\newpage
\section{Class \tkzClass{path}}

This class was developed in response to a question posted on \code{tex.stackexchange.com} [\href{https://tex.stackexchange.com/questions/741663/fill-a-space-made-by-three-arcs/741758#741758}{fill-a-space-made-by-three-arcs}].

The concept was missing from the package, although in some cases it was possible to work around this using \TIKZ{} paths. When creating the \code{conic} class, I frequently had to manipulate tables of coordinates, so it became natural to formalize these as \code{paths}.

\directlua{
 z.O = point(0, 0)
 z.A = point(8, 0)
 z.E = z.O:rotation(2*math.pi / 3, z.A)
 z.C = z.O:rotation(-2*math.pi / 3, z.A)
 z.F, z.D, z.B = z.O:rotation(math.pi / 3, z.A, z.E, z.C)
 L.AC = line(z.A, z.C)
 L.OB = line(z.O, z.B)
 L.OD = line(z.O, z.D)
 L.CE = line(z.C, z.E)
 L.AE = line(z.A, z.E)
 L.OF = line(z.O, z.F)
 z.G = intersection(L.AC, L.OB)
 z.H = intersection(L.CE, L.OD)
 z.I = intersection(L.AE, L.OF)
 C.GA = circle(z.G, z.A)
 C.FE = circle(z.F, z.E)
 C.BA = circle(z.B, z.A)
 C.IE = circle(z.I, z.E)
 C.DE = circle(z.D, z.E)
 C.FA = circle(z.F, z.A)
 C.HE = circle(z.H, z.E)
 C.GC = circle(z.G, z.C)
 _,z.J = intersection(C.GA, C.FE)
 _,z.K = intersection(C.BA, C.IE)
 _,z.L = intersection(C.DE, C.IE)
 _,z.M = intersection(C.FA, C.HE)
 z.N = intersection(C.BA, C.HE)
 z.P = intersection(C.DE, C.GC)
 % the paths
 local nb = 40
 PA.th1 = C.IE:path(z.E, z.L, nb) + C.DE:path(z.L, z.O, nb) - C.FA:path(z.E, z.O, nb)
 PA.th2 = C.IE:path(z.G, z.A, nb) - C.FA:path(z.M, z.A, nb) - C.HE:path(z.G, z.M, nb)
 PA.th3 = C.IE:path(z.H, z.K, nb) + C.BA:path(z.K, z.C, nb) - C.GC:path(z.H, z.C, nb)
 PA.th4 = C.GA:path(z.A, z.J, nb) + C.FA:path(z.J, z.O, nb) - C.BA:path(z.A, z.O, nb)
 PA.th5 = C.HE:path(z.C, z.N, nb) + C.BA:path(z.N, z.O, nb) - C.DE:path(z.C, z.O, nb)
 PA.th6 = C.HE:path(z.I, z.E, nb) - C.DE:path(z.P, z.E, nb) - C.GA:path(z.I, z.P, nb)}
\begin{center}
  \begin{tikzpicture}[scale = .5, rotate = -30]
    \tkzGetNodes
    \tkzFillCircle[fill=yellow!20](O,A)
    \tkzDrawCircle[ultra thick,black](O,A)
    \tkzClipCircle(O,A)
    \tkzDrawCoordinates[draw,ultra thick,fill=green!15](PA.th1)
    \tkzDrawCoordinates[draw,ultra thick,fill=green!15](PA.th2)
    \tkzDrawCoordinates[draw,ultra thick,fill=blue!15](PA.th3)
    \tkzDrawCoordinates[draw,ultra thick,fill=blue!15](PA.th4)
    \tkzDrawCoordinates[draw,ultra thick,fill=red!15](PA.th5)
    \tkzDrawCoordinates[draw,ultra thick,fill=red!15](PA.th6)
    \tkzDrawCircle[ultra thick,black](O,A)
    \end{tikzpicture}
\end{center}

\begin{verbatim}

\directlua{
 z.O = point(0, 0)
 z.A = point(8, 0)
 z.E = z.O:rotation(2*math.pi / 3, z.A)
 z.C = z.O:rotation(-2*math.pi / 3, z.A)
 z.F, z.D, z.B = z.O:rotation(math.pi / 3, z.A, z.E, z.C)
 L.AC = line(z.A, z.C)
 L.OB = line(z.O, z.B)
 L.OD = line(z.O, z.D)
 L.CE = line(z.C, z.E)
 L.AE = line(z.A, z.E)
 L.OF = line(z.O, z.F)
 z.G = intersection(L.AC, L.OB)
 z.H = intersection(L.CE, L.OD)
 z.I = intersection(L.AE, L.OF)
 C.GA = circle(z.G, z.A)
 C.FE = circle(z.F, z.E)
 C.BA = circle(z.B, z.A)
 C.IE = circle(z.I, z.E)
 C.DE = circle(z.D, z.E)
 C.FA = circle(z.F, z.A)
 C.HE = circle(z.H, z.E)
 C.GC = circle(z.G, z.C)
 _,z.J = intersection(C.GA, C.FE)
 _,z.K = intersection(C.BA, C.IE)
 _,z.L = intersection(C.DE, C.IE)
 _,z.M = intersection(C.FA, C.HE)
 z.N = intersection(C.BA, C.HE)
 z.P = intersection(C.DE, C.GC)
 % the paths
 local nb = 40
 PA.th1 = C.IE:path(z.E, z.L, nb) + C.DE:path(z.L, z.O, nb) - C.FA:path(z.E, z.O, nb)
 PA.th2 = C.IE:path(z.G, z.A, nb) - C.FA:path(z.M, z.A, nb) - C.HE:path(z.G, z.M, nb)
 PA.th3 = C.IE:path(z.H, z.K, nb) + C.BA:path(z.K, z.C, nb) - C.GC:path(z.H, z.C, nb)
 PA.th4 = C.GA:path(z.A, z.J, nb) + C.FA:path(z.J, z.O, nb) - C.BA:path(z.A, z.O, nb)
 PA.th5 = C.HE:path(z.C, z.N, nb) + C.BA:path(z.N, z.O, nb) - C.DE:path(z.C, z.O, nb)
 PA.th6 = C.HE:path(z.I, z.E, nb) - C.DE:path(z.P, z.E, nb) - C.GA:path(z.I, z.P, nb)}
\begin{center}
  \begin{tikzpicture}[scale = .5, rotate = -30]
    \tkzGetNodes
    \tkzFillCircle[fill=yellow!20](O,A)
    \tkzDrawCircle[ultra thick,black](O,A)
    \tkzClipCircle(O,A)
    \tkzDrawCoordinates[draw,ultra thick,fill=green!15](PA.th1)
    \tkzDrawCoordinates[draw,ultra thick,fill=green!15](PA.th2)
    \tkzDrawCoordinates[draw,ultra thick,fill=blue!15](PA.th3)
    \tkzDrawCoordinates[draw,ultra thick,fill=blue!15](PA.th4)
    \tkzDrawCoordinates[draw,ultra thick,fill=red!15](PA.th5)
    \tkzDrawCoordinates[draw,ultra thick,fill=red!15](PA.th6)
    \tkzDrawCircle[ultra thick,black](O,A)
    \end{tikzpicture}
\end{center}

\end{verbatim}

\medskip

In addition to manually constructing paths from a list of points, you can also automatically generate paths using geometric classes. This is particularly useful when drawing arcs, segments, or interpolated curves. Currently, the following classes provide built-in methods to generate path objects:

\begin{itemize}
\item \tkzClass{line} — segment interpolation between two points See [\ref{ssub:line_path}]
\item \tkzClass{triangle} — triangle outline as a path. See [\ref{ssub:method_tkzmeth_triangle_path}]
\item \tkzClass{circle} — circular arcs between two points on the circumference See [\ref{ssub:circle_path}]
\item \tkzClass{conic} — arcs or full curves parameterized. See [\ref{ssub:conic_path}]
\end{itemize}

These methods typically return a \tkzClass{path} object with a default or user-defined number of points. For instance:

\begin{verbatim}
PA.arc = C.OA:path(z.A, z.B, 40) -- circular arc from A to B on circle OA
PA.seg = L.AB:path(10) -- line segment from A to B with 10 points
PA.tri = T.ABC:path() -- triangle outline path
PA.co = CO.EL:points(0, 1, 50) -- conic arc from t=0 to t=1
\end{verbatim}

These generated paths can be combined, reversed, filled, or used for \TIKZ{} decorations.


\subsection{Overview}

The path object represents an ordered sequence of 2D complex points, stored as TikZ-compatible strings in the form "(x,y)". It supports geometric operations such as translation, homothety, and rotation, and can be combined with other paths using Lua operator overloading.

The variable \code{PA} holds a table used to store paths. It is optional, and you are free to choose the variable name. However, using \code{PA} is a recommended convention for clarity and consistency. If you use a custom variable (e.g., Paths), you must initialize it manually. The \tkzFct{tkz-elements}{init\_elements()} function reinitializes the \code{PA} table if used.

When working with path objects—especially when several intermediate curves are involved—it is strongly recommended to use indexed names such as \code{PA.p1}, \code{PA.p2}, etc., rather than a generic name.

\begin{mybox}
\textbf{Rule — Prefer explicit, indexed names:}
Using \code{PA.p1}, \code{PA.p2}, \code{PA.result}, etc., improves clarity, avoids accidental overwriting, and ensures that all geometric data can be reset with a single call to \tkzFct{init\_elements()}.
\end{mybox}

This approach aligns with the overall philosophy of \tkzNamePack{tkz-elements}, where named tables like \code{PA}, \code{T}, \code{C}, and \code{L} are used to store structured geometric information. Indexing also helps make diagrams more readable, reproducible, and easier to debug.


It supports arithmetic operations for combining and manipulating paths, as well as geometric transformations such as translation, rotation, homothety, and reversal.

\subsection{Notes}

\begin{itemize}
\item Points are internally stored as strings like "(x,y)", and parsing is done via a utility function parse\_point.

\begin{verbatim}
local x, y = utils.parse_point("(3.5, -2)")
-- x = 3.5, y = -2.0
\end{verbatim}


\item Number formatting (e.g., for \TIKZ{} compatibility) should be handled by checknumber, assumed to be defined globally or in a utility module.

\item The class is designed to be lightweight and compatible with TikZ/LuaLaTeX workflows.

\end{itemize}

\subsection{Constructor}

\begin{center}
  \begin{verbatim}
 PA.name = path(table_of_points) -- Creates a new path object.
  \end{verbatim}
\end{center}


If data is provided, the input should be a table of points written as strings, e.g., |{ "(0,0)", "(1,0)"|, |"(1,1)" }| otherwise creates an empty path.

Here is a \tkzname{path} representing a simple triangle:

  \begin{mybox}
    \begin{verbatim}
      PA.triangle = path({ "(0, 0)", "(1, 0)","(1, 1)" })
    \end{verbatim}
  \end{mybox}



\subsection{Operator Overloading; metamethods}

\subsubsection{Table of metamethods}
\bgroup
  \catcode`_=12
  \small
  \captionof{table}{Methods of the class vector.}\label{path:metamethods}
  \begin{tabular}{ll}
  \toprule
   \textbf{Metamethods} & \textbf{Reference} \\
   \midrule
  \tkzMeta{path}{add(path1,path2)}   & [\ref{ssub:method_imeth_vector_add}]  \\
  \tkzMeta{path}{sub(path1,path2)}   & [\ref{ssub:method_imeth_vector_sub}]  \\
  \tkzMeta{path}{unm(path1)}     &  []                                   \\
  \tkzMeta{vector}{tostring(path1)}   & [\ref{ssub:method_vector_mul}]        \\
  \bottomrule
  \end{tabular}
\egroup

\subsubsection{Metamethod \tkzMeta{path}{add}}

If p1 and p2 are two paths, we can obtain a third path with
  |p3 = p1:add(p2)|

 More easily with an operator

 \code{p3 = p1 + p2} — \textbf{Concatenation} \\
  Returns a new path by appending \code{p2} after \code{p1}.

\subsubsection{Metamethod \tkzMeta{path}{unm}}
|p2 = p1:unm()|

 More easily with an operator

  \code{-p} — \textbf{Reversal} \\
  Returns a copy of the path in reverse order.

\subsubsection{Metamethod \tkzMeta{path}{sub}}
If p1 and p2 are two paths, we can obtain a third path with
|p3 = p1:sub(p2)|

 More easily with an operator

  \code{p1 - p2} — \textbf{Subtraction} \\
  Equivalent to \code{p1 + (-p2)} (concatenates \code{p1} with the reversed \code{p2}).

\subsubsection{Metamethod \tkzMeta{path}{tostring}}

String Representation

The \code{tostring} metamethod provides a readable form, mainly for debugging:

\begin{verbatim}
tostring(p) --> path: { (x1,y1) , (x2,y2) , ... }
\end{verbatim}

An example:

\begin{tkzexample}[latex=.5\textwidth]
  \directlua{
    local function print_section(title)
      tex.print("\\par\\noindent
      \\textbf{" .. title .. "}\\par")
    end
    local A = point(1,0)
    local B = point(2,2)
    local C = point(3,3)
    local p1 = path()
    local p2 = path()
    p1 = path({ "(0, 0)", "(1, 0)","(1, 1)" })
    p2 = path({"(0,1)"})
    tex.print("\\noindent p1 = ",tostring(p1))
    tex.print("\\\\")
    tex.print("p2 = ", tostring(p2))
    print_section("operator +")
    local p3 = p1 + p2
    tex.print("p3 = p1 + p2 = ",tostring(p3))
    print_section("operator -")
    local p4 = p2 - p1
    tex.print("p4 = p2 - p1 = ",tostring(p4))}
\end{tkzexample}


\subsection{Methods}

\bgroup
  \catcode`\_ = 12
  \small
  \captionof{table}{Methods of the \texttt{path} class.}\label{path:methods}
  \begin{tabular}{ll}
    \toprule
    \textbf{Method} & \textbf{Reference} \\
    \midrule
\tkzMeth{path}{add\_point(z)} & See [\ref{ssub:method_tkzmeth_path_add__point}; \ref{thm:circle-point}]\\

\tkzMeth{path}{copy()}       & See [\ref{ssub:method_tkzmeth_path_copy}] \\

\tkzMeth{path}{translate(dx, dy)} & See [\ref{ssub:method_tkzmeth_path_translate}] \\

\tkzMeth{path}{homothety(pt, k)} & See [\ref{ssub:method_tkzmeth_path_homothety}] \\

\tkzMeth{path}{rotate(pt, an)}  & See [\ref{ssub:method_tkzmeth_path_rotate}] \\

\tkzMeth{path}{close()}  & See [\ref{ssub:method_tkzmeth_path_close}]\\

\tkzMeth{path}{sub(i1, i2)} & See [\ref{ssub:method_tkzmeth_path_sub_i1_i2}] \\

\tkzMeth{path}{show()}  & See [\ref{ssub:method_tkzmeth_path_show}] \\

\tkzMeth{path}{add\_pair\_to\_path(z1, z2, n)} & or add\_pair  See [\ref{ssub:subsubsection_name}] \\

\tkzFct{path}{path\_from\_pairs(pt, pt, n)} & See [\ref{ssub:subsubsection_name}; \ref{ssub:method_tkzmeth_path_path_from_pairs}] \\

\tkzFct{path}{stock\_paths(...}  & See [\ref{ssub:method_tkzmeth_path_stock__paths}
; \ref{ssub:subsubsection_name}] \\

\tkzMeth{path}{concat(sep)}       & See [\ref{ssub:method_tkzmeth_path_concat}]\\
\bottomrule
\end{tabular}
\egroup



\subsubsection{Method \tkzMeth{path}{add\_point(pt,<n>)}}
\label{ssub:method_tkzmeth_path_add__point}

Adds a point (given as a complex table) to the path. Let's take one of the first examples:

| PA.triangle = path({ "(0, 0)", "(1, 0)","(1, 1)" })|and use the method to build the path automatically

\begin{tkzexample}[latex=.3\textwidth]
\directlua{
  PA.triangle = path()
  z.A = point(0, 0)
  z.B = point(1, 0)
  z.C = point(1, 1)
  PA.triangle:add_point(z.A, 0)
  PA.triangle:add_point(z.B, 0)
  PA.triangle:add_point(z.C, 0)
  tex.print(tostring(PA.triangle))}
\end{tkzexample}

\subsubsection{Method \tkzMeth{path}{copy()}}
\label{ssub:method_tkzmeth_path_copy}

Returns a deep copy of the path

\begin{tkzexample}[latex=.3\textwidth]
\directlua{
  PA.p1 = path({ "(0, 0)", "(1, 0)","(1, 1)" })
  PA.p2 = PA.p1:copy()
  PA.p2:add_point(point(0, 0),0)}
\begin{tikzpicture}
  \tkzDrawCoordinates(PA.p2)
\end{tikzpicture}
\end{tkzexample}

\subsubsection{Method \tkzMeth{path}{translate(dx,dy)}}
\label{ssub:method_tkzmeth_path_translate}

Translates the path by a given vector \((dx,dy)\)

\begin{tkzexample}[latex=.3\textwidth]
\directlua{
  PA.p1 = path({ "(0,0)", "(1,0)", "(1,1)" })
  PA.p2 = PA.p1:translate(2,1)}
\begin{tikzpicture}
\tkzDrawCoordinates(PA.p2)
\end{tikzpicture}
\end{tkzexample}

\subsubsection{Method \tkzMeth{path}{homothety(pt,r)}}
\label{ssub:method_tkzmeth_path_homothety}

Applies a homothety centered at \(pt\) with ratio \(k\)


\begin{tkzexample}[latex=.3\textwidth]
\directlua{
   z.O = point(0, 0)
   PA.base = path({ "(0,0)", "(1,0)", "(1,1)" })
   PA.scaled = PA.base:homothety(z.O, -2)}
\begin{tikzpicture}
  \tkzGetNodes
  \tkzDrawCoordinates(PA.base)
  \tkzDrawCoordinates(PA.scaled)
  \tkzDrawPoint(O)
\end{tikzpicture}
\end{tkzexample}

\subsubsection{Method \tkzMeth{path}{rotate(pt, an)}}
\label{ssub:method_tkzmeth_path_rotate}

 Rotates the path around \(pt\) by angle \(\theta\) (radians)

\begin{tkzexample}[latex=.3\textwidth]
\directlua{
   z.O = point(1, 1)
   PA.base = path({ "(1,0)", "(2,0)", "(2,1)" })
   PA.rotated = PA.base:rotate(z.O, math.pi / 2)}
\begin{tikzpicture}
  \tkzGetNodes
  \tkzDrawCoordinates(PA.base)
  \tkzDrawCoordinates(PA.rotated)
  \tkzDrawPoint(O)
\end{tikzpicture}
\end{tkzexample}

\subsubsection{Method \tkzMeth{path}{close()}}
\label{ssub:method_tkzmeth_path_close}

Closes the path by repeating the first point at the end

\begin{tkzexample}[latex=.3\textwidth]
\directlua{
  PA.p = path({ "(0,0)", "(1,0)", "(1,1)" })
  PA.closed = PA.p:close()}
\begin{tikzpicture}
  \tkzDrawCoordinates[fill=green!20](PA.closed)
\end{tikzpicture}
\end{tkzexample}

\subsubsection{Method \tkzMeth{path}{sub(i1, i2)}}
\label{ssub:method_tkzmeth_path_sub_i1_i2}

Returns a subpath between two indices


\begin{tkzexample}[latex=.3\textwidth]
\directlua{
  PA.full = path({ "(0,0)", "(1,0)", "(1,1)", "(0,1)", "(0,0)" })
  PA.part = PA.full:sub(2, 4)}
\begin{tikzpicture}
  \tkzDrawCoordinates[fill=green!20](PA.part)
\end{tikzpicture}
\end{tkzexample}


\subsubsection{Method \tkzMeth{path}{show()}}
\label{ssub:method_tkzmeth_path_show}

Prints the path in the terminal (for debugging)

\begin{tkzexample}[latex=.45\textwidth]
\directlua{
  PA.p = path({ "(0,0)", "(1,0)", "(1,1)" })
  PA.p:show()}
\end{tkzexample}

% ADD(AM, 4.25) some complements
\subsubsection{Method \tkzMeth{path}{add\_pair\_to\_path(p, p, n)} or \tkzMeth{path}{add\_pair(p, p, n)}}
\label{ssub:subsubsection_name}
\begin{verbatim}
  path:add_pair(z1, z2, decimals)
\end{verbatim}

or equivalently:

\begin{verbatim}
 path:add_pair_to_path(z1, z2, decimals)
\end{verbatim}

\paragraph{Description}
 The coordinates of \code{z1} and \code{z2} are extracted, formatted with a fixed number of decimals $n$, and combined into a string \code{"x1/y1/x2/y2"} appended to the path.
This string is then inserted into the current path table.This method appends to a \tkzname{path} object. a formatted point pair.
It is mainly used to store segments, by keeping together the coordinates of two points.



\paragraph{Arguments}
\begin{itemize}
  \item \code{z1}, \code{z2} – points given as tables \code{\{re, im\}}.
  \item \code{decimals} (optional) – number of decimals (default: \code{5}).
\end{itemize}

\paragraph{Returns} Nothing (modifies the path in place).



\paragraph{Example}
\begin{verbatim}
   local PA.p = path() -- empty path
   local z.A = point(1, 2)
   local z.B = point(3, 4)
   PA.p:add_pair(z.A, z.B, 2)
   -- PA.p = {"1.00/2.00/3.00/4.00"}
\end{verbatim}


\subsubsection{Function \tkzFct{path}{path\_from\_pairs(p, p, n)}}
\label{ssub:method_tkzmeth_path_path_from_pairs}

Creates a new \tkzname{path} object by combining pairs of points taken from two parallel paths. Each resulting entry is a string \code{"x1/y1/x2/y2"}.

\paragraph{Arguments}
\begin{itemize}
  \item \code{p1}, \code{p2} – tables of point strings \code{"(x,y)"}.
  \item \code{decimals} (optional) – number of decimals to retain (default: \code{5}).
\end{itemize}

\paragraph{Returns} A new \tkzname{path} object.

\paragraph{Errors} Raises an error if the paths do not have the same length.

\paragraph{Example}
\begin{verbatim}
local PA.p1 = {point(0,0), point(1,1)}
local PA.p2 = {point(1,0), point(2,1)}
local PA.result = path_from_pairs(PA.p1, PA.p2, 1)
-- PA.result = {"0.0/0.0/1.0/0.0", "1.0/1.0/2.0/1.0"}
\end{verbatim}



\subsubsection{Function \tkzFct{path}{stock\_paths(...)}}
\label{ssub:method_tkzmeth_path_stock__paths}

Combines an arbitrary number of paths into a single \tkzname{path}. Each entry contains the concatenated coordinates of corresponding points from all input paths, formatted as \code{"x1/y1/x2/y2/..."}.

\paragraph{Arguments}
\begin{itemize}
  \item \code{...} – one or more tables of TikZ-style points \code{"(x,y)"}. An optional final argument may specify the number of decimals.
\end{itemize}

\paragraph{Returns} A new \tkzname{path} object.

\paragraph{Example}
\begin{verbatim}
local PA.A = {point(1,2), point(2,2)}
local PA.B = {point(1,3), point(2,3)}
local PA.path = stock_paths(PA.A, PA.B, 2)
-- PA.path = {"1.00/2.00/1.00/3.00", "2.00/2.00/2.00/3.00"}
\end{verbatim}

\paragraph{Errors} All paths must have the same number of points.

\subsubsection{Method \tkzMeth{path}{concat(<'sep'>)}}
\label{ssub:method_tkzmeth_path_concat}

Concatenates the path into a string (default sep: space)

\begin{tkzexample}[latex=.3\textwidth]
\directlua{
 PA.p = path({ "(0,0)", "(1,0)", "(1,1)" })
 PA.p:concat(" -- ")}
\begin{tikzpicture}
  \tkzDrawCoordinates(PA.p)
\end{tikzpicture}
\end{tkzexample}


\subsubsection{Example : Director circle }
\label{ssub:director_circle}


Conics are drawn using the \tkzClass{path} tool.

\begin{minipage}{.55\textwidth}
  \begin{verbatim}
   \directlua{
    init_elements()
    z.O = point(0, 0)
    z.F1 = point(4, 1)
    z.F2 = point(-4, -1)
    z.H = point(4 * math.sqrt(2), 0)
    local a = tkz.length(z.O, z.H)
    CO.EL = conic(EL_bifocal(z.F2, z.F1, a))
    PA.curve = CO.EL:points(0, 1, 50)
    z.A = CO.EL.covertex
    T.HOA = triangle(z.H, z.O, z.A)
    z.P = T.HOA:parallelogram()
    C.OP = circle(z.O, z.P)
    z.L = C.OP:point(0.25)
    T.LJ, T.LK = CO.EL:tangent_from(z.L)
    z.J = T.LJ.pb
    z.K = T.LK.pb}
  \end{verbatim}
\end{minipage}
\begin{minipage}{.45\textwidth}
   \directlua{
    init_elements()
    z.O = point(0, 0)
    z.F1 = point(4, 1)
    z.F2 = point(-4, -1)
    z.H = point(4 * math.sqrt(2), 0)
    local a = tkz.length(z.O, z.H)
    CO.EL = conic(EL_bifocal(z.F2, z.F1, a))
    PA.curve = CO.EL:points(0, 1, 50)
    z.A = CO.EL.covertex
    T.HOA = triangle(z.H, z.O, z.A)
    z.P = T.HOA:parallelogram()
    C.OP = circle(z.O, z.P)
    z.L = C.OP:point(0.25)
    T.LJ, T.LK = CO.EL:tangent_from(z.L)
    z.J = T.LJ.pb
    z.K = T.LK.pb}
\begin{center}
  \begin{tikzpicture}[scale = .4]
   \tkzGetNodes
   \tkzDrawPoints(F1,F2,O)
   \tkzDrawCircles[teal](O,P)
   \tkzDrawPolygon(H,O,A,P)
   \tkzDrawCoordinates[smooth,cyan](PA.curve)
   \tkzDrawSegments[orange](O,P O,L L,J L,K)
   \tkzDrawPoints(F1,F2,O,H,A,P,L,J,K)
   \tkzLabelPoints(F1,F2,O,H,A,P,L,J,K)
  \end{tikzpicture}
\end{center}
\end{minipage}

\begin{verbatim}
\begin{tikzpicture}
 \tkzGetNodes
 \tkzDrawPoints(F1,F2,O)
 \tkzDrawCircles[teal](O,P)
 \tkzDrawPolygon(H,O,A,P)
 \tkzDrawCoordinates[smooth,cyan](PA.curve)
 \tkzDrawSegments[orange](O,P O,L L,J L,K)
 \tkzDrawPoints(F1,F2,O,H,A,P,L,J,K)
 \tkzLabelPoints(F1,F2,O,H,A,P,L,J,K)
\end{tikzpicture}
\end{verbatim}



\subsubsection{Classic parabola}
\label{ssub:classic_parabola}

This example moves away from the concept of Euclidean geometry, but the method used can be reused.
Here we want to draw a parabola whose axis of symmetry is parallel to the y-axis.
Knowing three of its points, we can determine the equation. The set of points for tracing the parabola is obtained using the tools provided by the \tkzClass{path} class.

\begin{minipage}{.55\textwidth}
\begin{verbatim}
\directlua{
 init_elements()
 z.a = point(1, 0)
 z.b = point(3, 2)
 z.c = point(0, 2)
 local A, B, C = parabola(z.a, z.b, z.c)

 function f(t0, t1, n)
 	local PA.tbl = path()
 	for t = t0, t1, (t1 - t0) / n do
 		local y = A * t ^ 2 + B * t + C
    local pt = point(t, y)
    PA.tbl:add_point(pt)
 	end
 	return PA.tbl
 end}
 \end{verbatim}
\end{minipage}
\begin{minipage}{.45\textwidth}
  \directlua{
   init_elements()
   z.a = point(1, 0)
   z.b = point(3, 2)
   z.c = point(0, 2)
   local A, B, C = tkz.parabola(z.a, z.b, z.c)

   function def_curve(t0, t1, n)
   local p = path()
   for t = t0,t1,(t1-t0)/n  do
       y = A*t^2+B*t +C
       pt = point(t, y)
       p:add_point(pt)
    end
    return p
   end

   PA.curve = def_curve(-1,3,100)}

  \begin{center}
    \begin{tikzpicture}
       \tkzGetNodes
       \tkzInit[xmin = -2,xmax=4,ymin =-1,ymax=6]
       \tkzDrawX\tkzDrawY
       \tkzDrawPoints[red,size=4pt](a,b,c)
      \tkzDrawCoordinates[smooth,purple](PA.curve)
    \end{tikzpicture}
  \end{center}
\end{minipage}


\begin{verbatim}
\begin{tikzpicture}
 \tkzGetNodes
 \tkzInit[xmin = -2,xmax=4,ymin =-1,ymax=6]
 \tkzDrawX\tkzDrawY
 \tkzDrawPoints[red,size=4pt](a,b,c)
 \draw[smooth] plot coordinates {%
 \directlua{tex.print(f(-1,3,100))}};
\end{tikzpicture}
\end{verbatim}

\subsection{Example with several paths}

\begin{tkzexample}[latex=.45\textwidth]
\directlua{
  z.O = point(0, 0)
  z.A = point(5, 0)
  C.OA = circle(z.O, z.A)
  z.S = C.OA.south
  C.SO = circle(z.S, z.O)
  z.B, z.C = intersection(C.OA, C.SO)
  C.BC = circle(z.B, z.C)
  z.D = intersection(C.OA, C.BC)
  C.CD = circle(z.C, z.D)

  local p1 = C.SO:path(z.C, z.B, 50)
  local p2 = C.BC:path(z.C, z.D, 50)
  local p3 = C.CD:path(z.D, z.B, 50)
  thepath = (-p1) + p2 + p3 }
\begin{tikzpicture}[scale=.5]
  \tkzGetNodes
  \tkzDrawCircles(O,A S,O)
  \tkzDrawArc(B,C)(D)
  \tkzDrawArc(C,D)(B)
  \tkzDrawCoordinates[fill = purple!20,
                      opacity=.4](thepath)
  \tkzDrawCoordinates[smooth,red,thick](thepath)
  \tkzDrawPoints(A,O,B,C,S,D)
  \tkzLabelPoints(A,B,C)
\end{tikzpicture}
\end{tkzexample}
\endinput