[Tex/LaTex] Can one use tikz-3dplot with perspective

tikz-3dplot

Consider the following 3D surface graphs created using the tikz-3dplot package:

\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{caption}
\usepackage{subcaption}
\usepackage[outline]{contour}
\contourlength{0.05em}

\begin{document}

\begin{figure}[ht]
 \hfill
 \begin{subfigure}{.333333\linewidth}
  \centering
  \tdplotsetmaincoords{70}{-22.5}
  \begin{tikzpicture}[scale=2.8,tdplot_main_coords]
  \foreach \index in {1,...,9}
   {\draw[domain=0:1,smooth]
      plot
        (\x,\index/10,
         {\x*\index/10*(\x+\index/10-\x*\index/10)});
    \draw[domain=0:1,smooth]
      plot
        (\index/10,\x,
         {\x*\index/10*(\x+\index/10-\x*\index/10)});}
  \draw
    (0,1,0) -- (1,1,1) -- (1,0,0);
  \draw[dashed]
    (0,0,1) node[left] {$1$} --
    (1,0,1) -- (1,1,1) -- (0,1,1) -- cycle;
  \draw[dashed]
    (1,0,1) -- (1,0,0) node[below] {$1\mathstrut$};
  \draw[dashed]
    (0,1,1) -- (0,1,0) node[below] {$1\mathstrut$};
  \draw[thick,latex-latex]
    (0,1.2,0) node[below] {$v\mathstrut$} --
    (0,0,0)   node[below] {$O\mathstrut$} --
    (1.2,0,0) node[below] {$u\mathstrut$};
  \draw[thick,-latex]
    (0,0,0) --
    (0,0,1.2) node[above] {\contour{white}{$C_{-1}(u,v)$}};
  \end{tikzpicture}
  \caption{$C_{-1}(u,v)$.}
 \end{subfigure}%
 \hfill
 \begin{subfigure}{.333333\linewidth}
  \centering
  \tdplotsetmaincoords{70}{-22.5}
  \begin{tikzpicture}[scale=2.8,tdplot_main_coords]
  \foreach \index in {1,...,9}
   {\draw[domain=0:1,smooth]
      plot
        (\x,\index/10,
         {\x*\index/10*(2-\x-\index/10+\x*\index/10)});
    \draw[domain=0:1,smooth]
      plot
        (\index/10,\x,
         {\x*\index/10*(2-\x-\index/10+\x*\index/10)});}
  \draw
    (0,1,0) -- (1,1,1) -- (1,0,0);
  \draw[dashed]
    (0,0,1) node[left] {$1$} --
    (1,0,1) -- (1,1,1) -- (0,1,1) -- cycle;
  \draw[dashed]
    (1,0,1) -- (1,0,0) node[below] {$1\mathstrut$};
  \draw[dashed]
    (0,1,1) -- (0,1,0) node[below] {$1\mathstrut$};
  \draw[thick,latex-latex]
    (0,1.2,0) node[below] {$v\mathstrut$} --
    (0,0,0)   node[below] {$O\mathstrut$} --
    (1.2,0,0) node[below] {$u\mathstrut$};
  \draw[thick,-latex]
    (0,0,0) --
    (0,0,1.2) node[above] {\contour{white}{$C_1(u,v)$}};
  \end{tikzpicture}
  \caption{$C_1(u,v)$.}
 \end{subfigure}%
 \hfill\strut
\caption{Plots of the Farlie--Gumbel--Morgenstern copulae~$C_{-1}$ and~$C_1$.}
\end{figure}

\end{document}

Is there a method to create the above 3D graphs, with perspective? More precisely, can anyone create a perspective drawing with two vanishing points on the horizon, which automates all necessary computations?

This answer by @JanHlavacek attempted to draw a cube with perspective, but I doubt the application to my situation.

Best Answer

All the conceptual issues have been solved by Max in this fantastic answer, which I just copied in the updated answer. In my original answer, I used a more clumsy syntax, but this is no longer necessary since Max has fixed it.

UPDATED ANSWER: Max' coordinate parser works, according to what I find, just great. ~~The only exception is when the coordinates contain a newline, i.e. stretch over more than one line.~~ FIXED BY MAX It is just Max' stellar transformations + your code + tpp cs: added to all coordinates.

\documentclass{article}
% Max preamble
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepgfmodule{nonlineartransformations}
\usepackage{mathtools}
% Ruixi packages
\usepackage{caption}
\usepackage{subcaption}
\usepackage[outline]{contour}
\contourlength{0.05em}
% Max magic
\makeatletter 
% the first part is not in use here
\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}{
           \pgfmathsetmacro\tpp@x{#1} %<- Max' fix
            \pgfmathsetmacro\tpp@y{#2}
            \pgfmathsetmacro\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


\begin{document}
\begin{figure}[ht]
 \hfill
 \begin{subfigure}{.333333\linewidth}
  \centering
  \tdplotsetmaincoords{70}{-22.5}
  \begin{tikzpicture}[scale=pi,%tdplot_main_coords
  view={\tdplotmaintheta}{\tdplotmainphi},
            perspective={
                p = {(4,0,1.5)},
                q = {(0,4,1.5)},
            }
  ]
  \foreach \index in {1,...,9}
   {\draw[domain=0:1,smooth]
      plot
        (tpp cs:\x,\index/10,{\x*\index/10*(\x+\index/10-\x*\index/10)});
    \draw[domain=0:1,smooth]
      plot
        (tpp cs:\index/10,\x,{\x*\index/10*(\x+\index/10-\x*\index/10)});}
  \draw
    (tpp cs:0,1,0) -- (tpp cs:1,1,1) -- (tpp cs:1,0,0);
  \draw[dashed]
    (tpp cs:0,0,1) node[left] {$1$} --
    (tpp cs:1,0,1) -- (tpp cs:1,1,1) -- (tpp cs:0,1,1) -- cycle;
  \draw[dashed]
    (tpp cs:1,0,1) -- (tpp cs:1,0,0) node[below] {$1\mathstrut$};
  \draw[dashed]
    (tpp cs:0,1,1) -- (tpp cs:0,1,0) node[below] {$1\mathstrut$};
  \draw[thick,latex-latex]
    (tpp cs:0,1.2,0) node[below] {$v\mathstrut$} --
    (tpp cs:0,0,0)   node[below] {$O\mathstrut$} --
    (tpp cs:1.2,0,0) node[below] {$u\mathstrut$};
  \draw[thick,-latex]
    (tpp cs:0,0,0) --
    (tpp cs:0,0,1.2) node[above] {\contour{white}{$C_{-1}(u,v)$}};
  \end{tikzpicture}
  \caption{$C_{-1}(u,v)$.}
 \end{subfigure}%
 \hfill
 \begin{subfigure}{.333333\linewidth}
  \centering
  \tdplotsetmaincoords{70}{-22.5}
  \begin{tikzpicture}[scale=pi,%tdplot_main_coords]
  view={\tdplotmaintheta}{\tdplotmainphi},
            perspective={
                p = {(4,0,1.5)},
                q = {(0,4,1.5)},
            }
  ]
  \foreach \index in {1,...,9}
   {\draw[domain=0:1,smooth]
      plot
        (tpp cs:\x,\index/10,{\x*\index/10*(2-\x-\index/10+\x*\index/10)});
    \draw[domain=0:1,smooth]
      plot
        (tpp cs:\index/10,\x,{\x*\index/10*(2-\x-\index/10+\x*\index/10)});}
  \draw
    (tpp cs:0,1,0) -- (tpp cs:1,1,1) -- (tpp cs:1,0,0);
  \draw[dashed]
    (tpp cs:0,0,1) node[left] {$1$} --
    (tpp cs:1,0,1) -- (tpp cs:1,1,1) -- (tpp cs:0,1,1) -- cycle;
  \draw[dashed]
    (tpp cs:1,0,1) -- (tpp cs:1,0,0) node[below] {$1\mathstrut$};
  \draw[dashed]
    (tpp cs:0,1,1) -- (tpp cs:0,1,0) node[below] {$1\mathstrut$};
  \draw[thick,latex-latex]
    (tpp cs:0,1.2,0) node[below] {$v\mathstrut$} --
    (tpp cs:0,0,0)   node[below] {$O\mathstrut$} --
    (tpp cs:1.2,0,0) node[below] {$u\mathstrut$};
  \draw[thick,-latex]
    (tpp cs:0,0,0) --
    (tpp cs:0,0,1.2) node[above] {\contour{white}{$C_1(u,v)$}};
  \end{tikzpicture}
  \caption{$C_1(u,v)$.}
 \end{subfigure}%
 \hfill\strut
\caption{Plots of the Farlie--Gumbel--Morgenstern copulae~$C_{-1}$ and~$C_1$.}
\end{figure}
\end{document}

The perspective is defined in

perspective={
                p = {(4,0,1.5)},
                q = {(0,4,1.5)},
            }

which you may adjust to your needs.

Related Solutions

[Tex/LaTex] Drawing arcs in tikz-3dplot

Is this what you want?

If so, what's the problem using the angles library? It seems to work for me:

\documentclass[tikz,multi,border=10pt]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{arrows.meta,angles,quotes,calc}
\begin{document}
\tdplotsetmaincoords{60}{110}
\begin{tikzpicture}[>=latex,tdplot_main_coords,font=\footnotesize]%,scale=1.5]
  \coordinate (0) at (0,0,0);
  \coordinate (R) at (2.5,2.5,0);
  \coordinate (V) at (2.5,2.5,3);
  \path (V) -- (R) coordinate [pos=.05] (tl2);
  \draw[thick,->] (0,0,0) coordinate (O) -- (3,0,0) coordinate (X) node[anchor=north east]{$x$};
  \draw[thick,->] (0,0,0) -- (0,3,0) coordinate (Y) node[anchor=north west]{$y$};
  \draw[thick,->] (0,0,0) -- (0,0,3) coordinate (Z) node[anchor=south]{$z$};
  \draw[thick,->,color=red,opacity=0.5] (0,0,0) -- (2.5,2.5,3) coordinate (V) node[ right] {$P(\alpha,\beta,\gamma)$};
  \draw[dashed,magenta!50] (2.5,2.5,3) -- (2.5,2.5,0) -- (0,0,0);
  \draw[thick,blue!50] (2,0,0) -- (2.5,2.5,0);
  \draw[thick,blue!50] (2.5,2.5,0) -- (0,2,0);
  \draw[very thick,red!80!black,-{Triangle[fill=white]}] (0,0,0) -- (2,0,0)node [left=1ex] {$Q(\alpha,0,0)$} node[midway,left]{$\color{black}\alpha$};
  \draw[very thick,red!80!black,-{Triangle[fill=white]}] (0,0,0) -- (0,2,0)node [above] {$Q^{'}(0,\beta,0)$} node[midway,above]{$\color{black}\beta$};
  \draw[very thick,red!80!black,-{Triangle[fill=white]}] (0,0,0) -- (0,0,2)node [left] {$Q(0,0,\gamma)$};
  \draw[dashed,red!80!black,-{Triangle[fill=white]}] (0,0,0) -- (2.5,2.5,0)node  (b1)[right] {$R(\alpha,\beta,0)$};
  \draw[dashed,red!80!black,-{Triangle[fill=white]}] (2.5,2.5,3) -- (2.5,2.5,0)node [left,yshift=2.75cm]{$\color{black}\gamma$};
  \pic [blue, draw, thick, "$\theta$", angle radius=4mm] {angle = V--O--X };
  \pic [red, draw, thick, "$\phi$", angle eccentricity=1.3] {angle = Y--O--V };
  \pic [magenta, draw, thick, "$\psi$", angle radius=6mm, angle eccentricity=1.3] {angle = V--O--Z };
\end{tikzpicture}
\end{document}

[Tex/LaTex] set z-axis to point to the right with tikz-3dplot

I was finally able to do that with \tdplotsetrotatedcoords{-90}{-90}{0}. I also had to put \tdplotsetrotatedthetaplanecoords{\phivec} instead of \tdplotsetthetaplanecoords{\phivec}.

\documentclass{article} 

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

\begin{document}

\tdplotsetmaincoords{60}{110}
\tdplotsetrotatedcoords{-90}{-90}{0}
%
\begin{tikzpicture}[scale=5,tdplot_rotated_coords]
  \coordinate (O) at (0,0,0);
  \draw[thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
  \draw[thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
  \draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
  \def\rvec{1}
  \def\thetavec{45}
  \def\phivec{30}
  \tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec}
  \draw[-stealth,color=red] (O) -- (P) node[above right] {$P$};
  \draw[dashed, color=red] (O) -- (Pxy);
  \draw[dashed, color=red] (P) -- (Pxy);
%  \draw[dashed, color=red] (P) -- (Pyz);
%  \draw[dashed, color=red] (O) -- (Pxz);
%  \draw[dashed, color=red] (P) -- (Pxz);
  \tdplotdrawarc[tdplot_rotated_coords]{(O)}{0.2}{0}{\phivec}{anchor=north east}{$\phi$}
  \tdplotsetrotatedthetaplanecoords{\phivec}
  \tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{0.5}{0}{\thetavec}{anchor=south west}{$\theta$}
\end{tikzpicture}

\end{document}

Best Answer

Related Solutions

[Tex/LaTex] Drawing arcs in tikz-3dplot

[Tex/LaTex] set z-axis to point to the right with tikz-3dplot

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