\newpage
\section{Class \tkzClass{point}} % (fold)
\label{sec:class_point}
The variable \tkzVar{point}{z} holds a table used to store points. It is \textbf{mandatory} and is automatically initialized by the package (e.g.,\tkzname{ z = \{\}}).
\vspace{1em}
The \tkzClass{point} class forms the foundation of the entire framework. It is a hybrid class, representing both points in the plane and complex numbers. The underlying principle is as follows:
\begin{itemize}
\item The plane is equipped with an orthonormal basis (OCCS See [\ref{sec:orthonormal_cartesian_coordinate_system}]), allowing us to determine a point's position via its abscissa and ordinate.
\item Similarly, any complex number can be seen as an ordered pair of real numbers (its real and imaginary parts).
\item Therefore, the plane can be identified with the complex plane, and a complex number $x + iy$ is represented by a point in the plane with coordinates $(x, y)$.
\end{itemize}
Thus, a point such as $A$ is stored as the object \code{z.A}, with its coordinates and associated properties encapsulated within this object.
\vspace{1em}
The \code{point} object possesses several attributes:
\begin{itemize}
\item \code{re} --> the real part (abscissa),
\item \code{im} --> the imaginary part (ordinate),
\item \code{type} --> the type of the object (in this case, always \code{"point"}),
\item \code{arg} --> the argument of the complex number (angle with respect to the x-axis),
\item \code{mod} --> the modulus of the complex number (distance from the origin).
\end{itemize}
\subsection{Creating a Point} % (fold)
\label{sub:creating_a_point}
Points are created by providing their coordinates in the current orthonormal Cartesian coordinate system (OCCS). The recommended form is:
\begin{mybox}
\code{z.A = point(x, y)}
\end{mybox}
where \code{x} and \code{y} are real numbers corresponding to the $x$ and $y$ coordinates of the point.
\medskip
\noindent
Internally, this creates a complex number $x + i y$ and stores it in the table \tkzVar{point}{z} under the key \code{"A"}. The table \tkzVar{point}{z} is used to reference all points by their label.
\medskip
\noindent
Alternatively, the more explicit syntax is also available:
\begin{center}
\code{z.A = point:new(x, y)}
\end{center}
\noindent
Both forms are equivalent. The shorter syntax is available since version 4 and preferred for readability and consistency.
% subsection creating_a_point (end)
\def\size{42mm}
\begin{tikzpicture}[remember picture]
\node[ draw, fill=red!10] (tbl) {%
\centering
\begin{minipage}{\size}
\hspace{\fill} \texttt{Arguments}\hspace{\fill}
\tikz\node[minimum width=\size,font=\small,
draw, fill=cyan!10,
rectangle split, rectangle split parts=6
] {
\texttt{re (real)}
\nodepart{two}\texttt{im (real)}
\nodepart{three}\texttt{type = 'point'}
\nodepart{four}\texttt{argument (rad)}
\nodepart{five}\texttt{modulus (cm)}
\nodepart{six}\texttt{mtx (matrix)}
};
\hspace{\fill} \texttt{Methods}\hspace{\fill}
\tikz\node[minimum width=\size,font=\small,
draw, fill=orange!20,sharp corners,
rectangle split, rectangle split parts=4
] {
\texttt{homothety(coeff,obj)}
\nodepart{two}\texttt{rotation (angle,object)}
\nodepart{three}\texttt{symmetry (object)}
\nodepart{four}\texttt{\ldots}
};
\end{minipage}};
\node[ draw, fill=red!10,,minimum height = 2em,
rounded corners,anchor=south] (tc) at (tbl.north){Class |Point|};
\end{tikzpicture}
\hspace{5cm}\begin{tikzpicture}[remember picture]
\node[ draw, fill=red!10] (tbl) {%
\centering
\begin{minipage}{\size}
\hspace{\fill} \texttt{Arguments}\hspace{\fill}
\tikz\node[minimum width=\size,font=\small,
draw, fill=cyan!10,
rectangle split, rectangle split parts=6
] {
\texttt{re = 1}
\nodepart{two}\texttt{im = 2}
\nodepart{three}\texttt{type = 'point'}
\nodepart{four}\texttt{argument = atan(2)}
\nodepart{five}\texttt{modulus = $\sqrt{5}$}
\nodepart{six}\texttt{mtx = \{\{1\},\{2\}\}}
};
\hspace{\fill} \texttt{Methods}\hspace{\fill}
\tikz\node[minimum width=\size,font=\small,
draw, fill=orange!20,sharp corners,
rectangle split, rectangle split parts=4
] {
\texttt{homothety(coeff,obj)}
\nodepart{two}\texttt{rotation (angle,object)}
\nodepart{three}\texttt{symmetry (object)}
\nodepart{four}\texttt{\ldots}
};
\end{minipage}
};
\node[ draw, fill=red!10,remember picture,minimum height = 2em,
rounded corners,anchor=south] (to) at (tbl.north){object |z.A|};
\end{tikzpicture}
\begin{tikzpicture}[remember picture,overlay]
\draw [thick,->](tc.east) -- (to.west);
\end{tikzpicture}
\subsection{Attributes of a point} % (fold)
\label{sub:attributes_of_a_point}
% Method \tkzMeth{point}{new}
\begin{mybox}
Creation \\
|z.A = point (1,2) |
\end{mybox}
The point $A$ has coordinates $x=1$ and $y=2$. If you use the notation |z.A|, then $A$ will be referenced as a node in \TIKZ\ or in \tkzNamePack{tkz-euclide}.
This is the creation of a fixed point with coordinates $1$ and $2$ and which is named $A$. The notation \tkzname{z.A} indicates that the coordinates will be stored in a table assigned to the variable \tkzVar{point}{z} (reference to the notation of the affixes of the complex numbers) that $A$ is the name of the point and the key allowing access to the values.
\begin{center}
\bgroup
\small
\catcode`_=12
\captionof{table}{Point attributes.}\label{point:att}
\begin{tabular}{lll}
\toprule
\textbf{Attributes} & \textbf{Description}& \textbf{Example / Reference} \\
\midrule
\tkzAttr{point}{type} & Object type name, always \texttt{"point"} & \\
\tkzAttr{point}{re} & Real part (i.e., $x$-coordinate) & [\ref{ssub:methods}] \\
\tkzAttr{point}{im} & Imaginary part (i.e., $y$-coordinate) & [\ref{ssub:methods}] \\
\tkzAttr{point}{argument} & Argument of the affix (angle in radians) & $\approx$ \code{0.785398...} \quad [\ref{ssub:example_point_attributes}] \\
\tkzAttr{point}{modulus} & Modulus of the affix (distance to origin) & $\sqrt{5} \approx$ \code{2.2360...} \quad [\ref{ssub:example_point_attributes}] \\
\tkzAttr{point}{mtx} & Matrix representation as column vector & \code{z.A.mtx = {{1},{2}}} \quad [\ref{ssub:example_point_attributes}] \\
\bottomrule
\end{tabular}
\egroup
\end{center}
\subsubsection{Example: point attributes} % (fold)
\label{ssub:example_point_attributes}
\directlua{
init_elements()
z.M = point(1,2)}
\hspace*{\fill}
\begin{verbatim}
\directlua{
init_elements()
z.M = point(1,2)}
\end{verbatim}
\pgfkeys{/pgf/number format/.cd,std,precision=2}
\let\pmpn\pgfmathprintnumber
\DeleteShortVerb{\|}
\begin{verbatim}
\begin{tikzpicture}[scale = 1]
\pgfkeys{/pgf/number format/.cd,std,precision=2}
\let\pmpn\pgfmathprintnumber
\tkzDefPoints{2/4/M,2/0/A,0/0/O,0/4/B}
\tkzLabelPoints(O)
\tkzMarkAngle[fill=gray!30,size=1](A,O,M)
\tkzLabelAngle[pos=1,right](A,O,M){%
$\theta \approx \pmpn{\tkzUseLua{z.M.argument}}$ rad}
\tkzDrawSegments(O,M)
\tkzLabelSegment[above,sloped](O,M){%
$|z_M| =\sqrt{5}\approx \pmpn{%
\tkzUseLua{z.M.modulus}}$ cm}
\tkzLabelPoint[right](M){$M: z_M = 1 + 2i$}
\tkzDrawPoints(M,A,O,B)
\tkzPointShowCoord(M)
\tkzLabelPoint[below,teal](A){$\tkzUseLua{z.M.re}$}
\tkzLabelPoint[left,teal](B){$\tkzUseLua{z.M.im}$}
\tkzDrawSegments[->,add = 0 and 0.25](O,B O,A)
\end{tikzpicture}
\end{verbatim}
\begin{center}
\begin{tikzpicture}
\pgfkeys{/pgf/number format/.cd,std,precision=2}
\let\pmpn\pgfmathprintnumber
\tkzDefPoints{2/4/M,2/0/A,0/0/O,0/4/B}
\tkzLabelPoints(O)
\tkzMarkAngle[fill=gray!30,size=1](A,O,M)
\tkzLabelAngle[pos=1,right](A,O,M){%
$\theta \approx \pmpn{\tkzUseLua{z.M.argument}}$ rad}
\tkzDrawSegments(O,M)
\tkzLabelSegment[above,sloped](O,M){%
$|z_M| =\sqrt{5}\approx \pmpn{\tkzUseLua{z.M.modulus}}$ cm}
\tkzLabelPoint[right](M){$M: z_M = 1 + 2i$}
\tkzDrawPoints(M,A,O,B)
\tkzPointShowCoord(M)
\tkzLabelPoint[below,teal](A){$\tkzUseLua{z.M.re}$}
\tkzLabelPoint[left,teal](B){$\tkzUseLua{z.M.im}$}
\tkzDrawSegments[->,add = 0 and 0.25](O,B O,A)
\begin{scope}[every annotation/.style={fill=lightgray!15,anchor = east}]
\node [annotation,font =\small,text width=6cm] at (current bounding box.west) {
Attributes of \texttt{z.M}
\begin{mybox}{}
\begin{itemize}
\item \texttt{z.M.re} = 1
\item \texttt{z.M.im} = 2
\item \texttt{z.M.type} = 'point'
\item \texttt{z.M.argument} = $\theta \approx \pmpn{\tkzUseLua{z.M.argument}}$ rad
\item \texttt{z.M.modulus} = $|z_M| =\sqrt{5}\approx \pmpn{\tkzUseLua{z.M.modulus}}$ cm
\item \texttt{z.M.mtx} = \tkzUseLua{z.M.mtx:print()}
\end{itemize}
\end{mybox}
};
\end{scope}
\end{tikzpicture}
\end{center}
\MakeShortVerb{\|}
\subsubsection{Attribute \tkzAttr{point}{mtx}} % (fold)
\label{ssub:attribute_iattr_point_mtx}
This attribute allows the point to be used in conjunction with matrices.
\vspace{1em}
\begin{minipage}{.5\textwidth}
\begin{verbatim}
\directlua{
z.A = point(2,-1)
z.A.mtx:print()}
\end{verbatim}
\end{minipage}
\begin{minipage}{.5\textwidth}
\begin{center}
\directlua{
z.A = point(2,-1)
z.A.mtx:print()}
\end{center}
\end{minipage}
% subsubsection attribute_iattr_point_mtx (end)
% subsubsection example_point_attributes (end)
% subsection attributes_of_a_point (end)
\subsubsection{Argand diagram} % (fold)
\label{ssub:argand_diagram}
\normalsize
\begin{minipage}{\textwidth}
\begin{verbatim}
\directlua{
init_elements()
z.A = point(2, 3)
z.O = point(0, 0)
z.I = point(1, 0)}
\begin{tikzpicture}
\tkzGetNodes
\tkzInit[xmin=-4,ymin=-4,xmax=4,ymax=4]
\tkzDrawCircle[dashed,red](O,A)
\tkzPointShowCoord(A)
\tkzDrawPoint(A)
\tkzLabelPoint[above right](A){\normalsize $a+ib$}
\tkzDrawX\tkzDrawY
\tkzDrawSegment(O,A)
\tkzLabelSegment[above,anchor=south,sloped](O,A){ OA = modulus of $z_A$}
\tkzLabelAngle[anchor=west,pos=.5](I,O,A){$\theta$ = argument of $z_A$}
\end{tikzpicture}
\end{verbatim}
\end{minipage}
\begin{minipage}{\textwidth}
\directlua{
init_elements()
z.A = point(2, 3)
z.O = point(0, 0)
z.I = point(1, 0)}
\begin{center}
\begin{tikzpicture}
\tkzGetNodes
\tkzInit[xmin=-4,ymin=-4,xmax=4,ymax=4]
\tkzDrawCircle[dashed,red](O,A)
\tkzPointShowCoord(A)
\tkzDrawPoint(A)
\tkzLabelPoint[above right](A){\normalsize $a+ib$}
\tkzDrawX\tkzDrawY
\tkzDrawSegment(O,A)
\tkzLabelSegment[above,anchor=south,sloped](O,A){ OA = modulus of $z_A$}
\tkzLabelAngle[anchor=west,pos=.5](I,O,A){$\theta$ = argument of $z_A$}
\end{tikzpicture}
\end{center}
\end{minipage}
% subsubsection argand_diagram (end)
\subsection{Methods of the class point} % (fold)
\label{sub:methods_of_the_class_point}
The methods listed in the following table are standard and commonly used throughout the examples at the end of this documentation. Each of these methods returns a \tkzNameObj{point} object.
\medskip
\noindent
For more advanced operations using complex numbers and operator overloading, see section~\ref{sub:complex_numbers}, which describes the available metamethods.
\vspace{1em}
\begin{center}
\bgroup
\catcode`_=12
\small
\captionof{table}{Functions and Methods of the \code{class} point.}\label{point:met}
\begin{tabular}{lll}
\toprule
\textbf{Methods} & \textbf{Reference} \\
\midrule
\textbf{Creation} &\\
\midrule
\tkzMeth{point}{new(r,r)} &[\ref{ssub:method_point_new}; \ref{ssub:method_normalize}] \\
\tkzMeth{point}{polar(d,an)} &[\ref{ssub:method_point_polar}; \ref{sub:archimedes}] \\
\tkzMeth{point}{polar\_deg(d,an)} &[\ref{ssub:method_point_polar_deg}] \\
\tkzMeth{point}{get()} &[\ref{ssub:method_point_get}] \\
\midrule
\textbf{Directional Shifts} & \\
\midrule
\tkzMeth{point}{north(r)} & \ref{ssub:methods}] \\
\tkzMeth{point}{south(r)} & \\
\tkzMeth{point}{east(r)} & \\
\tkzMeth{point}{west(r)} & \\
\midrule
\textbf{Geometry} & \\
\midrule
\tkzMeth{point}{normalize()} & [\ref{ssub:method_normalize}] \\
\tkzMeth{point}{normalize\_from(pt)} & [\ref{ssub:method_normalize}] \\
\tkzMeth{point}{orthogonal(d)} & [\ref{ssub:orthogonal_method}]\\
\tkzMeth{point}{at()} & [\ref{ssub:_point_at_method}] \\
\midrule
\textbf{Transformations} & \\
\midrule
\tkzMeth{point}{symmetry(obj)} & [\ref{ssub:object_symmetry}] \\
\tkzMeth{point}{rotation(an, obj)} & [\ref{ssub:object_rotation}] \\
\tkzMeth{point}{homothety(r,obj)} & [\ref{ssub:method_point_homothety_k_obj}] \\
\midrule
\textbf{Utilities} &\\
\midrule
\tkzMeth{point}{print()} & [\ref{ssub:object_symmetry} ]\\
\bottomrule %
\end{tabular}
\egroup
\end{center}
\subsubsection{Method \tkzMeth{point}{new(r, r)}} % (fold)
\label{ssub:method_point_new}
This method creates a point in the plane using Cartesian coordinates. The shorter syntax \tkzMeth{point}{(r, r)} is available since version 4 and preferred for readability and consistency.
\medskip
\noindent
It takes two real numbers as arguments: the first represents the \emph{abscissa} (real part), and the second the \emph{ordinate} (imaginary part). Internally, the point is treated as a complex number and stored in the global table \tkzname{z}.
\medskip
\noindent
The resulting object is of type \tkzNameObj{point}, and can be used in further geometric constructions or displayed with \tkzNamePack{tkz-euclide}.
\medskip
\noindent
\smallbf{Note:} The default unit is the centimeter, in accordance with the conventions of \tkzNamePack{tkz-euclide}. All coordinates are interpreted in an orthonormal Cartesian coordinate system.
\vspace{1em}
\begin{tkzexample}[latex = 7cm]
\directlua{
init_elements()
z.A = point(0, 0)
z.B = point(2, 1)}
\begin{tikzpicture}[gridded]
\tkzGetNodes
\tkzDrawPoints(A,B)
\tkzLabelPoints(A,B)
\end{tikzpicture}
\end{tkzexample}
% subsubsection method_point_new (end)
\subsubsection{Method \tkzMeth{point}{get()}} % (fold)
\label{ssub:method_point_get}
The \code{get} method is used to retrieve the Cartesian coordinates of a point.
\medskip
\noindent
Let $I$ be the intersection point of two lines. You can obtain its coordinates in two equivalent ways:
\begin{itemize}
\item using the point’s attributes directly:
\begin{center}
\code{x = z.I.re} \quad and \quad \code{y = z.I.im}
\end{center}
\item or using the \code{get()} method:
\begin{center}
\code{x, y = z.I:get()}
\end{center}
\end{itemize}
\noindent
This method improves code readability and makes it easier to pass coordinates to functions that expect numerical values.
\vspace{1em}
\begin{minipage}{.5\textwidth-1cm}
\directlua{
init_elements()
z.A, z.B = point(0, 0), point(5, -1)
z.C, z.D = point(1, -4), point(4, 2)
L.AB = line(z.A, z.B)
L.CD = line(z.C, z.D)
z.I = intersection(L.AB, L.CD)
x, y = z.I:get()
tex.sprint("$z_I = $"..x.." ".."$y_I = $"..y)}
\end{minipage}
\begin{minipage}{.5\textwidth}
\begin{tkzexample}[code only]
\directlua{
init_elements()
z.A, z.B = point(0, 0), point(5, -1)
z.C, z.D = point(1, -4), point(4, 2)
L.AB = line(z.A, z.B)
L.CD = line(z.C, z.D)
z.I = intersection(L.AB, L.CD)
x, y = z.I:get()
tex.print("$x_I = $"..x.." ".."$y_I = $"..y)}
\end{tkzexample}
\end{minipage}
% subsubsection method_point_get (end)
\subsubsection{Method \tkzMeth{point}{polar(r, an)}} % (fold)
\label{ssub:method_point_polar}
This method creates a point in the plane using polar coordinates.
\medskip
\noindent
It takes two arguments:
\begin{itemize}
\item \code{r} —> the modulus (distance from the origin),
\item \code{an} —> the argument (angle in radians).
\end{itemize}
\noindent
Internally, the point is represented as a complex number: \code{r * exp(i * an)}. This method is particularly useful for constructing points on circles or for defining points in terms of angle and distance.
\medskip
\noindent
\textbf{Note:} The default unit is the centimeter, consistent with the conventions of \tkzNamePack{tkz-euclide}. All coordinates are interpreted in an orthonormal Cartesian coordinate system.
\vspace{1em}
\begin{mybox}
\begin{verbatim}
z.B = point:polar(2, math.pi / 4)
better
z.B = point(polar(2, math.pi / 4))
\end{verbatim}
\end{mybox}
\begin{tkzexample}[latex = 7cm]
\directlua{
init_elements()
z.O = point(0, 0)
z.A = point(3, 0)
z.F = point(polar(3, math.pi / 3))}
\begin{center}
\begin{tikzpicture}[scale=.75]
\tkzGetNodes
\tkzDrawCircle(O,A)
\tkzDrawSegments[new](O,A)
\tkzDrawSegments[purple](O,F)
\tkzDrawPoints(A,O,F)
\tkzLabelPoints[below right=6pt](A,O)
\tkzLabelPoints[above](F)
\end{tikzpicture}
\end{center}
\end{tkzexample}
% subsubsection method_point_polar (end)
\subsubsection{Method \tkzMeth{point}{polar\_deg(d,an)}} % (fold)
\label{ssub:method_point_polar_deg}
\begin{tkzexample}[latex=7cm]
\directlua{
init_elements()
z.A = point(0, 0)
z.B = point(polar_deg(3, 60))
z.C = point(polar_deg(3, 0))}
\begin{center}
\begin{tikzpicture}[gridded]
\tkzGetNodes
\tkzDrawPolygon(A,B,C)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,C)
\tkzLabelPoints[above](B)
\end{tikzpicture}
\end{center}
\end{tkzexample}
% subsubsection method_point_polar_deg(end)
\subsubsection{Method \tkzMeth{point}{north(d)}} % (fold)
\label{ssub:example_method_point_north_d}
This method creates a new point located at a vertical distance from the given point, along the line passing through it and directed upward (toward the north).
\medskip
\noindent
It is particularly useful when you want to construct a point offset by a specific distance above a reference point—for example, to place a label or construct a geometric configuration with a known height.
\medskip
\noindent
The optional argument \code{d} represents the vertical distance. If omitted, a default value of $1$ is used.
\vspace{1em}
\begin{tkzexample}[latex = 7cm]
\directlua{
init_elements()
z.O = point(0, 0)
z.A = z.O:east()
z.Ap= z.O:east(1):north(1)
z.B = z.O:north()
z.C = z.O:west()
z.D = z.O:south()}
\begin{center}
\begin{tikzpicture}
\tkzGetNodes
\tkzDrawPolygon(A,B,C,D)
\tkzDrawPoints(A,B,C,D,O,A')
\end{tikzpicture}
\end{center}
\end{tkzexample}
% subsubsection example_method_point_north_d (end)
\subsubsection{Method \tkzMeth{point}{normalize()}} % (fold)
\label{ssub:method_normalize}
This method returns a new point located on the segment from the origin to the current point, at a distance of $1$ from the origin. It is typically used to extract the direction of a vector and normalize its length to one.
\medskip
\noindent
You can also use this method to construct a point at a fixed distance from another point along a given direction. For example:
\begin{center}
\code{z.U = (z.C - z.B):normalize() + z.B}
\end{center}
\noindent
Here, the vector $\overrightarrow{BU}$ has length $1$, and $U$ lies on the segment $[BC]$ in the direction from $B$ to $C$.
\medskip
\noindent
There are two equivalent ways to achieve the same result:
\begin{mybox}
\begin{verbatim}
z.U = z.C:normalize_from(z.B)
z.U = L.BC:normalize()
\end{verbatim}
\end{mybox}
\noindent
The second approach requires prior creation of the line object \code{L.BC}.
\vspace{1em}
\begin{tkzexample}[latex=8cm]
\directlua{
init_elements()
z.A = point(0, 0)
z.B = point(4, 3)
z.C = point(1, 5)
L.AB = line(z.A, z.B)
L.BC = line(z.B, z.C)
z.N = z.B:normalize()
z.U = z.C:normalize_from(z.B)}
\begin{tikzpicture}
\tkzGetNodes
\tkzDrawLines(A,B B,C)
\tkzDrawPoints(A,B,C,U,N)
\tkzLabelPoints(A,B,C,U,N)
\tkzDrawSegment(B,C)
\end{tikzpicture}
\end{tkzexample}
% subsubsection method_normalize (end)
\subsubsection{Method \tkzMeth{point}{orthogonal(d)}} % (fold)
\label{ssub:orthogonal_method}
Let $O$ be the origin of the plane, and let $A$ be a point distinct from $O$. This method constructs a new point $B$ such that the vectors $\overrightarrow{OB}$ and $\overrightarrow{OA}$ are orthogonal:
\[
\overrightarrow{OB} \perp \overrightarrow{OA}
\]
\medskip
\noindent
By default, the point $B$ is chosen so that $|OB| = |OA|$. If the optional argument \code{d} is provided, then the point $B$ is constructed so that $|OB| = d$.
\medskip
\noindent
This method is useful for constructing perpendicular vectors or generating points on circles orthogonal to given directions.
\vspace{1em}
\begin{tkzexample}[latex=7cm]
\directlua{
init_elements()
z.A = point(3, 1)
z.B = z.A:orthogonal(1)
z.O = point(0, 0)
z.C = z.A:orthogonal()}
\begin{center}
\begin{tikzpicture}[gridded]
\tkzGetNodes
\tkzDrawSegments(O,A O,C)
\tkzDrawPoints(O,A,B,C)
\tkzLabelPoints[below right](O,A,B,C)
\end{tikzpicture}
\end{center}
\end{tkzexample}
% subsubsection orthogonal_method(end)
\subsubsection{Method \tkzMeth{point}{at(pt)}} % (fold)
\label{ssub:_point_at_method}
This method complements the \code{orthogonal} method. Instead of constructing a point $B$ such that $\overrightarrow{OB} \perp \overrightarrow{OA}$ (with $O$ as the origin), it constructs a point $B$ such that:
\[
\overrightarrow{AB} \perp \overrightarrow{OA}
\]
\noindent
In this case, the reference direction remains $\overrightarrow{OA}$, but the orthogonal vector is constructed from point $A$, not the origin. The result is a point $B$ lying on a line orthogonal to $(OA)$ and passing through $A$.
\medskip
\noindent
This method is useful when working with local orthogonal directions, such as when constructing altitudes in a triangle or defining perpendicular vectors anchored at a given point.
\vspace{1em}
\begin{tkzexample}[latex=7cm]
\directlua{
init_elements()
z.O = point(0,0 )
z.A = point(3, 2 )
z.B = z.A:orthogonal(1)
z.C = z.A+z.B
z.D =(z.C - z.A):orthogonal(2):at(z.C)}
\begin{center}
\begin{tikzpicture}[gridded]
\tkzGetNodes
\tkzLabelPoints[below right](O,A,C)
\tkzLabelPoints[above](B,D)
\tkzDrawSegments(O,A A,B A,C C,D O,B)
\tkzDrawPoints(O,A,B,C,D)
\end{tikzpicture}
\end{center}
\end{tkzexample}
% subsubsection _point_at_method (end)
\subsubsection{Method \tkzMeth{point}{rotation(obj)}} — First example % (fold)
\label{ssub:example_rotation_of_points}
This method performs a rotation of one or more points around the current point, which serves as the center of rotation.
\medskip
\noindent
The arguments are:
\begin{itemize}
\item the angle of rotation, expressed in radians;
\item a point or a list of points to rotate.
\end{itemize}
\noindent
The result is a point (or a list of points) obtained by rotating each given point around the center by the specified angle.
\medskip
\noindent
In the following example, a list of points is rotated about a given center.
\vspace{1em}
\begin{tkzexample}[latex=7cm]
\directlua{
init_elements()
z.a = point(0, -1)
z.b = point(4, 0)
z.o = point(6, -2)
z.ap,
z.bp = z.o:rotation(math.pi / 2, z.a, z.b)}
\begin{center}
\begin{tikzpicture}[scale = .5]
\tkzGetNodes
\tkzDrawLines(o,a o,a' o,b o,b')
\tkzDrawPoints(a,a',b,b',o)
\tkzLabelPoints(b,b',o)
\tkzLabelPoints[below left](a,a')
\tkzDrawArc(o,a)(a')
\tkzDrawArc(o,b)(b')
\end{tikzpicture}
\end{center}
\end{tkzexample}
% subsubsection example_rotation_of_points (end)
\subsubsection{Method \tkzMeth{point}{rotation(an, obj)}} — Second example % (fold)
\label{ssub:object_rotation}
This method rotates a geometric object by a given angle around a specified point.
\medskip
\noindent
In this example, the triangle is rotated by an angle of~$\pi/3$ around the point~$O$, which serves as the center of rotation.
\medskip
\noindent
The arguments are:
\begin{itemize}
\item \code{an} —> the angle of rotation, in radians;
\item \code{obj} —> the geometric object to be rotated (e.g., a triangle).
\end{itemize}
\noindent
The method returns a new object of the same type, rotated accordingly.
\vspace{1em}
\begin{tkzexample}[latex=7cm]
\directlua{
init_elements()
z.O = point(-1, -1)
z.A = point(2, 0)
z.B = point(5, 0)
L.AB = line(z.A, z.B)
T.ABC = L.AB:equilateral()
S.fig = L.AB:square()
_,_,z.E,z.F = S.fig:get()
S.new = z.O:rotation(math.pi / 3, S.fig)
_,_,z.Ep,z.Fp = S.new:get()
z.C = T.ABC.pc
T.ApBpCp = z.O:rotation(math.pi / 3, T.ABC)
z.Ap,z.Bp,
z.Cp = T.ApBpCp:get()}
\begin{center}
\begin{tikzpicture}[scale = .6]
\tkzGetNodes
\tkzDrawPolygons(A,B,C A',B',C' A,B,E,F)
\tkzDrawPolygons(A',B',E',F')
\tkzDrawPoints(A,B,C,A',B',C',O)
\tkzLabelPoints(A,B,C,A',B',C',O)
\begin{scope}
\tkzDrawArc[delta=0,->,dashed,red](O,A)(A')
\tkzDrawSegments[dashed,red](O,A O,A')
\end{scope}
\end{tikzpicture}
\end{center}
\end{tkzexample}
% subsubsection object_rotation (end)
\subsubsection{Method \tkzMeth{point}{symmetry(obj)}} % (fold)
\label{ssub:object_symmetry}
This method performs a central symmetry (point reflection) of a geometric object with respect to the given point.
\medskip
\noindent
The argument \code{obj} can be a point, a line, a triangle, a circle, or any other supported geometric object. Each element of the object is reflected through the center point, producing a new object of the same type.
\medskip
\noindent
The following example shows how to apply central symmetry to an object (e.g., a triangle) using a reference point as the center.
\vspace{1em}
\begin{tkzexample}[latex=7cm]
\directlua{
init_elements()
z.a = point(0,-1)
z.b = point(2, 0)
L.ab = line(z.a,z.b)
C.ab = circle(z.a,z.b)
z.o = point(1,1)
z.ap, z.bp = z.o:symmetry(C.ab):get()}
\begin{center}
\begin{tikzpicture}
\tkzGetNodes
\tkzDrawCircles(a,b a',b')
\tkzDrawLines(a,a' b,b')
\tkzDrawLines[red](a,b a',b')
\tkzDrawPoints(a,a',b,b',o)
\tkzLabelPoints(a,a',b,b',o)
\end{tikzpicture}
\end{center}
\end{tkzexample}
% subsubsection object_symmetry (end)
\subsubsection{Method \tkzMeth{point}{homothety(k, obj)}} % (fold)
\label{ssub:method_point_homothety_k_obj}
This method performs a homothety (dilation or contraction) of a geometric object with respect to the current point, which serves as the center of the transformation.
\medskip
\noindent
The arguments are:
\begin{itemize}
\item \code{k} — the homothety ratio (a real number),
\item \code{obj} — the object to be transformed, which can be:
\begin{enumerate}
\item a single point,
\item a list of points,
\item or a geometric object (line, triangle, circle, etc.).
\end{enumerate}
\end{itemize}
\noindent
A positive ratio \code{k} produces a scaling centered at the point, while a negative ratio also reflects the object through the center.
\vspace{1em}
\begin{tkzexample}[latex=7cm]
\directlua{
init_elements()
z.A = point(0, 0)
z.B = point(1, 2)
z.E = point(-3, 2)
z.C, z.D = z.E:homothety(2, z.A, z.B)}
\begin{center}
\begin{tikzpicture}[scale = .5]
\tkzGetNodes
\tkzDrawPoints(A,B,C,E,D)
\tkzLabelPoints(A,B,C,E)
\tkzDrawCircles(A,B C,D)
\tkzDrawLines(E,C E,D)
\end{tikzpicture}
\end{center}
\end{tkzexample}
This method converts the point’s coordinates to a formatted string that can be displayed directly in the text.
\medskip
\noindent
The number of decimal places is controlled by the global variable \code{tkz\_dc}, which is set to $2$ by default in the \code{init\_elements()} function. You can override it by assigning a new value before calling \code{print()}:
\begin{verbatim}
tkz_dc = 0
\end{verbatim}
\medskip
\noindent
This is particularly useful when displaying coordinates, sums, products, or intermediate results in mathematical expressions.
\medskip
\noindent
Example:
\begin{verbatim}
\directlua{
init_elements()
z.A = point(1, 2)
z.B = point(1, -1)
z.a = z.A + z.B
z.m = z.A * z.B
tkz_dc = 0}
The respective affixes of points $A$ and $B$ being
\tkzUseLua{z.A:print()} and \tkzUseLua{z.B:print()},
their sum is \tkzUseLua{z.a:print()} and
their product \tkzUseLua{z.m:print()}.
\end{verbatim}
\directlua{
init_elements()
z.A = point(1, 2)
z.B = point(1, -1)
z.a = z.A + z.B
z.m = z.A * z.B
tkz_dc = 0
}
The respective affixes of points $A$ and $B$ being
\tkzUseLua{z.A:print()} and \tkzUseLua{z.B:print()},
their sum is \tkzUseLua{z.a:print()} and
their product \tkzUseLua{z.m:print()}.
% subsubsection _point_print (end)
% subsubsection method_point_homothety_k_obj (end)
% subsection methods_of_the_class_point (end)
% section class_point (end)
\endinput