[Tex/LaTex] Aligning a text in 3d diagram

graphicsinkscapeluatextikz-pgf

This question led to a new library in TikZ:
perspective (PGF manual, chapter 63)

I have a 3d object created with inkscape as shown below:

Now I am trying to place a text in the same orientation as the light-blue colored area. However, I want to insert the text with latex instead of directly embedding it in the pdf (hence, I save it with pdf+Latex option in inkscape).

But when I try to insert the text in the 3d area, I get:

How can I get OMEAG aligned parallely to the light-blue region.

I could not attach the pdf of it, however the MWE is below:

%&lualatex
% !TeX program = lualatex
\documentclass[11pt,a4paper]{article}
%\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{pgfplots}
\usepackage{pstricks}    %for embedding pspicture.
\pgfplotsset{compat=newest}
\usepackage{tikz}
\begin{document}
    \begin{figure}[h]
    \centering{
        \input{drawing4.pdf_tex}
        \caption{Top view.}
        \label{fig:aktomnpView}
    }
\end{figure}
\end{document}

and \input{drawing4.pdf_tex} is below:

\begingroup%
  \makeatletter%
  \providecommand\color[2][]{%
    \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
    \renewcommand\color[2][]{}%
  }%
  \providecommand\transparent[1]{%
    \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
    \renewcommand\transparent[1]{}%
  }%
  \providecommand\rotatebox[2]{#2}%
  \newcommand*\fsize{\dimexpr\f@size pt\relax}%
  \newcommand*\lineheight[1]{\fontsize{\fsize}{#1\fsize}\selectfont}%
  \ifx\svgwidth\undefined%
    \setlength{\unitlength}{303.69978591bp}%
    \ifx\svgscale\undefined%
      \relax%
    \else%
      \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
    \fi%
  \else%
    \setlength{\unitlength}{\svgwidth}%
  \fi%
  \global\let\svgwidth\undefined%
  \global\let\svgscale\undefined%
  \makeatother%
  \begin{picture}(1,1.35089637)%
    \lineheight{1}%
    \setlength\tabcolsep{0pt}%
    \put(0,0){\includegraphics[width=\unitlength,page=1]{drawing4.pdf}}%
    \put(0.54334545,0.71256022){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}OMEAG\end{tabular}}}}%
  \end{picture}%
\endgroup%

Best Answer

Edit 3

I'm happy to announce that some of the code of this answer is now included in the Tikz package (v3.1.2) as the perspective library.

Edit 2

Using this awesome answer in combination with my tpp coordinate system, I managed to get an approximation of a nonlinear mapping to the side of the block.

Or without help lines:

Stationary image:

MWE:

\documentclass[tikz]{standalone}

\usepackage{tikz}
\usepackage{tikz-3dplot}

\usepgfmodule{nonlineartransformations}

\usepackage{mathtools}

\makeatletter

\def\tikz@scan@transform@one@point#1{%
  \tikz@scan@one@point\pgf@process#1%
  \pgf@pos@transform{\pgf@x}{\pgf@y}}
\tikzset{%
  grid source opposite corners/.code args={#1and#2}{%
   \pgfextract@process\tikz@transform@source@southwest{%
     \tikz@scan@transform@one@point{#1}}%
   \pgfextract@process\tikz@transform@source@northeast{%
     \tikz@scan@transform@one@point{#2}}%
  },
  grid target corners/.code args={#1--#2--#3--#4}{%
   \pgfextract@process\tikz@transform@target@southwest{%
     \tikz@scan@transform@one@point{#1}}%
   \pgfextract@process\tikz@transform@target@southeast{%
     \tikz@scan@transform@one@point{#2}}%
   \pgfextract@process\tikz@transform@target@northeast{%
     \tikz@scan@transform@one@point{#3}}%
   \pgfextract@process\tikz@transform@target@northwest{%
     \tikz@scan@transform@one@point{#4}}%
  }
}

\def\tikzgridtransform{%
  \pgfextract@process\tikz@current@point{}%
  \pgf@process{%
    \pgfpointdiff{\tikz@transform@source@southwest}%
      {\tikz@transform@source@northeast}%
  }%
  \pgf@xc=\pgf@x\pgf@yc=\pgf@y%
  \pgf@process{%
    \pgfpointdiff{\tikz@transform@source@southwest}{\tikz@current@point}%
  }%
  \pgfmathparse{\pgf@x/\pgf@xc}\let\tikz@tx=\pgfmathresult%
  \pgfmathparse{\pgf@y/\pgf@yc}\let\tikz@ty=\pgfmathresult%
  %
  \pgfpointlineattime{\tikz@ty}{%
    \pgfpointlineattime{\tikz@tx}{\tikz@transform@target@southwest}%
      {\tikz@transform@target@southeast}}{%
    \pgfpointlineattime{\tikz@tx}{\tikz@transform@target@northwest}%
      {\tikz@transform@target@northeast}}%
}

% Initialize H matrix for perspective view
\pgfmathsetmacro\H@tpp@aa{1}\pgfmathsetmacro\H@tpp@ab{0}\pgfmathsetmacro\H@tpp@ac{0}%\pgfmathsetmacro\H@tpp@ad{0}
\pgfmathsetmacro\H@tpp@ba{0}\pgfmathsetmacro\H@tpp@bb{1}\pgfmathsetmacro\H@tpp@bc{0}%\pgfmathsetmacro\H@tpp@bd{0}
\pgfmathsetmacro\H@tpp@ca{0}\pgfmathsetmacro\H@tpp@cb{0}\pgfmathsetmacro\H@tpp@cc{1}%\pgfmathsetmacro\H@tpp@cd{0}
\pgfmathsetmacro\H@tpp@da{0}\pgfmathsetmacro\H@tpp@db{0}\pgfmathsetmacro\H@tpp@dc{0}%\pgfmathsetmacro\H@tpp@dd{1}

%Initialize H matrix for main rotation
\pgfmathsetmacro\H@rot@aa{1}\pgfmathsetmacro\H@rot@ab{0}\pgfmathsetmacro\H@rot@ac{0}%\pgfmathsetmacro\H@rot@ad{0}
\pgfmathsetmacro\H@rot@ba{0}\pgfmathsetmacro\H@rot@bb{1}\pgfmathsetmacro\H@rot@bc{0}%\pgfmathsetmacro\H@rot@bd{0}
\pgfmathsetmacro\H@rot@ca{0}\pgfmathsetmacro\H@rot@cb{0}\pgfmathsetmacro\H@rot@cc{1}%\pgfmathsetmacro\H@rot@cd{0}
%\pgfmathsetmacro\H@rot@da{0}\pgfmathsetmacro\H@rot@db{0}\pgfmathsetmacro\H@rot@dc{0}\pgfmathsetmacro\H@rot@dd{1}

\pgfkeys{
    /three point perspective/.cd,
        p/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#1))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ba{#2/#1}
                \pgfmathsetmacro\H@tpp@ca{#3/#1}
                \pgfmathsetmacro\H@tpp@da{ 1/#1}
                \coordinate (vp-p) at (#1,#2,#3);
            \fi
        },
        q/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#2))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ab{#1/#2}
                \pgfmathsetmacro\H@tpp@cb{#3/#2}
                \pgfmathsetmacro\H@tpp@db{ 1/#2}
                \coordinate (vp-q) at (#1,#2,#3);
            \fi
        },
        r/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#3))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ac{#1/#3}
                \pgfmathsetmacro\H@tpp@bc{#2/#3}
                \pgfmathsetmacro\H@tpp@dc{ 1/#3}
                \coordinate (vp-r) at (#1,#2,#3);
            \fi
        },
        coordinate/.code args={#1,#2,#3}{
            \def\tpp@x{#1}
            \def\tpp@y{#2}
            \def\tpp@z{#3}
        },
}

\tikzset{
    view/.code 2 args={
        \pgfmathsetmacro\rot@main@theta{#1}
        \pgfmathsetmacro\rot@main@phi{#2}
        % Row 1
        \pgfmathsetmacro\H@rot@aa{cos(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ab{sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ac{0}
        % Row 2
        \pgfmathsetmacro\H@rot@ba{-cos(\rot@main@theta)*sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@bb{cos(\rot@main@phi)*cos(\rot@main@theta)}
        \pgfmathsetmacro\H@rot@bc{sin(\rot@main@theta)}
        % Row 3
        \pgfmathsetmacro\H@m@ca{sin(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cb{-cos(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cc{cos(\rot@main@theta)}
        % Set vector values
        \pgfmathsetmacro\vec@x@x{\H@rot@aa}
        \pgfmathsetmacro\vec@y@x{\H@rot@ab}
        \pgfmathsetmacro\vec@z@x{\H@rot@ac}
        \pgfmathsetmacro\vec@x@y{\H@rot@ba}
        \pgfmathsetmacro\vec@y@y{\H@rot@bb}
        \pgfmathsetmacro\vec@z@y{\H@rot@bc}
        % Set pgf vectors
        \pgfsetxvec{\pgfpoint{\vec@x@x cm}{\vec@x@y cm}}
        \pgfsetyvec{\pgfpoint{\vec@y@x cm}{\vec@y@y cm}}
        \pgfsetzvec{\pgfpoint{\vec@z@x cm}{\vec@z@y cm}}
    },
}

\tikzset{
    perspective/.code={\pgfkeys{/three point perspective/.cd,#1}},
    perspective/.default={p={(15,0,0)},q={(0,15,0)},r={(0,0,50)}},
}

\tikzdeclarecoordinatesystem{three point perspective}{
    \pgfkeys{/three point perspective/.cd,coordinate={#1}}
    \pgfmathsetmacro\temp@p@w{\H@tpp@da*\tpp@x + \H@tpp@db*\tpp@y + \H@tpp@dc*\tpp@z + 1}
    \pgfmathsetmacro\temp@p@x{(\H@tpp@aa*\tpp@x + \H@tpp@ab*\tpp@y + \H@tpp@ac*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@y{(\H@tpp@ba*\tpp@x + \H@tpp@bb*\tpp@y + \H@tpp@bc*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@z{(\H@tpp@ca*\tpp@x + \H@tpp@cb*\tpp@y + \H@tpp@cc*\tpp@z)/\temp@p@w}
    \pgfpointxyz{\temp@p@x}{\temp@p@y}{\temp@p@z}
}
\tikzaliascoordinatesystem{tpp}{three point perspective}

\makeatother

\definecolor{mydarkbluishgray}{RGB}{134 134 191}
\definecolor{mylightbluishgray}{RGB}{215 215 255}

\begin{document}
    \foreach \vp in {10,12.5,...,50}{
%    \foreach \vp in {10}{
    \begin{tikzpicture}[line join=round]

        \clip (-12,0) rectangle (12,10);

        \begin{scope}[
            view={85}{-40},
            perspective={
                p = {(\vp,0,5.5)},
                q = {(0,\vp,5.5)},
            }
        ]
            \fill[mydarkbluishgray]  (tpp cs:0,0,0) -- (tpp cs:0,0,10) -- (tpp cs:0,5,10) -- (tpp cs:0,5,0) -- cycle;
            \fill[mylightbluishgray] (tpp cs:0,0,0) -- (tpp cs:0,0,10) -- (tpp cs:15,0,10) -- (tpp cs:15,0,0) -- cycle;

            \begin{scope}[
                grid source opposite corners={(0cm,0cm) and (15cm,10cm)},
                grid target corners={(tpp cs:0,0,0)--(tpp cs:15,0,0)--(tpp cs:15,0,10)--(tpp cs:0,0,10)}
            ]
                \pgftransformnonlinear\tikzgridtransform

%                \draw[dotted] (0cm,4.75cm) -- (15cm,4.75cm);
%                \draw[red]    (0cm,5.00cm) -- (15cm,5.00cm);
%                \draw[dotted] (0cm,5.25cm) -- (15cm,5.25cm);
%                
%                \draw[dotted] (6.1cm,0cm) -- (6.1cm,10cm);
%                \draw[red]    (7.5cm,0cm) -- (7.5cm,10cm);
%                \draw[dotted] (8.9cm,0cm) -- (8.9cm,10cm);

                \foreach \char [count=\i from -2] in {O,M,E,A,G}{
                    \pgftransformshift{\pgfpointadd{\pgfpoint{7.5cm}{5cm}}{\pgfpoint{\i *0.6cm}{0cm}}}
                    \pgftransformscale{2}
                    \pgfnode{rectangle}{center}{\char}{}{}
                }
            \end{scope}

%            \begin{scope}[dotted,line width=0.2pt]
%                \node[label=right:p,fill,circle,inner sep = 2pt] (p) at (vp-p){};
%
%                \draw (tpp cs:0,0,10) -- (p.center);
%                \draw (tpp cs:0,0,0) -- (p.center);
%                \draw (tpp cs:0,5,10) -- (p.center);
%                \draw (tpp cs:0,5,0) -- (p.center);
%
%                \node[label=left:q,fill,circle,inner sep = 2pt] (q) at (vp-q){};
%
%                \draw (tpp cs:0,0,10) -- (q.center);
%                \draw (tpp cs:0,0,0) -- (q.center);
%                \draw (tpp cs:15,0,10) -- (q.center);
%                \draw (tpp cs:15,0,0) -- (q.center);
%            \end{scope}
        \end{scope}
    \end{tikzpicture}
    }
\end{document}

Edit _{^{See previous edits for my former answer}}

I defined a new coordinate system three point perspective. It can be used as

\draw (three point perspective cs:0,0,0) -- (three point perspective cs:5,5,5);

Or slightly more convenient

\draw (tpp cs:0,0,0) -- (tpp cs:5,5,5);

To turn perspective view on, you can call the perspective={<options>} Tikz-key. The options are:

p={(p_x,p_y,p_z)} to set the vanishing point in x direction, to turn of set p_x to 0.
q={(q_x,q_y,q_z)} to set the vanishing point in y direction, to turn of set q_y to 0.
r={(r_x,r_y,r_z)} to set the vanishing point in z direction, to turn of set r_z to 0.

The default perspective is set to p={(15,0,0)},q={(0,15,0)},r={(0,0,50)}.

To change the viewing angle, I also added a view={<rotate about x>}{<rotate about z>} key. The latter ensures that I don't need the tikz-3dplot any longer.

The result is similar, but is easier to use.

\documentclass[tikz]{standalone}

\usepackage{tikz}
\usepackage{tikz-3dplot}

\usepackage{mathtools}

\makeatletter

% Initialize H matrix for perspective view
\pgfmathsetmacro\H@tpp@aa{1}\pgfmathsetmacro\H@tpp@ab{0}\pgfmathsetmacro\H@tpp@ac{0}%\pgfmathsetmacro\H@tpp@ad{0}
\pgfmathsetmacro\H@tpp@ba{0}\pgfmathsetmacro\H@tpp@bb{1}\pgfmathsetmacro\H@tpp@bc{0}%\pgfmathsetmacro\H@tpp@bd{0}
\pgfmathsetmacro\H@tpp@ca{0}\pgfmathsetmacro\H@tpp@cb{0}\pgfmathsetmacro\H@tpp@cc{1}%\pgfmathsetmacro\H@tpp@cd{0}
\pgfmathsetmacro\H@tpp@da{0}\pgfmathsetmacro\H@tpp@db{0}\pgfmathsetmacro\H@tpp@dc{0}%\pgfmathsetmacro\H@tpp@dd{1}

%Initialize H matrix for main rotation
\pgfmathsetmacro\H@rot@aa{1}\pgfmathsetmacro\H@rot@ab{0}\pgfmathsetmacro\H@rot@ac{0}%\pgfmathsetmacro\H@rot@ad{0}
\pgfmathsetmacro\H@rot@ba{0}\pgfmathsetmacro\H@rot@bb{1}\pgfmathsetmacro\H@rot@bc{0}%\pgfmathsetmacro\H@rot@bd{0}
\pgfmathsetmacro\H@rot@ca{0}\pgfmathsetmacro\H@rot@cb{0}\pgfmathsetmacro\H@rot@cc{1}%\pgfmathsetmacro\H@rot@cd{0}
%\pgfmathsetmacro\H@rot@da{0}\pgfmathsetmacro\H@rot@db{0}\pgfmathsetmacro\H@rot@dc{0}\pgfmathsetmacro\H@rot@dd{1}

\pgfkeys{
    /three point perspective/.cd,
        p/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#1))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ba{#2/#1}
                \pgfmathsetmacro\H@tpp@ca{#3/#1}
                \pgfmathsetmacro\H@tpp@da{ 1/#1}
                \coordinate (vp-p) at (#1,#2,#3);
            \fi
        },
        q/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#2))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ab{#1/#2}
                \pgfmathsetmacro\H@tpp@cb{#3/#2}
                \pgfmathsetmacro\H@tpp@db{ 1/#2}
                \coordinate (vp-q) at (#1,#2,#3);
            \fi
        },
        r/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#3))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ac{#1/#3}
                \pgfmathsetmacro\H@tpp@bc{#2/#3}
                \pgfmathsetmacro\H@tpp@dc{ 1/#3}
                \coordinate (vp-r) at (#1,#2,#3);
            \fi
        },
        coordinate/.code args={#1,#2,#3}{
            \def\tpp@x{#1}
            \def\tpp@y{#2}
            \def\tpp@z{#3}
        },
}

\tikzset{
    view/.code 2 args={
        \pgfmathsetmacro\rot@main@theta{#1}
        \pgfmathsetmacro\rot@main@phi{#2}
        % Row 1
        \pgfmathsetmacro\H@rot@aa{cos(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ab{sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ac{0}
        % Row 2
        \pgfmathsetmacro\H@rot@ba{-cos(\rot@main@theta)*sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@bb{cos(\rot@main@phi)*cos(\rot@main@theta)}
        \pgfmathsetmacro\H@rot@bc{sin(\rot@main@theta)}
        % Row 3
        \pgfmathsetmacro\H@m@ca{sin(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cb{-cos(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cc{cos(\rot@main@theta)}

        \pgfmathsetmacro\vec@x@x{\H@rot@aa}
        \pgfmathsetmacro\vec@y@x{\H@rot@ab}
        \pgfmathsetmacro\vec@z@x{\H@rot@ac}
        \pgfmathsetmacro\vec@x@y{\H@rot@ba}
        \pgfmathsetmacro\vec@y@y{\H@rot@bb}
        \pgfmathsetmacro\vec@z@y{\H@rot@bc}

        \pgfsetxvec{\pgfpoint{\vec@x@x cm}{\vec@x@y cm}}
        \pgfsetyvec{\pgfpoint{\vec@y@x cm}{\vec@y@y cm}}
        \pgfsetzvec{\pgfpoint{\vec@z@x cm}{\vec@z@y cm}}
    },
}

\tikzset{
    perspective/.code={\pgfkeys{/three point perspective/.cd,#1}},
    perspective/.default={p={(15,0,0)},q={(0,15,0)},r={(0,0,50)}},
}

\tikzdeclarecoordinatesystem{three point perspective}{
    \pgfkeys{/three point perspective/.cd,coordinate={#1}}
    \pgfmathsetmacro\temp@p@w{\H@tpp@da*\tpp@x + \H@tpp@db*\tpp@y + \H@tpp@dc*\tpp@z + 1}
    \pgfmathsetmacro\temp@p@x{(\H@tpp@aa*\tpp@x + \H@tpp@ab*\tpp@y + \H@tpp@ac*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@y{(\H@tpp@ba*\tpp@x + \H@tpp@bb*\tpp@y + \H@tpp@bc*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@z{(\H@tpp@ca*\tpp@x + \H@tpp@cb*\tpp@y + \H@tpp@cc*\tpp@z)/\temp@p@w}
    \pgfpointxyz{\temp@p@x}{\temp@p@y}{\temp@p@z}
}
\tikzaliascoordinatesystem{tpp}{three point perspective}

\makeatother

\definecolor{mydarkbluishgray}{RGB}{134 134 191}
\definecolor{mylightbluishgray}{RGB}{215 215 255}

\begin{document}
    \begin{tikzpicture}[line join=round]

        \begin{scope}[
            scale=10,
            view={85}{-40},
            perspective={
                p = {(1,0,0.55)},
                q = {(0,1,0.55)},
            }
        ]
            \fill[mydarkbluishgray]  (tpp cs:0,0,0) -- (tpp cs:0,0,1) -- (tpp cs:0,0.5,1) -- (tpp cs:0,0.5,0) -- cycle;
            \fill[mylightbluishgray] (tpp cs:0,0,0) -- (tpp cs:0,0,1) -- (tpp cs:1.5,0,1) -- (tpp cs:1.5,0,0) -- cycle;

            \path (tpp cs:0,0,0.5) -- node[sloped]{OMEAG} (tpp cs:1.5,0,0.5);

            \begin{scope}[dotted,line width=0.2pt]
                \node[label=right:p,fill,circle,inner sep = 2pt] (p) at (vp-p){};

                \draw (tpp cs:0,0,1) -- (p.center);
                \draw (tpp cs:0,0,0) -- (p.center);
                \draw (tpp cs:0,0.5,1) -- (p.center);
                \draw (tpp cs:0,0.5,0) -- (p.center);

                \node[label=left:q,fill,circle,inner sep = 2pt] (q) at (vp-q){};

                \draw (tpp cs:0,0,1) -- (q.center);
                \draw (tpp cs:0,0,0) -- (q.center);
                \draw (tpp cs:1.5,0,1) -- (q.center);
                \draw (tpp cs:1.5,0,0) -- (q.center);
            \end{scope}
        \end{scope}
    \end{tikzpicture}
\end{document}

Of course I had to make an animation to show different vanishing point distances:

MWE animation:

\documentclass[tikz]{standalone}

\usepackage{tikz}
\usepackage{tikz-3dplot}

\usepackage{mathtools}

\makeatletter

% Initialize H matrix for perspective view
\pgfmathsetmacro\H@tpp@aa{1}\pgfmathsetmacro\H@tpp@ab{0}\pgfmathsetmacro\H@tpp@ac{0}%\pgfmathsetmacro\H@tpp@ad{0}
\pgfmathsetmacro\H@tpp@ba{0}\pgfmathsetmacro\H@tpp@bb{1}\pgfmathsetmacro\H@tpp@bc{0}%\pgfmathsetmacro\H@tpp@bd{0}
\pgfmathsetmacro\H@tpp@ca{0}\pgfmathsetmacro\H@tpp@cb{0}\pgfmathsetmacro\H@tpp@cc{1}%\pgfmathsetmacro\H@tpp@cd{0}
\pgfmathsetmacro\H@tpp@da{0}\pgfmathsetmacro\H@tpp@db{0}\pgfmathsetmacro\H@tpp@dc{0}%\pgfmathsetmacro\H@tpp@dd{1}

%Initialize H matrix for main rotation
\pgfmathsetmacro\H@rot@aa{1}\pgfmathsetmacro\H@rot@ab{0}\pgfmathsetmacro\H@rot@ac{0}%\pgfmathsetmacro\H@rot@ad{0}
\pgfmathsetmacro\H@rot@ba{0}\pgfmathsetmacro\H@rot@bb{1}\pgfmathsetmacro\H@rot@bc{0}%\pgfmathsetmacro\H@rot@bd{0}
\pgfmathsetmacro\H@rot@ca{0}\pgfmathsetmacro\H@rot@cb{0}\pgfmathsetmacro\H@rot@cc{1}%\pgfmathsetmacro\H@rot@cd{0}
%\pgfmathsetmacro\H@rot@da{0}\pgfmathsetmacro\H@rot@db{0}\pgfmathsetmacro\H@rot@dc{0}\pgfmathsetmacro\H@rot@dd{1}

\pgfkeys{
    /three point perspective/.cd,
        p/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#1))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ba{#2/#1}
                \pgfmathsetmacro\H@tpp@ca{#3/#1}
                \pgfmathsetmacro\H@tpp@da{ 1/#1}
                \coordinate (vp-p) at (#1,#2,#3);
            \fi
        },
        q/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#2))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ab{#1/#2}
                \pgfmathsetmacro\H@tpp@cb{#3/#2}
                \pgfmathsetmacro\H@tpp@db{ 1/#2}
                \coordinate (vp-q) at (#1,#2,#3);
            \fi
        },
        r/.code args={(#1,#2,#3)}{
            \pgfmathparse{int(round(#3))}
            \ifnum\pgfmathresult=0\else
                \pgfmathsetmacro\H@tpp@ac{#1/#3}
                \pgfmathsetmacro\H@tpp@bc{#2/#3}
                \pgfmathsetmacro\H@tpp@dc{ 1/#3}
                \coordinate (vp-r) at (#1,#2,#3);
            \fi
        },
        coordinate/.code args={#1,#2,#3}{
            \def\tpp@x{#1}
            \def\tpp@y{#2}
            \def\tpp@z{#3}
        },
}

\tikzset{
    view/.code 2 args={
        \pgfmathsetmacro\rot@main@theta{#1}
        \pgfmathsetmacro\rot@main@phi{#2}
        % Row 1
        \pgfmathsetmacro\H@rot@aa{cos(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ab{sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@ac{0}
        % Row 2
        \pgfmathsetmacro\H@rot@ba{-cos(\rot@main@theta)*sin(\rot@main@phi)}
        \pgfmathsetmacro\H@rot@bb{cos(\rot@main@phi)*cos(\rot@main@theta)}
        \pgfmathsetmacro\H@rot@bc{sin(\rot@main@theta)}
        % Row 3
        \pgfmathsetmacro\H@m@ca{sin(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cb{-cos(\rot@main@phi)*sin(\rot@main@theta)}
        \pgfmathsetmacro\H@m@cc{cos(\rot@main@theta)}
        % Set vector values
        \pgfmathsetmacro\vec@x@x{\H@rot@aa}
        \pgfmathsetmacro\vec@y@x{\H@rot@ab}
        \pgfmathsetmacro\vec@z@x{\H@rot@ac}
        \pgfmathsetmacro\vec@x@y{\H@rot@ba}
        \pgfmathsetmacro\vec@y@y{\H@rot@bb}
        \pgfmathsetmacro\vec@z@y{\H@rot@bc}
        % Set pgf vectors
        \pgfsetxvec{\pgfpoint{\vec@x@x cm}{\vec@x@y cm}}
        \pgfsetyvec{\pgfpoint{\vec@y@x cm}{\vec@y@y cm}}
        \pgfsetzvec{\pgfpoint{\vec@z@x cm}{\vec@z@y cm}}
    },
}

\tikzset{
    perspective/.code={\pgfkeys{/three point perspective/.cd,#1}},
    perspective/.default={p={(15,0,0)},q={(0,15,0)},r={(0,0,50)}},
}

\tikzdeclarecoordinatesystem{three point perspective}{
    \pgfkeys{/three point perspective/.cd,coordinate={#1}}
    \pgfmathsetmacro\temp@p@w{\H@tpp@da*\tpp@x + \H@tpp@db*\tpp@y + \H@tpp@dc*\tpp@z + 1}
    \pgfmathsetmacro\temp@p@x{(\H@tpp@aa*\tpp@x + \H@tpp@ab*\tpp@y + \H@tpp@ac*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@y{(\H@tpp@ba*\tpp@x + \H@tpp@bb*\tpp@y + \H@tpp@bc*\tpp@z)/\temp@p@w}
    \pgfmathsetmacro\temp@p@z{(\H@tpp@ca*\tpp@x + \H@tpp@cb*\tpp@y + \H@tpp@cc*\tpp@z)/\temp@p@w}
    \pgfpointxyz{\temp@p@x}{\temp@p@y}{\temp@p@z}
}
\tikzaliascoordinatesystem{tpp}{three point perspective}

\makeatother

\definecolor{mydarkbluishgray}{RGB}{134 134 191}
\definecolor{mylightbluishgray}{RGB}{215 215 255}

\begin{document}
    \foreach \vp in {1,1.1,...,5}{
%    \foreach \vp in {1}{
    \begin{tikzpicture}[line join=round]

        \clip (-12,0) rectangle (12,10);

        \begin{scope}[
            scale=10,
            view={85}{-40},
            perspective={
                p = {(\vp,0,0.55)},
                q = {(0,\vp,0.55)},
            }
        ]
            \fill[mydarkbluishgray]  (tpp cs:0,0,0) -- (tpp cs:0,0,1) -- (tpp cs:0,0.5,1) -- (tpp cs:0,0.5,0) -- cycle;
            \fill[mylightbluishgray] (tpp cs:0,0,0) -- (tpp cs:0,0,1) -- (tpp cs:1.5,0,1) -- (tpp cs:1.5,0,0) -- cycle;

            \path[dotted] (tpp cs:0,0,0.5) -- node[sloped]{OMEAG} (tpp cs:1.5,0,0.5);

            \begin{scope}[dotted,line width=0.2pt]
                \node[label=right:p,fill,circle,inner sep = 2pt] (p) at (vp-p){};

                \draw (tpp cs:0,0,1) -- (p.center);
                \draw (tpp cs:0,0,0) -- (p.center);
                \draw (tpp cs:0,0.5,1) -- (p.center);
                \draw (tpp cs:0,0.5,0) -- (p.center);

                \node[label=left:q,fill,circle,inner sep = 2pt] (q) at (vp-q){};

                \draw (tpp cs:0,0,1) -- (q.center);
                \draw (tpp cs:0,0,0) -- (q.center);
                \draw (tpp cs:1.5,0,1) -- (q.center);
                \draw (tpp cs:1.5,0,0) -- (q.center);
            \end{scope}
        \end{scope}
    \end{tikzpicture}
    }
\end{document}

Appendix: Two point perspective theory

A perspective transformation with two vanishing points can be described with a four by four transformation matrix H which is a function of the two vanishing points p with p_x <> 0, and r with r_y <> 0.

You can build H as follows

To be able to transform a point x expressed in 3D with H, it must be expressed in a projected space, which can be written as

Any multiplication (elongation of the 4D vector) of a vector in projected space with a non-zero scalar alpha, still maps to the same point in 3D space. E.g., the following two points map to the same point in 3D space:

So to get the three coordinates of the 3D point, you can divide all entries by the fourth entry (we need this after multiplication with a transformation matrix as H).

Best Answer

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Edit 2

Edit See previous edits for my former answer

Appendix: Two point perspective theory

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