS tree or in your TEXMFHOME:

How to use it

To use the package tkz-elements, place the following lines in the preamble of your LaTeX document:

% !TEX TS-program = lualatex
\usepackage[mini]{tkz-euclide}
\usepackage{tkz-elements}
\begin{document}
\directlua{
    init_elements ()_
    your code
}
\begin{tikzpicture}
\tkzGetNodes
    your code
\end{tikzpicture}

If you use the xcolor package, load that package before tkz-euclide to avoid package conflicts.

It's possible to use the environment tkzelements istead of the directive \directlua but in this case, you need to load the package luacode.

Examples

Some examples will be stored on my site : http://altermundus.fr. An important example Golden Arbelos using the package is on the site. All the files of the documentation are on the site.

History

• version 4.10c
  • Package additions: Two new modules have been created: utils and tkz.

     - The utils module provides a collection of general-purpose utility functions. These functions are designed to support common tasks such as numerical rounding, type checking, floating-point comparisons, and table operations.

      `utils.parse_point`, `utils.format_number`, `utils.format_coord`, `utils.format_point`, `utils.checknumber, `utils.almost_equal`, `utils.wlog

     -  The tkz module provides a collection of auxiliary functions. These functions are not strictly required but serve to simplify many geometric operations and shorten code. Their use depends on the context, but in the spirit of the package, it's preferable to use objects. Most of the following functions already existed, but are now used with the tkz prefix. This is the case for length and midpoint. Others are new, such as tkz.solve, which solves equations up to degree 3.
  Note also the `tkz.angle_between_vectors` function, which determines the angle between two vectors.

     `tkz.solve`, `tkz.solve_linear_system`, `tkz.midpoint`, `tkz.length`, `tkz.round`, `tkz.length`, `tkz.midpoint`,
      `tkz.midpoints`, `tkz.is_linear`, `tkz.is_ortho`, `tkz.bisector`, `tkz.bisector`, `tkz.altitude`, `tkz.get_angle`, `tkz.angle_normalize`, `tkz.barycenter`, `tkz.dot_product`, `tkz.nb_dec`, `tkz.epsilon`, 
      `tkz.dc`, `tkz.phi`, `tkz.invphi`, `tkz.sqrtphi`, `tkz.reset_defaults()`, `tkz.set_nb_dec(n)`.

  The `path` class has been enriched with the ability to create `paths` directly using conics and triangles. The methods have been rewritten and now use the add_point method. In order to make the best use of paths, I've added macros that allow you to create them using TikZ. There are now : `\tkzDrawPointsFromPath`, `\tkzGetPointsFromPath`, `\tkzDrawCirclesFromPaths`, `\tkzDrawSegmentsFromPaths`

  New mathematical features include the creation of the Poncelet point for the triangle class (poncelet_point). This method also exists for the quadrilateral class. The orthopole method has been added to the triangle class. 

  The macro `\tkzDN` has become `\tkzPN` alias of `\tkzPrintNumber`.
  The method of the circle class, `orthogonal_through`, has been corrected and rewritten.

  Aliases with more evocative names are appearing:

  `line.parallel_from`     alias of      `line.ll_from`
  `line.orthogonal_from`    alias of      `line.ortho_from`
  `\tkzDrawPath `           alias of      `\tkzDrawCoordinates`
  `\tkzPN`                  alias of      `\tkzPrintNumber`

  The `\tkzEraseLuaObj` macro deletes a lua object
  
  - Documentation
   
      - The Cheat_sheet section has been removed. You can find it on my site. Two sections have been added: "maths tools" and "tkz".

Some examples have been added: Archimedean spiral, Poncelet point, Orthopole, examples for "paths".

• version 4.00c
  • Package additions:
    • The points class was dependent on an external class, but this is no longer the case. This has made it possible to add a __call metamethod to all clsses, and to take advantage of this to lighten object creation. There is now a Short Syntax for classes that allows you to omit new when creating a class.

     - An important new class has been added: path. It allows us to better manage the sets of points needed to build cones. It is also useful for certain TikZ-related decorations. Automatic path creation is currently possible for segments, circular arcs and triangles. This will be extended to other classes in the next version.

      - A `mini` macro for using Metapost with tkz-elements has been created.

      - The `known` and `near` options have been added to the intersction function.

      - Rewrite `barycentric_coordinates` and addition of `trilinear_coordinates`.

      - Added with `line` , `orthogonal_at(pt, k)` like `colinear_at(pt, k)` and remove `_east` , `_west`, etc.

      - Function `search_ellipse` (with five points)

      - news for the triangle class :
        `adams_points`, `adams_circle`, `lamoen_points`, `lamoen_circle`, `yiu_circles`, `yiu_triangle`, `yiu_points`, `reflection`, `circumcevian`, `excenter`, `brocard point`(first and second), kimberling (brocard midpoint)

      - Undocumented : `gauss_jordan`, `gauss_jordan_rect`, `solve_linear_ystem` ; `solve(...)`,  `solve_cubic(a, b, c, d)`

      - Correction:
         - the point method for a triangle has been corrected.
         - get_lengths corrected for rectangles.
         - the Gauss pivot method has been improved
  - Documentation
      - Rewriting a large part
      - Index correction
      - Indexing macros rewritten.

      - Sections added : Short contents, Getting started, Class path
                         LuaLaTeX for beginners, Global variables and constants,
                         Various functions, Module utils, Metapost

• version 3.34c
  • Package additions
    • added functions for creating circles through and diameter
      • adding centers with kimberling method X(55), X(56), X(371)
        • added kenmotu_circle and kenmotu_point methods
          • added start of error handling with tex.error
      • Documentation
        • Correction of some file headers (date and name)
          • start of documentation reformatting; classification of examples
          • addition in the examples, circles of Yff, Adams, Van Lamoen, Kenmotu
  • version 3.32c
    • Modification of the class occs. The first argument is the main axis through a focus and a vertex.
    • Additions:
      • Example in the folder examples.
        • The method swap_line for the class line and a example. This makes it possible to change the orientation.
        • Example for the method isogonal for the class triangle.

• version 3.30c
  • Major evolution of tkz-elements with the introduction of the conic class, which replaces the ellipse class.

    - The latter was based on the ellipse operation, whereas plot coordinates is now used to construct all conic sections: parabolas, hyperbolas, and ellipses. It is worth noting that the circle, although a conic section, is not included in this class. Its significance grants it a special status and a dedicated class of its own.

    - Another class has been introduced: the occs class (orthonormal Cartesian coordinate system). To simplify the construction of conic sections, it was necessary to use well-suited coordinate systems.

    - A major change is the removal of scaling within the Lua section. Initially, I was in favor of avoiding scaling in the  TikZ part, but since most calculations were already performed there, I realized that it was significantly simpler to apply scaling within the tikzpicture environment. Technical complexities sometimes arise when scaling is handled in the Lua section, so I decided to remove this option.

   - Modifications:

    - In the regular\_polygon class, I renamed the item table to vertices, which is more appropriate and I also removed the first and next items, as they were unnecessary.

    - Correction of the code for the intersection of two circles, which did not provide an appropriate response in cases where no intersection was possible.

    - Improvement of the code for the euler\_line method of the class triangle.

    - Improvement of the code for the is\_orthogonal method of the class line.

   - Additions:

    - Major additions: the conic and occs classes.

    - An object of the conic class is created using the following arguments: `focus`, `directrix`, and `eccentricity`.
    - The available methods are: `points`, `point`, `antipode`, `tangent_at`, `tangent_from`, `intersection`, `in_out`, `orthopedic`, and `asymptotes`.

    - The 'points' method, common in many classes, allows creating a set of coordinates defining an object (e.g., a conic), extending the 'point' method which creates individual points.

    - The functions `EL_points`, `EL_bifocal`, `HY_bifocal`, `PA_dir`, and `PA_focus` provide the necessary arguments depending on the given data and the conic section being constructed.

    - The transformations projection_ll and affinity are now available for the line class.

    - The creation of an object from the occs class is done using  the data of a line and a point. This point will be the origin of the new coordinate system, while the line will define the direction of the new y-axis.

    - The 'kimberling' method allows the creation of some points using this notation with the 'triangle' class.

    - The methods: `steiner_line`, `simson_line`, `fermat_axis`, `brocard_axis`, `lemoine_axis`, `orthic_axis` and `orthic_axis_points` complete the methods of the triangle class, as well as the `anticomplementary` or `anti` method, the `taylor_circle` and the `taylor_points` methods.

    - Two macros for the 'tikzpicture' part have been created: `\tkzDrawCoordinates` for obtaining a curve from a table of coordinates and `\tkzDrawPointOnCurve` for placing a point on such a curve.

   - About documentation:

    - Removal of all “overfull boxes”.
    - Added examples concerning new features.
    - Corrected some examples, such as the Euler line.
   

• version 3.10c
  • Most of the functions have been optimized, and some have been commented on.
    • Object classes have been enhanced with new attributes. For a triangle, you can directly access the semiperimeter, area, inradius and circumradius. In some classes, the exradius attribute is replaced by circumradius.
      • For rectangle, square and circle, perimeter and area have been added.
      • For line, new methods appear: is_parallel, is_orthogonal and is_equidistant. The latter allows you to determine whether a point is equidistant from the two points defining the line. The swap argument is available for all triangle creations. The result is now a single triangle, the second is obtained with swap.
      • It is now possible to define an isosceles triangle from a straight line (segment) with length isosceles_s. You can use isosceles_a or the old isosceles method if you're using an angle. I've added a new test for triangles: is_acute. The two_angles method is identical to asa.
      • The line , circle and triangle classes are complemented by methods with complicated names: c_l_pp, c_ll_p, c_c_pp and c_cc_p. These methods allow you to determine, from a line or circle, one or more circles tangent to lines or circles and passing through points. So c_l_pp means to create a circle tangent to a line (l) and passing through two points (pp). The first c reminds us that we're looking for a circle, the second group between _ and _ indicates the tangent objects (c or l) and the last indicates the points through which the circle passes.
      • In the documentation, I've added a section on important geometry theorems (
      Viviani, Reuschle, Thébault, Varignon, Wittenbauer, Soddy, Six circles ... to be completed ...). Examples of new methods and attributes have also been added.
  • version 3.00c
    • It is now possible to use the directlua primitive to perform lua code. In this case, tables and scaling can be reset using the init_elements function. You can still use the tkzelements environment, but only if you load the luacode package.
      • Examples have been added to the transfers section.
    • version 2.30c
      • New version of the macro \tkzGetNodes written by Sanskar Singh. This version now fixes a bug that prevented a figure from being centred with centering or the center environment.
      • Adding methods bevan_circle, symmedial_circle.
      • Correction of the methods function triangle: bevan_point () and function triangle: mittenpunkt_point ().
      • Adding function triangle: similar ()
      • Adding function line : perpendicular_bisector () which is similar to function line : mediator ()
      • Correction of documentation.
  • version 2.25c
    • French documentation at my site: http://altermundus.fr
      • Added colinear_at a new method for the classe line
      • Added cevian, pedal, conway_circle, conway_points new methods to the class triangle.

• version 2.20c
  • Package:
    • Added class matrix; methods are mainly of order 2, sometimes of order 3.
    • Added function solve_quadratic. This function can be used to solve second-degree equations with real or complex numbers.
    • Added method print for the class point. Example z.A : print ()
    • Correction of the macrotkzDN. I deleted a spurious space
    • Modification of vector class attributes. Attributes h and t become head and tail.
    • The mtx attribute is introduced for point and vector. z.A.mtx represents the column matrix whose coefficients are the point's coordinates. Same for vectors.
    • Documentation:
    • Rewriting of all texts
    • Correction of example: pentagon
    • Documentation about matrices
  • version 2.00c
    • class development vector
      • added attribute vec
      • added at and orthogonal methods to the class point
      • rewriting the function angle_normalize_
      • modification of the slope attribute for the line, now the result is normalized.
      • the angles of a triangle are also normalized
      • added function format_number(number,decimal) sets the number of digits in the decimal part.
      • added \tkzDN a macro pour formater les nombres dans la partie TikZ \tkzDNnbdecimal{number}
      • added the macro \tkzDrawLuaEllipse draw an ellipse in tikz knowing its center, vertex and covertex.
      • correction de la documentation
  • version 1.82c
    • Point object : name like z.App now gives a node with name A''
      • Modification of methods north, south
      • Added the function length(z.A,z.B) shortcut for point.abs(z.A-z.B).
      • Line object added some methods
      • Added method in_out_segment
      • (sacred triangle)
        • gold
        • sublime or euclide
        • cheops
        • divine
        • pythagoras or isis or egyptian
        • golden
      • (classic triangles)
        • two_angles (side between)
        • sss (three sides)
        • ssa (two sides and an angle)
        • sas (an angle between two sides)
        • school (30°, 60° and 90°)
        • half right triangle in A with AB= 2AC
      • Circle object
        • added method common_tangent (gives the common tangents of two circles)
        • Correction for a bug and an oversight in the circles_position method.
        • Rewriting the radical_axis methods
      • Triangle object
        • method trilinear (to use trilinear coordinates)
        • method barycentric (to use barycentric coordinates)
      • Added some functions
        • bisector (a,b,c) altitude (a,b,c) bisector_ext(a,b,c) equilateral (a,b) midpoint (a,b) to avoid creating unnecessary objects.
      • Added new examples and a cheat sheet in the documentation

• version 1.72c
  • added a line method (apollonius) set of points M with MA/MB = k
    • example with line : apollonius
      • example: three circle
      • example: pentagons on golden arbelos
      • descriptions of several cases with 'midcircle'
      • added soddy method and examples
      • added example with circlesposition
      • correction of the documentation
  • version 1.60c

 - added Internal and external tangents common to two circles:
 - function circle : `external_tangent(C)`
 - function circle : `internal_tangent(C)`
 - `radical_center` and `radical_circle` are also valid for two circles
 - function `radical_center (C1,C2,C3)`
 - function `radical_circle (C1,C2,C3)`
 - function `circles_position (C1,C2)`
 - function `midcircle (C1,C2)` powerful tool for working with inversions
 - Bug corrected in midarc now use `get_angle` instead of` get_angle_`
 - Modification of a triangle attribute `ca` replaces `ac` to designate the line passing through the third and first points
 - The center of symmetry of a parallelogram is named center instead of `i`.
 - Correction documentation 
 - Correction of examples using the circle:point (k) method, where k is now a real number rather than an angle.

• version 1.50c Correction of the documentation

- Added `swap` option to create triangles from the line object.
- `iscyclic` is a new method to know if a quadrilateral is inscribable in a circle.
- Added function `diameter` to create a circle.
- Added function `swap` to swap two points.
- Correction method `gold` of object rectangle.
- Correction method `in_circle_` of object triangle.
- Correction method `incentral_tr_` of object triangle.
- Added method `soddy_center` of object triangle.
- Added option `swap` for method `square` of object line.
- Added method `report` for  object line. Transfer a defined length from a point
- Added option `swap` to the function square : side 

• Version 1.40c Restructuring objects

- New version for all transformations. Now, they accept all objects as parameters.  
- Symmetry_axial has changed its name to reflection.    
- Added scale to north south etc.. (point object).
- Change the point method of the objects  circle  and ellipse. now the parameter is un real t (between 0 and 1) and not an angle
- Added the method `check_equilateral` to know if a triangle is equilateral.
- Added option indirect to the method equilateral for a  line object.
- Correction of the documentation. (Added sections).
  

• Version 1.20 Memory management: tables are emptied when the tkzelements environment is opened.

 - `set_lua_to_tex` has been replaced by `tkzUseLua` to transfer data between the `tkzelements` and `tikzpicture` environments.
 - New version of `inversion` with respect to a circle method. It selects the correct algorithm based on the object passed as a parameter.
 - Added an `in_out_disk` method for the `circle` object, which indicates whether or not a point is in the disk. `in_out` is for the circle. 
 - Added two methods: `radical_center (C1,C2,C3)`  radical center of three circles.
      `radical_circle (C1,C2,C3)` orthogonal circle of three circles. 
 - Added function `circle : radius` to define a circle with a centre and a radius. 
 -  Added methods `normalize` and  `normalize_inv`  for `line`.
 - Added methods `translation` and `set_translation` to the `line` object. 
 - Added  an example to illustrate combinations of methods and attributes.
 

• First version 1.00b

Author

Alain Matthes, 5 rue de Valence, Paris 75005, al (dot) ma (at) mac (dot) com