Plot the cusp catastrophe surface

Question

I'm trying to draw (plot) the Cusp catastrophe surface into latex using any software logic. I am completely stuck because I'm definitely not a mathematician. The surface has the potential:

V(x) = F(x,u,v) = x^4 + ux^2 + vx

so it should be given by this implicit equation:

4x^3 + 2ux + v = 0

But I don't know how to implement this. Apparently gnuplot can't do this (even if I find something approaching) so I should go with a CAS like Octave or scripting something in Maxima first, then feed gnuplot or export the graphic. Ideally, it would be great if it was as easy as this Mathematica example:

F[x_, u_, v_] :=  x^4 + u x^2 + v*x

Needs["Graphics`ContourPlot3D`"]

ContourPlot3D[Evaluate[D[F[x, u, v], x]],
  {u, -2.5, 2}, {v, -2.2, 3}, {x, -1.4, 1.3}, 
  PlotPoints -> 7, ViewPoint -> {-1.5, 1.5, 1.4}, 
  Axes -> True, ContourStyle -> {EdgeForm[]}, 
  AxesLabel -> TraditionalForm /@ {u, v, x}] // Timing

---

edit / updates

So I dig I lot, apparently, I can't do it perfectly without the ability of a mathematician (that's for sure) si I decided I should stick with an implicit solving method. I can translate the Mathematica code into Sage:

u, v, x = var('u, v, x')
f(u,v,x) = x^4 + u*x^2 + v*x
g = f.diff(x)
implicit_plot3d(g, (u, -2, 2), (v, -2, 2), (x, -2, 2))

It produces neat results:

You can also check the result on cloud.sagemath (subscription required).

But unfortunately 3d plot can't be exported in vector format, maybe there's a method for raw (x,y,z) points export, dunno... I tried an other solution with a the experimental psplotImpIIID feature of ps-tricks, it's ugly and produce a lot of artefacts.

And to end up with pstricks, one could do something like this:

\begin{pspicture}(-6,-4)(6,5)
    \pstThreeDCoor[xMin=-4,xMax=4,yMin=-4,yMax=4,zMin=-4,zMax=4,RotY=90,RotX=25]
    \psplotThreeD[plotstyle=line,hiddenLine,% does not work in my case (xelatex?)
    yPlotpoints=50,xPlotpoints=50,linewidth=.5pt,algebraic](-3,3)(-3,3){4*x^3 - 2*y*x}
\end{pspicture}

The advantage being that we can easily rotate the axis, the disadvantage (from the perspective of a pstricks noob) being styling, plus I need to clip the result image. There's also a solution that uses a R package which I tried and and it works great and outputs vector graphics. But honestly, I think I loop the web (I even scripted it as a Blender Python) and there is definitely no easy straight solution for this.

Answer 1

You can actually do this via pgfplots but to make it a little nicer you have to switch to LuaLaTeX since PDFLaTeX chokes up on memory limitations (due to many samples per axis. You don't need that much but why not :P).

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.10}
\begin{document}
\begin{tikzpicture}
\begin{axis}[y domain=-4:4,
          domain=-2:2,
          restrict z to domain=-5:5,
          samples=100,
          view={43}{8},
          mesh/interior colormap name=hot,
          colormap/blackwhite,
          xlabel=$x$,ylabel=$u$,zlabel=$v$,grid=both
         ]
\addplot3[surf] {-4*x^3 - 2*y*x};
\end{axis}
\end{tikzpicture}
\end{document}

If you decide the hide the rough edges you can get a lo-fi version which LaTeX can also handle

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.10}
\begin{document}
\begin{tikzpicture}
\begin{axis}[y domain=-3:3,
          domain=-2:2,zmin=-3,zmax=3,
          restrict z to domain=-5:5,
          samples=45,
          view={48}{15},
          mesh/interior colormap name=hot,
          colormap/blackwhite,
          xlabel=$x$,ylabel=$u$,zlabel=$v$,grid=both
         ]
\addplot3[surf] {-4*x^3 - 2*y*x};
\end{axis}
\end{tikzpicture}
\end{document}

Answer 2

I think what you are asking for it is precisely in ePiX's, (http://bay.uchicago.edu/tex-archive/graphics/epix/samples/butterfly.xp) examples.

If I understand the code, the strategy is to draw the surface in three patches. It would be fun to translate this into pgfplots and asymptote. Maybe asymptote can even define this curve implicitly (it is very good at that).

Here it is an image of the result (I didn't make it):

Answer 3

Asymptote can plot an implicitly defined surface using the smoothcontour3 module. This module has been incorporated into the (just-released) version 2.33 of Asymptote, but if you don't have an earlier version, you can install the module using the link above. (Just copy the file into the directory containing your image.) Here's how the surface you describe might be drawn:

\documentclass{standalone}
\usepackage{asypictureB}
\begin{document}
\begin{asypicture}{name=catastrophe_surface}
settings.outformat = "png";
settings.render = 32;
size(8cm);
import smoothcontour3;
import graph3;

currentprojection=perspective(3*(4,-6,3));

real f(real v, real u, real x) { return 4x^3 + 2u*x + v; }
draw(implicitsurface(f,(-2,-2,-2),(2,2,2), n=15, overlapedges=true), 
    surfacepen=material(diffusepen=gray, 
    ambientpen=blue));

zaxis3("$x$", Bounds, InTicks, zmin=-2, zmax=2);
xaxis3("$v$", Bounds, InTicks);
yaxis3("$u$", Bounds, InTicks(beginlabel=false));
\end{asypicture}
\end{document}

It can even produce a vector graphics version if you change the first two lines inside the asypicture environment to

settings.outformat = "pdf";
settings.render = 0;

but the resulting pdf file will be enormous and will look bad in most viewers (in my experience, it looks okay in Adobe Reader, but not in any other I've tried).

Either way, for this particular problem, I think you're better off using the small bit of math

4x^3 + 2ux + v = 0
v = -4x^3 - 2ux

to turn this into the plot of a function rather than an implicitly defined surface. Implicitly defined surfaces generally don't look as nice.

Here's a non-implicit version with Asymptote. (Note that it requires my crop3D module.)

settings.outformat = "png";
settings.render = 16;
size(8cm);

import graph3;
import crop3D;
currentprojection=perspective(3*(4,-6,3));

triple F(pair ux) {
  real u = ux.x;
  real x = ux.y;
  return (-4x^3 - 2*u*x, u, x);
}
surface s = surface(F, (-2,-2), (2,2), nu=20, usplinetype=Spline);
draw(crop(s,(-2,-2,-2),(2,2,2)), surfacepen=material(diffusepen=gray, ambientpen=blue));

zaxis3("$x$", Bounds, InTicks, zmin=-2, zmax=2);
xaxis3("$v$", Bounds, InTicks);
yaxis3("$u$", Bounds, InTicks(beginlabel=false));

It compiles more quickly than the implicit version and gives, in my opinion, a nicer result:

Answer 4

\documentclass[pstricks]{standalone}
\usepackage{pst-solides3d}
\begin{document}

\begin{pspicture}(-4,-6)(5,9)
\psset{viewpoint=100 30 40 rtp2xyz,lightsrc=viewpoint, Decran=120}
\psSurface[ngrid=.15 .15,incolor=yellow,hue=0 1,linewidth=0.1\pslinewidth,
  algebraic,axesboxed](-1,-2)(1,2){ 4*x^3 - 2*y*x}
\end{pspicture}

\end{document}

The z max/min values can be handled only on PostScript level (will be changed in the future)

\documentclass[pstricks]{standalone}
\usepackage{pst-solides3d}
\begin{document}

\begin{pspicture}(-4,-6)(5,9)
\psset{viewpoint=100 30 40 rtp2xyz,lightsrc=viewpoint, Decran=120}
\psSurface[ngrid=0.1 0.15,incolor=yellow,hue=0 1,linewidth=0.1\pslinewidth,axesboxed, 
 % algebraic](-2,-2)(2,2){4*x^3 -2*y*x}
  ](-2,-2)(1,3){ 4 x 3 exp mul 2 y mul x mul sub 
    dup 4 gt { pop 4 } if 
    dup -4 lt { pop -4 } if 
    }
\end{pspicture}

\end{document}

However, the values are not ignored, they are only set to the max/min.