Velocity Field / 3D Vector Fields in Tikz or PGFplots

Question

I would like to draw "Velocity field" or "Vector Field" like:

I have tried using PGFplots "quiver" but this only works for vector fields on SURFACES:

\begin{tikzpicture}
\begin{axis}[
domain=0:1,
xmax=1,
ymax=1,
]
\addplot3[cyan,/pgfplots/quiver,
quiver/u=y,
quiver/v=z,
quiver/w=x,
quiver/scale arrows=0.1,
-stealth,samples=10] ({x},{y},{x+y});
\end{axis}
\end{tikzpicture}

Is there a mechanism for drawing vectors on a 3D LATTICE? Something which does

for i from 1 to 10
   for j from 1 to 10
      for k from 1 to 10
            draw vector (i,j,k) -- f(i,j,k);
      end do;
   end do;
end do;

in PGFplots or Tikz? (As done here: 3D Vector Fields in Asymptote )

Answer 1

You could unroll the layers in z direction by hand using \pgfplotsinvokeforeach like in the hedgehog example below. I could not use your example because the parametric function f = (x,y,x+y) actually is a surface.

\documentclass{article}
\usepackage{pgfplots}
\begin{document}

\begin{tikzpicture}
  \begin{axis}[
    domain=-1:1,
    samples=10,
    xmin=-1,xmax=1,
    ymin=-1,ymax=1,
    zmin=-1,zmax=1,
    ]
    \pgfplotsinvokeforeach{-1,-.5,0,.5,1}{
      \addplot3[cyan,quiver,-stealth,
      point meta={sqrt((x)^2+(y)^2+(z)^2)},
      quiver={
        u={x/sqrt((x)^2+(y)^2+(z)^2)},
        v={y/sqrt((x)^2+(y)^2+(z)^2)},
        w={z/sqrt((x)^2+(y)^2+(z)^2)},
        colored,scale arrows=.1}]
      (x,y,#1);
    }
  \end{axis}
\end{tikzpicture}

\end{document}

Answer 2

This is an illustration with 3D Asymptote. I normalize the length of the velocity (cyan means small, magenta means large for simplicity). Of course, we also can smoothly change color from cyan to magenta.

size(10cm);
import three;
currentprojection=orthographic(4,2,1,zoom=.9);
triple f(real x, real y, real z){
return (y,z,x);
}

dot(O,2mm+red);
int n=3;
triple A=(0,0,0), B=(n,n,n);
draw(box(-B,B),gray);

for(int i=-n; i<n; ++i)
for(int j=-n; j<n; ++j)
for(int k=-n; k<n; ++k)
{
triple Ms=(i,j,k), Me=f(i,j,k);
pen p;  
if (abs(Me-Ms)<4) p=cyan+opacity(.5);
  else p=magenta+opacity(.5);
draw(Me--Me+.5*unit(Ms-Me),p,Arrow3);
}

Answer 3

Following your own idea:

\documentclass[margin=1cm]{standalone}
\usepackage[pdftex]{graphicx}
\usepackage{pgfplots,tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{decorations.markings,arrows}
\pgfplotsset{compat=newest}
\usepackage{amsmath}


\tdplotsetmaincoords{60}{120}

\begin{document}

\begin{tikzpicture}[scale=1.4]
% scale
\pgfmathsetmacro{\s}{0.15}

  \foreach \i in {-0.8,-0.6,...,0.8}
     \foreach \j in {-0.8,-0.6,...,0.8}
        \foreach \k in {-0.8,-0.6,..., 0.8}
            \draw[->, color=cyan, line width=0.2pt] 
              (\i, \j, \k) -- (\i+ \s*\j, \j + \s*\k, \k  + \s*\i);

  % draw the cube
  \draw[] (-1,1,1)--(1,1,1);
  \draw[] (-1,1,-1)--(1,1,-1);
  \draw[] (-1,1,1)--(-1,1,-1);
  \draw[] (1,1,1)--(1,1,-1);

  \draw[] (-1,-1,1)--(1,-1,1);
  \draw[] (-1,-1,-1)--(1,-1,-1);
  \draw[] (-1,-1,1)--(-1,-1,-1);
  \draw[] (1,-1,1)--(1,-1,-1);

  \draw[] (-1,1,1)--(-1,-1,1);
  \draw[] (-1,1,-1)--(-1,-1,-1);
  \draw[] (-1,1,1)--(-1,-1,1);
  \draw[] (1,1,1)--(1,-1,1);
  \draw[] (1,1,-1)--(1,-1,-1);


\end{tikzpicture}
\end{document}

And the figure