\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}
\usepackage{pgfplots}
\usepackage{units}
\pgfplotsset{compat=1.3}% <-- moves axis labels near ticklabels
                        % (respects tick label widths)
\usepackage{tikz}
\usetikzlibrary{arrows, positioning, calc, intersections, decorations.markings}

\usepackage{xcolor}
\definecolor{horizontalLineColor}{HTML}{008000}
\definecolor{verticalLineColor}{HTML}{FF0000}

\begin{document}

% Define this as a command to ensure that it is same in both cases
\newcommand*{\ShowIntersection}[2]{
\fill
    [name intersections={of=#1 and #2, name=i, total=\t}]
    [red, opacity=1, every node/.style={above left, black, opacity=1}]
    \foreach \s in {1,...,\t}{(i-\s) circle (2pt)
        node [above left] {\s}};
}

\begin{preview}
\begin{tikzpicture}
    \begin{axis}[
        grid=both,
        minor tick num=1,
        xlabel=$x$,
        ylabel=$y$,
        zlabel=$z$,
        %label distance=0mm,
        %width=8cm, height=7cm,     % size of the image
        axis lines=left,
        %tick style={draw=none},
        %xticklabels={,,},
        %yticklabels={,,}
    ]
      \addplot3[surf, colormap name=viridis] {-(x*x/16+y*y/4-1)};
    \end{axis}
\end{tikzpicture}
\end{preview}
\end{document}