\newpage
\section{Orthormal cartesian coordinate system} % (fold)
\label{sec:orthormal_cartesian_coordinate_system}

This new class is still experimental. I mainly use it for drawing conics.

It's also an opportunity to remind you of certain notions concerning this package. Although the aim of the package is not to use equations, to draw with \TIKZ{} it is necessary to use a coordinate system. So as not to complicate development, I've chosen to work only with an orthonormal Cartesian coordinate system (occs).

For now, the goal is to move from one orthonormal Cartesian coordinate system to another, also orthonormal. The new coordinate system is obtained from the first by translation and rotation. The arguments are a straight line that indicates the direction of the new abscissa axis and a point that will be the center of the new coordinate system.

Let's look at a concrete example. We are looking for the intersections of a parabola with a straight line. The parabola is defined by a directrix and a focus. In a reference frame centered on the vertex $S$ of the parabola, with the x-axis being a parallel passing through this vertex to the directrix, the equation of the parabola is $y =\dfrac{x^2}{2p}$, where $p$ is the value of the \code{latus rectum}. This corresponds to the length of the segment originating from the focus, parallel to the directrix, and ending on the parabola. $p$ is also the distance from the focus to the directrix.

To find the intersections, we simply determine the equation of the line in the new reference frame and use $y =\dfrac{x^2}{2p}$ as the equation of the curve. In this example, two internal functions are used. It is best not to use them at this stage, as their names may change. The function \code{param\_line} is used to determine the coefficients of the line, and \code{solve\_para\_line} finds the intersection points.

If solutions exist, the result provides two values, $r_1$ and $r_2$, which are the abscissas of the intersection points in the new reference frame. After computing the ordinates, one could use a function to determine the coordinates in \TIKZ{} within the new coordinate system. However, there is another approach: we determine the points $x$ and $y$ corresponding to the values found ($r_1$, $r_2$) on the abscissa axis of the new reference frame \code{OCCS.abscissa}. Then, geometrically, we locate the intersection points on the parabola.

\begin{Verbatim}
\directlua{
 init_elements ()
z.O       = point : new (0,0)
z.i       = point : new (1,0)
z.j       = point : new (0,1)
z.A       = point : new (-1 , 0)
z.B       = point : new (5 , 4)
L.dir     = line  : new (z.A , z.B)
z.F       = point : new (0 , 3)        
PA        = conic : new (z.F,L.dir,1)
curve     = PA    : points (-3,3,50)
local p   = PA.p
z.P       = L.dir : report (p,z.K)
z.X       = PA : point (p)
z.H       = L.dir : projection (z.X)
z.K       = PA.K
z.S       = PA.vertex
L.KF      = PA.major_axis
% new occs          
OCCS      = occs  : new (L.dir,z.S)
z.u       = OCCS.x
z.v       = OCCS.y
% line a,b
z.a       = point : new (-1,1)
z.b       = point : new (3,5)
L.ab      = line  : new (z.a,z.b)
% % coordinates in the new occs
X,Y       = OCCS : coordinates (z.F)
Xa,Ya     = OCCS : coordinates (z.a)
Xb,Yb     = OCCS : coordinates (z.b)
% solve in the new occs
local r,s = param_line (Xa,Ya,Xb,Yb)
r1,r2     = solve_para_line (p,r,s)
z.x       = OCCS.abscissa :report(r1,z.K)
z.y       = OCCS.abscissa :report(r2,z.K)
L1        = L.dir : ortho_from (z.x)
L2        = L.dir : ortho_from (z.y)
z.s1      = intersection (L.ab,L1)
z.s2      = intersection (L.ab,L2)
}

\begin{tikzpicture}
 \tkzGetNodes
 \tkzDrawCoordinates[smooth,orange,thick](curve)
 \tkzDrawLines(A,B)
 \tkzDrawLines[add = 1 and 1](K,F)
 \tkzDrawSegments[add = .5 and .5,blue](a,b)
 \tkzDrawSegments[dashed](s1,x s2,y)
 \tkzDrawPoints(A,B,F,K,S)
 \tkzDrawPoints[blue,size=2](a,b)
 \tkzDrawPoints[blue,size=2](s1,s2,x,y)
 \tkzLabelPoints[blue](a,b)
 \tkzLabelPoints[blue,above left](s1,s2)
 \tkzLabelPoints(O,i,u,S,A,B,x,y)
 \tkzLabelPoints[left](j,v)
 \tkzLabelPoints[right](F,K)
 \tkzDrawSegments[->,red,thick](O,i O,j)
 \tkzDrawSegments[->,purple,thick](S,u S,v)
 \tkzLabelSegment[below,sloped,pos=.7](A,B){Directrix}
\end{tikzpicture}
\end{Verbatim}

\directlua{
 init_elements ()
z.O       = point : new (0,0)
z.i       = point : new (1,0)
z.j       = point : new (0,1)
z.A       = point : new (-1 , 0)
z.B       = point : new (5 , 4)
L.dir     = line  : new (z.A , z.B)
z.F       = point : new (0 , 3)        
PA        = conic : new (z.F,L.dir,1)
curve     = PA    : points (-3,3,50)
local p   = PA.p
z.P       = L.dir : report (p,z.K)
z.X       = PA : point (p)
z.H       = L.dir : projection (z.X)
z.K       = PA.K
z.S       = PA.vertex
L.KF      = PA.major_axis
% new occs          
OCCS      = occs  : new (L.dir,z.S)
z.u       = OCCS.x
z.v       = OCCS.y
% line a,b
z.a       = point : new (-1,1)
z.b       = point : new (3,5)
L.ab      = line  : new (z.a,z.b)
% % coordinates in the new occs
X,Y       = OCCS : coordinates (z.F)
Xa,Ya     = OCCS : coordinates (z.a)
Xb,Yb     = OCCS : coordinates (z.b)
% solve in the new occs
local r,s = param_line (Xa,Ya,Xb,Yb)
r1,r2     = solve_para_line (p,r,s)
z.x       = OCCS.abscissa :report(r1,z.K)
z.y       = OCCS.abscissa :report(r2,z.K)
L1        = L.dir : ortho_from (z.x)
L2        = L.dir : ortho_from (z.y)
z.s1      = intersection (L.ab,L1)
z.s2      = intersection (L.ab,L2)
}

\begin{center}
  \begin{tikzpicture}
   \tkzGetNodes
   \tkzDrawCoordinates[smooth,orange,thick](curve)
   \tkzDrawLines(A,B)
   \tkzDrawLines[add = 1 and 1](K,F)
   \tkzDrawSegments[add = .5 and .5,blue](a,b)
   \tkzDrawSegments[dashed](s1,x s2,y)
   \tkzDrawPoints(A,B,F,K,S)
   \tkzDrawPoints[blue,size=2](a,b)
   \tkzDrawPoints[blue,size=2](s1,s2,x,y)
   \tkzLabelPoints[blue](a,b)
   \tkzLabelPoints[blue,above left](s1,s2)
   \tkzLabelPoints(O,i,u,S,A,B,x,y)
   \tkzLabelPoints[left](j,v)
   \tkzLabelPoints[right](F,K)
   \tkzDrawSegments[->,red,thick](O,i O,j)
   \tkzDrawSegments[->,purple,thick](S,u S,v)
   \tkzLabelSegment[below,sloped,pos=.7](A,B){Directrix}
  \end{tikzpicture}
\end{center}

\subsection{Attribtes of an occs} % (fold)
\label{sub:attribtes_of_an_occs}

\begin{mybox}
   Creation |sys = occs: new (L.AB,z.S) |
\end{mybox}

\bgroup
\catcode`_=12
\small
\captionof{table}{occs attributes.}\label{occs:att}
\begin{tabular}{lll}
\toprule
\textbf{Attributes}     & \textbf{Application} &\\
\Iattr{occs}{type}  & |sys.type = "occs"| & \\
\Iattr{occs}{origin} & |sys.origin = z.S| &\\
\Iattr{occs}{x}    &  |sys.x = z.I|   &  |SI = 1|\\
\Iattr{occs}{y}    &  |sys.x = z.J|   &  |SJ = 1|\\
\Iattr{occs}{abscissa}    &  |sys.abscissa = L.SI|   &  \\
\Iattr{occs}{ordinate}    &  |sys.ordinate = L.SJ|   &  \\
\bottomrule %
\end{tabular}
\egroup

\subsubsection{Example: attributes of occs} % (fold)
\label{ssub:example_attributes_of_occs}

  \begin{minipage}{.5\textwidth}
\begin{Verbatim}
  \directlua{
  init_elements ()
  z.O       = point : new (0,0)
  z.i       = point : new (1,0)
  z.j       = point : new (0,1)
  z.A       = point : new (-1 , -1)
  z.B       = point : new (4 , 2)
  L.AB      = line  : new (z.A , z.B)
  z.S       = point : new (0 , 3)        
  % new occs          
  sys       = occs  : new (L.AB,z.S)
  z.u       = sys.x
  z.v       = sys.y
  z.ax      = sys.abscissa.pa
  z.bx      = sys.ordinate.pa
  z.ay      = sys.abscissa.pb
  z.by      = sys.ordinate.pb
  }
  \begin{tikzpicture}
   \tkzGetNodes
   \tkzDrawLines(A,B)
   \tkzDrawLines[purple,add=2 and 4](ax,ay bx,by)
   \tkzDrawSegments[->,red,thick](O,i O,j)
   \tkzDrawSegments[->,purple,thick](S,u S,v)
   \tkzLabelSegment[below,sloped,pos=.7](A,B){L.AB}
   \tkzLabelSegment[below,sloped,pos=3](ax,ay){abscissa}
   \tkzLabelSegment[below,sloped,pos=3](bx,by){ordinate}
   \tkzLabelPoints(O,S)
   \tkzLabelPoints[left](j,v)
   \tkzLabelPoints[below right](i,u)
  \end{tikzpicture}
\end{Verbatim}
\end{minipage}
  \begin{minipage}{.5\textwidth}
    \directlua{
    init_elements ()
    z.O       = point : new (0,0)
    z.i       = point : new (1,0)
    z.j       = point : new (0,1)
    z.A       = point : new (-1 , -1)
    z.B       = point : new (4 , 2)
    L.AB      = line  : new (z.A , z.B)
    z.S       = point : new (0 , 3)        
    % new occs          
    sys       = occs  : new (L.AB,z.S)
    z.u       = sys.x
    z.v       = sys.y
    z.ax      = sys.abscissa.pa
    z.bx      = sys.ordinate.pa
    z.ay      = sys.abscissa.pb
    z.by      = sys.ordinate.pb
    }
    \begin{center}
      \begin{tikzpicture}
       \tkzGetNodes
       \tkzDrawSegment(A,B)
       \tkzDrawLines[purple,add=2 and 4](ax,ay bx,by)
       \tkzDrawSegments[->,red,thick](O,i O,j)
       \tkzDrawSegments[->,purple,thick](S,u S,v)
       \tkzLabelSegment[below,sloped,pos=.7](A,B){L.AB}
       \tkzLabelSegment[below,sloped,pos=3](ax,ay){abscissa}
       \tkzLabelSegment[below,sloped,pos=3](bx,by){ordinate}
       \tkzLabelPoints(O,S)
       \tkzLabelPoints[left](j,v)
       \tkzLabelPoints[below right](i,u)
      \end{tikzpicture}
    \end{center}
  \end{minipage}


  
% subsubsection example_attributes_of_occs (end)
% subsection attribtes_of_an_occs (end)

\subsection{Methods of the class occs} % (fold)
\label{sub:methods_of_the_class_occs}

Currently, there are only two methods. \code{new} for creation and \code{coordinates} for obtaining the new coordinates of a point.

The affix of $S$ in the original system is $\tkzUseLua{tostring(z.S)}$.
The new coordinates of $S$ are $\tkzUseLua{XS}$ and $\tkzUseLua{YS}$.
The affix of $u$ in the original system is $\tkzUseLua{tostring(z.u)}$.
The new coordinates of $u$ are $\tkzUseLua{Xu}$ and $\tkzUseLua{Yu}$.

The point $M$ whose coordinates are $(2,6)$ in the original system has coordinates $(4.6305; 1.8865)$ in the new one.

\begin{Verbatim}
  \directlua{
  z.O       = point : new (0,0)
  z.i       = point : new (1,0)
  z.j       = point : new (0,1)
  z.A       = point : new (-1 , -1)
  z.B       = point : new (4 , 2)
  L.AB      = line  : new (z.A , z.B)
  z.S       = point : new (-1 , 2)        
  % new occs          
  sys       = occs  : new (L.AB,z.S)
  z.u       = sys.x
  z.v       = sys.y
  z.ax      = sys.abscissa.pa
  z.bx      = sys.ordinate.pa
  z.ay      = sys.abscissa.pb
  z.by      = sys.ordinate.pb
  XS,YS     = sys : coordinates (z.S)
  Xu,Yu     = sys : coordinates (z.u)
  Xv,Yv     = sys : coordinates (z.v)
  z.M       = point : new (2,6)
  XM,YM     = sys : coordinates (z.M)
  z.xm = sys.abscissa :projection (z.M) 
  z.ym = sys.ordinate :projection (z.M) 
  }

  \begin{tikzpicture}[gridded]
   \tkzGetNodes
   \tkzDrawLines(A,B)
   \tkzDrawLines[purple,add=4 and 4](ax,ay bx,by)
   \tkzDrawSegments[->,red,thick](O,i O,j)
   \tkzDrawSegments[->,purple,thick](S,u S,v M,xm M,ym)
   \tkzDrawPoint(M)
   \tkzLabelPoint[above](M){$M:$ ($\pmpn{\tkzUseLua{XM}};\pmpn{\tkzUseLua{YM}})$}
   \tkzLabelPoints(O,S)
   \tkzLabelPoints[left](j,v)
   \tkzLabelPoints[below right](i,u)
  \end{tikzpicture}
\end{Verbatim}

  \directlua{
  z.O       = point : new (0,0)
  z.i       = point : new (1,0)
  z.j       = point : new (0,1)
  z.A       = point : new (-1 , -1)
  z.B       = point : new (4 , 2)
  L.AB      = line  : new (z.A , z.B)
  z.S       = point : new (-1 , 2)        
  % new occs          
  sys       = occs  : new (L.AB,z.S)
  z.u       = sys.x
  z.v       = sys.y
  z.ax      = sys.abscissa.pa
  z.bx      = sys.ordinate.pa
  z.ay      = sys.abscissa.pb
  z.by      = sys.ordinate.pb
  XS,YS     = sys : coordinates (z.S)
  Xu,Yu     = sys : coordinates (z.u)
  Xv,Yv     = sys : coordinates (z.v)
  z.M       = point : new (2,6)
  XM,YM     = sys : coordinates (z.M)
  z.xm = sys.abscissa :projection (z.M) 
  z.ym = sys.ordinate :projection (z.M) 
  }

  \begin{center}
    \begin{tikzpicture}[gridded]
     \tkzGetNodes
     \tkzDrawLines(A,B)
     \tkzDrawLines[purple,add=4 and 4](ax,ay bx,by)
     \tkzDrawSegments[->,red,thick](O,i O,j)
     \tkzDrawSegments[->,purple,thick](S,u S,v M,xm M,ym)
     \tkzDrawPoint(M)
     \tkzLabelPoint[above](M){$M:$ ($\pmpn{\tkzUseLua{XM}};\pmpn{\tkzUseLua{YM}})$}
     \tkzLabelPoints(O,S)
     \tkzLabelPoints[left](j,v)
     \tkzLabelPoints[below right](i,u)
    \end{tikzpicture}
  \end{center}


% subsection methods_of_the_class_occs (end)
% section orthormal_cartesian_coordinate_system (end)
\endinput