TikZ-PGF – How to Draw a Paraboloid and Its Contours in TikZ

Question

I have two similar images in which paraboloid and its contours are drawn.

What I've ever tried is in the following code that is very different from the figure.

\documentclass{standalone}
\usepackage{tikz, tikz-3dplot}
\usetikzlibrary{arrows.meta}
\tikzset{reverseclip/.style={insert path={(current bounding box.north
    east) rectangle (current bounding box.south west)}}}
\begin{document}
\tdplotsetmaincoords{120}{-25}
\begin{tikzpicture}[tdplot_main_coords]
\draw [thick] (0,0,0) -- (4,0,0) node {y} (0,0,0) -- (0,4,0) node {x} (0,0,0) -- (0,0,10) node {z};
\tdplotsetrotatedcoords{0}{0}{-25}
\draw [red, thick, tdplot_rotated_coords] (0,0,0) -- (4,0,0) node {y} (0,0,0) -- (0,4,0) node {x} (0,0,0) -- (0,0,10) node {z};
\fill [cyan, tdplot_rotated_coords, canvas is xy plane at z=0, opacity=0.67] (0,0) circle [radius=3];
\fill [cyan, tdplot_rotated_coords, canvas is xz plane at y=0, opacity=0.67] plot [variable=\x, domain=-3:3, samples=50] ({\x},{9-\x*\x});
\foreach \r in {1,2,3}
\draw [red, very thick, canvas is xy plane at z=0] (0,0) circle [radius=\r];
\node [below] at (0,3,0) {10};
\node [below] at (3,0,0) {10};
\node [left] at (0,0,9) {100};

\begin{scope}
\clip [canvas is xy plane at z=6] (0,0) circle [radius={sqrt(3)}] [reverseclip];
\fill [orange, canvas is xy plane at z=6] (-4,-4) rectangle (4,4);
\end{scope}
\end{tikzpicture}
\end{document}

How to draw more realistic figure?

Answer 1

Why not using pgfplots; something like that:

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
xlabel=$x$, ylabel=$y$, zlabel=$z$, 
xmin=-19, xmax=19,
ymin=-19, ymax=19,
zmin=-1, zmax=110,
x={(-0.125cm,-0.05cm)}, y={(0.125cm,-0.05cm)}, z={(0cm,0.05cm)},
axis lines=middle,
every axis x label/.style={  at={(ticklabel* cs:1.05)},  },
every axis y label/.style={  at={(ticklabel* cs:1.05)},  },
every axis z label/.style={  at={(ticklabel* cs:1.05)},  },
]
% Paraboloid
\addplot3[surf,
shader=flat, draw=lightgray, fill=green, ultra thin, 
left color=green, right color=green, middle color=green!25, 
opacity=0.5, fill opacity=0.5,
data cs=polar, domain=0:360, 
y domain=0:10,
restrict z to domain=0:101, 
](x, y, 100-y^2);

% Plane 
\addplot3[surf, shader=faceted,
color=orange, 
opacity=0.01, fill opacity=0.4, 
domain=-10:10, 
](x,y,75);

% Circle at plane
\addplot3[red, smooth, 
domain=0:360, variable=\t
]({5*cos(\t)},{5*sin(\t)},{75});

% Circles at xy-plane
\pgfplotsinvokeforeach{10, 7, 2.5}{%%
\pgfmathsetmacro\Radius{#1}
\addplot3[red, smooth, 
%no markers,
%samples=55,% samples y=0, 
domain=0:360, variable=\t
]({\Radius*cos(\t)},{\Radius*sin(\t)},{0});
}%%

% Annotations 1/2
\coordinate[label=](A) at ({7*cos(135)},{7*sin(135)},{0});
\coordinate[label=](B) at (8, -8, 75);
\coordinate[label=](C) at (-4, 5, 40);
\coordinate[label=](D) at({5*cos(300)},{5*sin(300)},{75});
\end{axis}

% Annotations 2/2
\draw[semithick] (A) -- +(45:2) node[right, align=left]{
$f(x,y)=51$\\ (a typical\\ level curve in\\ the function's\\ domain)};

\draw[semithick] (B) -- +(135:0.75) node[above, align=left]{
Plane $z=75$};

\draw[semithick] (C) -- +(45:3) node[above]{$z=100-x^2-y^2$};

\draw[semithick] (D) -- +(122:2.5) node[above, align=left, xshift=2cm]{The contour curve $f(x,y)=100-x^2-y^2=75$ \\ is the circle $x^2+y^2=25$ in the plane $z=75$.};
\end{tikzpicture}   
\end{document}