Como fazer uma grade de superfície 2D no Tikz?

Question

Eu quero fazer algo assim:

E foi isso que fiz até agora:

\documentclass[a4paper, 11pt, oneside]{article}
\usepackage{tikz}
\usepackage{pstricks}
\usetikzlibrary{scopes}
\usetikzlibrary{backgrounds}

\begin{document}

 \begin{tikzpicture}[x=(10:4cm),y=(90:4cm),z=(225:4cm),>=latex]
 \coordinate (O) at (0,0,0); 
 \node[circle,color=white] at (0.15,0.1,0.05) {$P$};
 \draw [-stealth] (O) -- (1,0,0) node [at end, right] {$u_2$ axis};
 \draw [-stealth] (O) -- (0,1,0) node [at end, above] {$u_3$ axis};
 \draw [-stealth] (O) -- (0,0,1) node [at end, left]  {$u_1$ axis};

 \draw [draw=black, >=latex, thick] (O) to [bend left=8] 
   coordinate [pos=7/8] (q2n) 
   (1,-1/4,0) coordinate (q2) node [right] {$u_2$};
 \draw [draw=black, >=latex, thick] (O) to [bend right=8] 
   coordinate [pos=7/8] (q3n) 
   (0,1,1/2) coordinate (q3) node [left] {$u_3$};
 \draw [draw=black, >=latex, thick] (O) to [bend right=8] 
   coordinate [pos=7/8] (q1n) 
   (1/4,0,1) coordinate (q1) node [right] {$u_1$};

 \begin{pgfonlayer}{background}
 \begin{scope}
 \clip (O) to [bend left=8] (q2) -- (1,1,0) -- (q3n) to [bend right=8] (O);
 \shade [left color=black!60, right color=black!60!white, shading angle=135]
   (O) to [bend left] (q3n) to [bend left=16] (3/4,1/2,0) to [bend left=16 (q2n) -- cycle;
 \end{scope}

 \begin{scope}
 \clip (O) to [bend left=8] (q2) -- (1,0,1) -- (q1) to [bend left=8] (O);
 \shade [left color=black!40, right color=black!40!white, shading angle=45]
   (O) to [bend right] (q1n) to [bend left=16] (1,0,1) to [bend left=16] 
   (q2n) to [bend right] (O);
 \end{scope}

 \begin{scope}
 \clip (O) to [bend right=8] (q1) -- (0,1,1) -- (q3) to [bend left=8] (O);
 \shade [left color=black!20, right color=black!20!white, shading angle=225] 
   (O) -- (q1n) to [bend right=16] (0,1,1) to [bend left=16] (q3n) 
 to [bend left] (O);
 \end{scope}
 \end{pgfonlayer}

 \node[color=white] at (1/3,1/3,0) {$u_1=c_1$};
 \node at (0,1/2,1/2) {$u_2=c_2$};
 \node at (1/2,0,1/3) {$u_3=c_3$};
 \end{tikzpicture}

\end{document}

Answer 1

Na segunda parte aesta respostaum sistema de coordenadas personalizado foi fornecido. Isto pode ser usado para traçar as grades (embora um pouco lentamente). Os outros requisitos (não feitos aqui) envolvem a reorientação dos vetores x, ye ze a alteração do conteúdo e posicionamento dos rótulos.

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{arrows}
\tikzset{declare function={f(\x)=sin(\x*100)/10;},
  non-linear cs/.cd,
    x/.store in=\nlx,y/.store in=\nly,z/.store in=\nlz,
    x=0,y=0,z=0}
\tikzdeclarecoordinatesystem{non-linear}{%
  \tikzset{non-linear cs/.cd,#1}%
  \pgfpointxyz{(\nlx)-f(\nly)*3+f(\nlz)}%
    {-f(\nlx)*2+(\nly)-f(\nlz)}{-f(\nlx)-f(\nly)+(\nlz)}}
\begin{document}
\begin{tikzpicture}[x=(10:3cm),y=(90:3cm),z=(215:2.75cm),
  >=latex,
  domain=0:1, samples=50, variable=\t]
\coordinate (O) at (0,0,0); 

\fill [black!10]
  (O) \foreach \x/\y in{\t/0,1/\t,1-\t/1,0/1-\t}{
     -- plot [smooth] (non-linear cs:x=\x, y=\y) };
\fill [black!30]  
  (O) \foreach \x/\z in{\t/0,1/\t,1-\t/1,0/1-\t}{
     -- plot [smooth] (non-linear cs:x=\x, z=\z) };
\fill [black!50]
  (O) \foreach \y/\z in{\t/0,1/\t,1-\t/1,0/1-\t}{
     -- plot [smooth] (non-linear cs:y=\y, z=\z) };

\foreach \i in {0.25,0.5,0.75}
  \draw [domain=0:1.1875, smooth]
  plot (non-linear cs:x=\i, y=\t)
  plot (non-linear cs:x=\t, y=\i)
  plot (non-linear cs:y=\i, z=\t)
  plot (non-linear cs:y=\t, z=\i)
  plot (non-linear cs:x=\i, z=\t)  
  plot (non-linear cs:x=\t, z=\i);

\draw [thick] (O) -- plot [domain=0:1.125, smooth] 
   (non-linear cs:x=\t) node [right] {$u_2$};
 \draw [thick] (O) -- plot [domain=0:1.125, smooth] 
   (non-linear cs:y=\t) node [above] {$u_3$};
 \draw [thick] (O) -- plot [domain=0:1.125, smooth] 
   (non-linear cs:z=\t) node [below] {$u_1$};

\draw [very thick, ->] (O) -- (3/2,0,0) node [at end, right] {$u_2$ axis};
\draw [very thick, ->] (O) -- (0,3/2,0) node [at end, above] {$u_3$ axis};
\draw [very thick, ->] (O) -- (0,0,3/2) node [at end, left]  {$u_1$ axis};

\node [inner sep=0, fill=black!10] at (1/2,1/2,0) {$u_1=\mbox{const}$};
\node [inner sep=0, fill=black!50] at (0,1/2,2/3) {$u_2=\mbox{const}$};
\node [inner sep=0, fill=black!30] at (3/4,0,3/4) {$u_3=\mbox{const}$};
\end{tikzpicture}
\end{document}