Make plot smoother in pgfplots

Question

I'm trying to plot a function using pgfplots:

\documentclass{scrartcl}

\usepackage{pgfplots}
\pgfplotsset{compat = 1.12}

\newcommand{\param}{2.0}

\begin{document}
        \begin{tikzpicture}
          \begin{axis}[view={40}{50}]
            \addplot3[surf, domain = 0:1, y domain = 0:1, unbounded coords=jump,
            samples = 50]
            {x^(-\param-1)*y^(-\param-1)*(x^(-\param)+y^(-\param)-1)^(-1/\param-2)+\param*x^(-\param-1)*y^(-\param-1)*(x^(-\param)+y^(-\param)-1)^(-1/\param-2)};
        \end{axis}
      \end{tikzpicture}
\end{document}

This results in:

Compared with (plotted in the OS X Grapher application)

the plot generated by pgfplots is a lot "rougher" near x=y=0. I have tried increasing the number of samples from the default to 50 but that hasn't really improved the plot much.

Answer 1

The default surface plot of pgfplots uses two triangles for each rectangular patch segment. Usually, the diagonal does not matter much -- but in this case, it really matters and the result is unsuitable.

Note that shader=interp appears to select the other diagonal (unintentionally, but it does). A simple solution would be to add shader=interp, unless you really need the grid lines.

\documentclass{standalone}

\usepackage{pgfplots}
\usepgfplotslibrary{patchplots}
\pgfplotsset{compat = 1.12}

\newcommand{\param}{2.0}

\begin{document}
        \begin{tikzpicture}
          \begin{axis}[view={40}{50}]
            \addplot3[surf, domain = 0:1, y domain = 0:1, unbounded coords=jump,
                shader=interp,
                samples = 25]
            {x^(-\param-1)*y^(-\param-1)*(x^(-\param)+y^(-\param)-1)^(-1/\param-2)+\param*x^(-\param-1)*y^(-\param-1)*(x^(-\param)+y^(-\param)-1)^(-1/\param-2)};
        \end{axis}
      \end{tikzpicture}
\end{document}

Answer 2

This answer is now an appendix (how to add gridlines) to C.F.'s (better) answer above: shader=interp is the surf equivalent of smooth, and it preserves the color scheme, unlike my original answer.

shader=interp removes gridlines, but you add some of them back in, because of the above fact, thus:

\documentclass{article}\usepackage{pgfplots}\usepgfplotslibrary{patchplots}
\newcommand{\pt}{2}
\begin{document}\begin{tikzpicture}
    \begin{axis}[3d box,width=8cm,view={147}{56},
    domain=0:0.4,y domain=0:0.4,samples=32,
    xlabel=$x$,ylabel=$y$,zlabel={$z$},]
    \addplot3[surf,domain=0:0.4,y domain=0:0.4,unbounded coords=jump,shader=interp]
{%
x^(-\pt-1)*y^(-\pt-1)*(x^(-\pt)+y^(-\pt)-1)^(-1/\pt-2)+\pt*x^(-\pt-1)*y^(-\pt-1)*(x^(-\pt)+y^(-\pt)-1)^(-1/\pt-2)%
};
    \addplot3[domain=0:0.4,y domain=0:0.4,unbounded coords=jump,smooth]
{%
x^(-\pt-1)*y^(-\pt-1)*(x^(-\pt)+y^(-\pt)-1)^(-1/\pt-2)+\pt*x^(-\pt-1)*y^(-\pt-1)*(x^(-\pt)+y^(-\pt)-1)^(-1/\pt-2)%
};
\end{axis}\end{tikzpicture}\end{document}

The gridlines themselves are:

Smoothing via cubic bézier curves is implemented in pgfplots. (See p.76 of the pgfplots manual.)

\addplot3[...,smooth,...] for line only or \addplot+[...,smooth,...] gives the above and fill it with blue points.

Consider

\documentclass{standalone}
\usepackage{pgfplots}
\begin{document}
    \begin{tikzpicture}
      \begin{axis}[view={40}{50}]
        \addplot3 [y domain = 0:2,smooth]{-0.7+4*exp(-0.5*(x+3))*(3*cos(4*x*180/pi)+2.5*cos(2*x*180/pi))+0.5*y*y*4};
    \end{axis}
  \end{tikzpicture}
\end{document}

and compare to unsmoothed version

\documentclass{standalone}
\usepackage{pgfplots}
\begin{document}
    \begin{tikzpicture}
      \begin{axis}[view={40}{50}]
        \addplot3 [y domain = 0:2]{-0.7+4*exp(-0.5*(x+3))*(3*cos(4*x*180/pi)+2.5*cos(2*x*180/pi))+0.5*y*y*4};
    \end{axis}
  \end{tikzpicture}
\end{document}

True, there is a problem with color. The equation in the question is 1/x^5*1/y^3+ 2/x^5*1/y^3+2*y^(9/2) reduced. It's very sharp and using 256 samples for example instead causes main memory to run out and TeX halts compiling... Smooth with around 100 samples and rotating the view may be the only option, when used with a compatible color scheme to return the surface fill.

Answer 3

Just for showing another plotting approach, here is the LaTeX-R-knitr solution.

\documentclass[10pt,letterpaper]{article}

\begin{document}

\begin{figure}[scale=2.5]
<<>>=
library(lattice)
param<-2
x<-seq(0,1,len=30)
y<-seq(0,1,len=30)
g<-expand.grid(x=x,y=y)
g$z<-(g$x^(-param-1)*g$y^(-param-1)*(g$x^(-param)
     +g$y^(-param)-1)^(-1/param-2)+param*g$x^(-param-1)*g$y^(-param-1)*(g$x^(-param)
     +g$y^(-param)-1)^(-1/param-2))
wireframe(z~x*y,g,drape=TRUE,aspect=c(1,1),colorkey=TRUE
          ,screen = list(x=-40,y=-60,z=-45))
@
%
\end{figure}
\end{document}