PGFPlots - how to avoid aliasing when plotting 3d surface on limited domain

Question

I'm trying to render the following plot so that the shading goes right up to the bounding curve without this jagged aliasing.

The function is $\frac{x}{y^5 + 1}$ in the domain $\sqrt{x} \le y \le 2$ and $0 \le x \le 4$.

I translated this to PGFPlots as

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
    unbounded coords=jump,
    clip=false,
    view={-30}{45},
    axis lines=middle,
    xmin=0, xmax=4,
    ymin=0, ymax=2.15,
    zmin=0, zmax=0.5,
    xlabel=$x$, ylabel=$y$, zlabel=$z$,
    xtick={1, 2, 3, 4}, ytick={1, 2},
    ztick={0.5}, zticklabels={$\frac 1 2$}
]
\addplot3[
    samples=100, samples y=250,
    domain=0:4, domain y=0:2,
    colormap/blackwhite, shader=interp, surf, z buffer=sort
] {y >= sqrt(x) ? x / (y^5 + 1) : inf};
\addplot3[
    samples=100, domain=0:8
](
    {(x < 4) ? x : 8-x},
    {(x < 4) ? sqrt(x) : 2},
    {(x < 4) ? x / (sqrt(x)^5 + 1) : 0}
);
\addplot3[
    samples=100, domain=0:8
](
    {(x < 4) ? x : 8-x},
    {(x < 4) ? sqrt(x) : 2},
    {0}
);
\end{axis}
\end{tikzpicture}
\end{document}

and rendered it with LuaLaTeX.

It takes too long to render when I set samples and samples y high enough to make the aliasing disappear. What I want to do is something like samples at and put a bunch of samples near the edges and fewer in the midst, but samples at only allows you to set $x$ samples and not $(x,y)$ samples.

Any ideas? Thanks folks!

Answer 1

Thanks for your MWE!

Now to the real problem. You wish to plot the function only in the domain y >= sqrt(x). This can be achieved by doing a parametric plot. I constructed a functions ycheat(x,y), which, for a fixed x returns values in the interval between sqrt(x) and 2, which is the maximal value of y in your example. That is, if y varies between 0 and 2, ycheat runs only between sqrt(x) and 2. You thus get rid of your filter in y >= sqrt(x) ? x / (y^5 + 1) : inf as now simply no "bad" y occurs, and solve at the same time the issue with the boundary.

Last but not least let me mention that if you add samples y=1 to the 1-dimensional plots, this reduces the compilation time drastically.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{width=12cm,compat=1.16}
\begin{document}
\begin{tikzpicture}[declare function={ycheat(\x,\y)=sqrt(\x)+\y*(1-sqrt(\x)/2);}]
\begin{axis}[
    unbounded coords=jump,
    clip=false,
    view={-30}{45},
    axis lines=middle,
    xmin=0, xmax=4,
    ymin=0, ymax=2.15,
    zmin=0, zmax=0.5,
    xlabel=$x$, ylabel=$y$, zlabel=$z$,
    xtick={1, 2, 3, 4}, ytick={1, 2},
    ztick={0.5}, zticklabels={$\frac{1}{2}$}
]
\addplot3[samples y=1,% <-added
    samples=101, domain=0:4
](
    {(x < 4) ? x : 8-x},
    {(x < 4) ? sqrt(x) : 2},
    {0}
);
\addplot3[opacity=0.7,% <- added just for fun
     samples=101, samples y=25,
     domain=0:4, domain y=0:2,
     colormap/blackwhite, shader=interp, surf, z buffer=sort
 ] ({x},{ycheat(x,y)}, {x / (pow(ycheat(x,y),5) + 1)} );
\addplot3[samples y=1,% <-added
    samples=101, domain=0:4
](
    {(x < 4) ? x : 8-x},
    {(x < 4) ? sqrt(x) : 2},
    {(x < 4) ? x / (sqrt(x)^5 + 1) : 0}
);
\end{axis}
\end{tikzpicture}
\end{document}