Pgfplot nested loops with calculated parameter

Question

I need to draw a finite number of cubes below a 3D surface. Thus I need nested loops. The following code is close to what I want, except that I want to modify the z limit for the \k parameter to have all the cubes under the surface plot. Basically, I would like to use \fun{\i}{\j}-\dl instead of \fun{\i}{\j}:

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
 \usetikzlibrary{calc}
\pgfplotsset{compat=1.16}


\begin{document}


 
\newcommand{\fun}[2]{{4*((#1/(#1+1))*(#2/(#2+1))+0.5)}}


\tikzset{
    mycube/.pic={
        \pgfmathsetmacro\size{{#1}}
        \draw[fill=gray!20] (0,0,\size) -- ++(0,\size,0) -- ++(\size,0,0) -- ++(0,-\size,0) -- cycle; % top
        \draw[fill=gray!10] (0,0,0) --++(\size,0,0)-- ++ (0,0,\size)--++(-\size,0,0)  -- cycle; % front
        \draw[fill=gray!40] (\size,0,0) --++(0,\size,0)-- ++ (0,0,\size)--++(0,-\size,0)  -- cycle; %side
}
}

\begin{tikzpicture}
\def\dl{1}
\begin{axis}[xlabel = $x$, ylabel = $y$, zlabel = {$z$},
    xmin=0, xmax=5,
    ymin=0, ymax = 5,
    zmin=0, 
    clip=false]

% WORKING but not satisfactory
\pgfplotsforeachungrouped \i in {0,0+\dl,...,5-\dl}{
    \pgfplotsforeachungrouped \j in {5-\dl,5-(2*\dl),...,0}{
        \pgfplotsforeachungrouped \k in {0,0+\dl,...,int(\fun{\i}{\j})}{
            \edef\temp{\noexpand \draw (\i,\j,\k) pic{mycube={\dl}};}\temp
        }
    }
}

% NOT working
% \pgfplotsforeachungrouped \i in {0,0+\dl,...,5-\dl}{
%     \pgfplotsforeachungrouped \j in {5-\dl,5-(2*\dl),...,0}{
%         \pgfplotsforeachungrouped \k in {0,0+\dl,...,int(\fun{\i}{\j}-\dl)}{
%             \edef\temp{\noexpand \draw (\i,\j,\k) pic{mycube={\dl}};}\temp
%         }
%     }
% }

% NOT working
% \pgfplotsforeachungrouped \i in {0,0+\dl,...,5-\dl}{
%     \pgfplotsforeachungrouped \j [evaluate=\j as \klim using {int(\fun{\i}{\j})}] in {5-\dl,5-(2*\dl),...,0}{
%         \pgfplotsforeachungrouped \k in {0,0+\dl,...,\klim}{
%             \edef\temp{\noexpand \draw (\i,\j,\k) pic{mycube={\dl}};}\temp
%         }
%     }
% }

    \addplot3[surf,domain=0:5,y domain=0:5,fill opacity = 0.3] {\fun{x}{y}};
\end{axis}
\end{tikzpicture}
 
\end{document}

I tried several trick to lower this z-limit but nothing works here. Any help would be appreciated!

---

Bonus questions:

• I'm not sure that defining my surface function as a separate macro is the best way to do. How to do it in a cleaner way?
• Why is it necessary to use int( ) in the \k loop parameters?

Thanks a lot.

Answer 1

Trying to understand what "lower the z limit" is to mean, I eventually guessed that the OP would like the boxes shifted downward in the z direction, while leaving the function intact, in a fashion so that the boxes do not exceed the function.

I adjusted the \k loop index to

\pgfplotsforeachungrouped \k in {0,0+\dl,...,(int(\fun{\i}{\j}) - 1.5)}

Why -1.5? Well, since \k starts drawing at 0, and the zero draw extends up to z=1, that accounts for subtracting 1. Then, I discovered that if I set the top limit on \z manually to, for example, 2.5, it rounds it up, as if it were a 3 and actually plots boxes up to z=4. Thus, there can be as large a difference between the actual function (z=2.5) and the stacked z-boxes (to z=4) of 4 - 2.5 = 1.5. That is why 1.5 must be subtracted off the total, in order to ensure that the box height never exceeds the function height.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
 \usetikzlibrary{calc}
\pgfplotsset{compat=1.16}

\begin{document}

\newcommand{\fun}[2]{{4*((#1/(#1+1))*(#2/(#2+1))+0.5)}}

\tikzset{
    mycube/.pic={
        \pgfmathsetmacro\size{{#1}}
        \draw[fill=gray!20] (0,0,\size) -- ++(0,\size,0) -- ++(\size,0,0) -- ++(0,-\size,0) -- cycle; % top
        \draw[fill=gray!10] (0,0,0) --++(\size,0,0)-- ++ (0,0,\size)--++(-\size,0,0)  -- cycle; % front
        \draw[fill=gray!40] (\size,0,0) --++(0,\size,0)-- ++ (0,0,\size)--++(0,-\size,0)  -- cycle; %side
}
}

\begin{tikzpicture}
\def\dl{1}
\begin{axis}[xlabel = $x$, ylabel = $y$, zlabel = {$z$},
    xmin=0, xmax=5,
    ymin=0, ymax = 5,
    zmin=0, 
    clip=false]

% WORKING but not satisfactory
\pgfplotsforeachungrouped \i in {0,0+\dl,...,5-\dl}{
    \pgfplotsforeachungrouped \j in {5-\dl,5-(2*\dl),...,0}{
    \pgfplotsforeachungrouped \k in {0,0+\dl,...,(int(\fun{\i}{\j}) - 1.5)}{
            \edef\temp{\noexpand \draw (\i,\j,\k) pic{mycube={\dl}};}\temp
        }
    }
}

    \addplot3[surf,domain=0:5,y domain=0:5,fill opacity = 0.3] {\fun{x}{y}};
\end{axis}
\end{tikzpicture}
 
\end{document}