How can I draw the surface f(x,y)=x^2+y^2 like my picture?

Question

I have been trying to get this surface drawn using pgfplot, but I can't make it look like the picture I'm trying to make the code for.

\documentclass{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
    
    \begin{axis}[view = {70}{-40},
        xmax = 4,
        xmin = -4,
        ymax = 4, 
        ymin = -4, 
        restrict z to domain = -2:5]
        
        \addplot3 [
        domain=-3:3,
        domain y = -3:3,
        surf] {x^2 + y^2};
        
    \end{axis}
\end{tikzpicture}
\end{document}

I also really want to draw some other surfaces together with my surface, but I do not have any clue how to do that. (See the pictures)

I really hope you can point me in the right direction.

Answer 1

It does not make sense to use a surf plot for a completely flat area - just use \fill like this:

\documentclass[border=1 cm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
view={45}{20},
xmin=-4, xmax=4,
ymin=-4, ymax=4, 
zmin=-2, zmax=5,
axis lines=middle,
axis on top,
]
\fill[red!80!black] plot[domain=-sqrt(2.5):sqrt(2.5),samples=100] (\x,\x,{2*(\x)^2}) --cycle;
\fill[red!80!pink] (0,0,5) circle[radius=sqrt(5)];
\end{axis}
\end{tikzpicture}
\end{document}

If you want it shaded with a grid or other things, you can use a surf plot like this:

\documentclass[border=1 cm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
colormap/hot2,
view={45}{20},
xmin=-4, xmax=4,
ymin=-4, ymax=4, 
zmin=-2, zmax=5,
restrict z to domain=-2:6,
axis lines=middle,
axis on top,
]
\addplot3[
surf,
samples=51,
] {x^2 + y^2};
\fill[white] (0,0,5) +(-2cm,0cm) rectangle +(2cm,1cm);
\draw[fill=red!80!pink] (0,0,5) circle[radius=sqrt(5)];
\end{axis}
\end{tikzpicture}
\end{document}

Answer 2

The first figure with Asymptote. Other figures can be drawn in a similar way.

import graph3;
currentprojection=orthographic(2,1.5,.5,zoom=.95);
unitsize(1cm);
//currentprojection=obliqueX(40);
size(8cm);
real a=4.5; // below z=a
triple f(pair M) {
real x = sqrt(a)*(M.x)*cos(M.y);
real y = sqrt(a)*(M.x)*sin(M.y);
return (x,y,x^2+y^2);
}
real rmax = 1, rmin =0;
real phimax =2*pi, phimin =0;
surface s=surface(f, (rmin,phimin), (rmax,phimax), Spline);
//draw(s, surfacepen=pink,meshpen=brown+thick());
draw(s,surfacepen=yellow+opacity(.7));

real b=2.5;
real x(real t) {return sqrt(b)*cos(t);}
real y(real t) {return sqrt(b)*sin(t);}
real z(real t) {return b;}
path3 g=graph(x,y,z,phimin,phimax,operator..);
draw(g,red+1pt);

draw(shift(-3,-3,b)*scale(6,6,0)*unitplane,blue+opacity(.5));

draw(Label("$x$",EndPoint),-5*X--5*X,Arrow3());
draw(Label("$y$",EndPoint),-5*Y--5*Y,Arrow3());
draw(Label("$z$",EndPoint),-Z--6*Z,Arrow3());

The second picture: we can play with opacity values, remove the red box. I choose the parabolic cylinder z=1+2y^2.

// http://asymptote.ualberta.ca/
unitsize(1cm);
size(8cm);
import graph3;
currentprojection=orthographic(2,1.4,.6,zoom=.95);
//currentprojection=obliqueX(40);

// the yellow paraboloid z=f(x,y)= x^2+y^2
real a=2.25; // below z=a^2
// to take domain is a disk (instead of a rectangle)
triple f(pair M) {
real x = a*(M.x)*cos(M.y);
real y = a*(M.x)*sin(M.y);
return (x,y,x^2+y^2);
}
real rmax=1, rmin=0;
real phimax=2*pi, phimin=0;
surface sf=surface(f, (rmin,phimin), (rmax,phimax), Spline);
//draw(sf, surfacepen=pink,meshpen=brown+thick());
draw(sf,surfacepen=yellow+opacity(1));

triple h(pair M) {
if (1+2M.y^2>a^2) return (M.x,M.y,a^2);
else return (M.x,M.y,1+2M.y^2);
}
real xmax=4, xmin=-4;
real ymax=1.5, ymin=-1.5;
surface sh=surface(h, (xmin,ymin), (xmax,ymax), Spline);
//draw(sh, surfacepen=pink,meshpen=brown+thick());
draw(sh,surfacepen=blue+opacity(.8));

triple A=(xmin,ymin,0), B=(xmax,ymax,a^2);
draw(box(A,B),red+opacity(.5));

// the red circle x^2+y^2=b^2 is the intersection of
// the blue plane z=b^2 and the paraboloid
real b=1.6;
real x(real t) {return b*cos(t);}
real y(real t) {return b*sin(t);}
real z(real t) {return b^2;}
path3 g=graph(x,y,z,phimin,phimax,operator..);
//draw(g,red+1pt);

//draw(shift(-3,-3,b^2)*scale(6,6,0)*unitplane,blue+opacity(1));

draw(Label("$x$",EndPoint),-5*X--5*X,Arrow3());
draw(Label("$y$",EndPoint),-5*Y--5*Y,Arrow3());
draw(Label("$z$",EndPoint),-Z--7*Z,Arrow3());

Answer 3

I offer a TikZ solution. I think that for this figure it's easier to draw in 2d, so I did it this way.

I'm not posting the maths, because it will be tedious without MathJax or something similar. But the key here is to find the envelope of the family of ellipses (circles in 3d) that are obtained cutting the paraboloid by horizontal planes.

Making the drawing in this way all the calculations are exact (as far as TikZ goes) and all the curves are TikZ paths, I'm not using plot (see the update 1 below).

The code:

\documentclass[tikz,border=2mm]{standalone}

\tikzset
{%
  axes/.style={thick,-latex},
  paraboloid front/.style={right color=red!80,left color=white,fill opacity=0.6},
  paraboloid back/.style={left color=red!80,fill opacity=0.6},
  plane/.style={blue,fill=blue!50,fill opacity=0.8},
}

\begin{document}
\begin{tikzpicture}[line cap=round,line join=round]
% axes x,y
\draw[axes]  (30:4)  -- (210:4) node [left]  {$x$};
\draw[axes]  (150:4) -- (330:4) node [right] {$y$};
\draw[thick] (0,0)   -- (0,2.25);
% paraboloid, bottom, back
\draw[paraboloid back] ({0.25*sqrt(51)},2) parabola bend (0,-0.125) ({-0.25*sqrt(51)},2)
  arc ({180+atan(1/sqrt(17))}:{-atan(1/sqrt(17))}:{0.75*sqrt(6)} and {0.75*sqrt(2)});
% paraboloid, bottom, front
\draw[paraboloid front] ({0.25*sqrt(51)},2) parabola bend (0,-0.125) ({-0.25*sqrt(51)},2)
  arc ({180+atan(1/sqrt(17))}:{360-atan(1/sqrt(17))}:{0.75*sqrt(6)} and {0.75*sqrt(2)});
% horizontal plane
\draw[plane] (210:3) ++ (0,2.25) ++ (330:3.5) --++ (30:6) --++ (150:7) --++ (210:6) -- cycle;
% paraboloid, top, back
\draw[paraboloid back] ({0.25*sqrt(93)},3.75) parabola bend (0,-0.125) ({0.25*sqrt(51)},2)
   arc ({-atan(1/sqrt(17))}:{180+atan(1/sqrt(17))}:{0.75*sqrt(6)} and {0.75*sqrt(2)})
   parabola bend (0,-0.125) ({-0.25*sqrt(93)},3.75)
   arc ({180+atan(1/sqrt(31))}:{-atan(1/sqrt(31))}:{sqrt(6)} and {sqrt(2)});
% axis z
\draw[axes] (0,2.25) -- (0,6) node [above] {$z$};
% paraboloid, top, front
\draw[paraboloid front] ({0.25*sqrt(93)},3.75) parabola bend (0,-0.125) ({0.25*sqrt(51)},2)
   arc ({-atan(1/sqrt(17))}:{-180+atan(1/sqrt(17))}:{0.75*sqrt(6)} and {0.75*sqrt(2)})
   parabola bend (0,-0.125) ({-0.25*sqrt(93)},3.75)
   arc ({180+atan(1/sqrt(31))}:{360-atan(1/sqrt(31))}:{sqrt(6)} and {sqrt(2)});
\end{tikzpicture}
\end{document}

Update 1: I didn't it see at first, but the above code doesn't get the visibility right. This is because the parabola bend path draws the parabola to the vertex despite the last point specified is before this vertex. So I don't think parabola is the right way to do it after all. I could do a clip, but it will not be easy to find the appropriate path. Perhaps spath3 library will do the trick, I'll take a look at it in the future (see update 2 below).

Meanwhile, the same drawing with plots:

\documentclass[tikz,border=2mm]{standalone}

\tikzset
{%
  axes/.style={thick,-latex},
  paraboloid front/.style={right color=red!80,left color=white,fill opacity=0.6},
  paraboloid back/.style={left color=red!80,fill opacity=0.6},
  plane/.style={blue,fill=blue!50,fill opacity=0.8},
}

\newcommand{\myparabola}[2]{-- plot [domain={#1}:{#2},samples=40] (\x,2/3*\x*\x-0.125)}

\begin{document}
\begin{tikzpicture}[line cap=round,line join=round]
% dimensions
\pgfmathsetmacro\angTC{atan(1/sqrt(31))} % angles, top circle
\pgfmathsetmacro\angBC{atan(1/sqrt(17))} % angles, bottom circle
\pgfmathsetmacro\TTx  {0.25*sqrt(93)}    % tangent point x, top circle
\pgfmathsetmacro\TBx  {0.25*sqrt(51)}    % tangent point x, bottom circle
% axes x,y
\draw[axes]  (30:4)  -- (210:4) node [left]  {$x$};
\draw[axes]  (150:4) -- (330:4) node [right] {$y$};
\draw[thick] (0,0)   -- (0,2.25);
% paraboloid, bottom, back
\draw[paraboloid back] (-\TBx,2)
  arc (180+\angBC:-\angBC:{0.75*sqrt(6)} and {0.75*sqrt(2)}) \myparabola{\TBx}{-\TBx};
% paraboloid, bottom, front
\draw[paraboloid front] (-\TBx,2)
  arc (180+\angBC:360-\angBC:{0.75*sqrt(6)} and {0.75*sqrt(2)}) \myparabola{\TBx}{-\TBx};
% horizontal plane
\draw[plane] (210:3) ++ (0,2.25) ++ (330:3.5) --++ (30:6) --++ (150:7) --++ (210:6) -- cycle;
% paraboloid, top, back
\draw[paraboloid back] (\TBx,2)
  arc (-\angBC:180+\angBC:{0.75*sqrt(6)} and {0.75*sqrt(2)}) \myparabola{-\TBx}{-\TTx}
  arc (180+\angTC:-\angTC:{sqrt(6)} and {sqrt(2)}) \myparabola{\TTx}{\TBx};
% axis z
\draw[axes] (0,2.25) -- (0,6) node [above] {$z$};
% paraboloid, top, front
\draw[paraboloid front] (\TBx,2)
  arc (-\angBC:-180+\angBC:{0.75*sqrt(6)} and {0.75*sqrt(2)}) \myparabola{-\TBx}{-\TTx}
  arc (180+\angTC:360-\angBC:{sqrt(6)} and {sqrt(2)}) \myparabola{\TTx}{\TBx};
\end{tikzpicture}
\end{document}

And the drawing (with the right visibility):

Update 2: This is my first attempt at Andrew Stacey's spath3 library. I'm don't know if I know what I'm doing, so use it at your own risk!! (It needs TL2021, I don't know about other LaTeX distros).

I'm following Andrew's excellent answer here (the second one) step by step. In fact the comments about spath3 are all his (I changed a word or two). And , of course, the errors are all mine.

The new code:

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{intersections,spath3}

\tikzset
{%
  axes/.style={thick,-latex},
  paraboloid front/.style={right color=red!80,left color=white,fill opacity=0.6},
  paraboloid back/.style={left color=red!80,fill opacity=0.6},
  plane/.style={blue,fill=blue!50,fill opacity=0.8},
}

\begin{document}
\begin{tikzpicture}[line cap=round,line join=round]
% creating the paths: parabola, two ellipses and two lines (for cutting)
\path[spath/save=P]  ({-0.25*sqrt(93)},3.75) parabola bend (0,-0.125) ({0.25*sqrt(93)},3.75);
\path[spath/save=EB] (0,2.25) ellipse ({0.75*sqrt(6)} and {0.75*sqrt(2)}); % ellipse, bottom
\path[spath/save=ET] (0,4)    ellipse ({sqrt(6)} and {sqrt(2)});           % ellipse, top
\path[spath/save=LB] (-3,2)    -- (3,2);                                   % line,    bottom
\path[spath/save=LT] (-3,3.75) -- (3,3.75);                                % line,    top
% spath3 operations
\tikzset
{% Ellipses have an "empty" component at the start which
 % moves from the centre to the rim; it can be irritating
 % when trying to count components later so this removes
 % any empty components
   spath/remove empty components={EB},
   spath/remove empty components={ET},
 % Now split each path where it intersects with the lines
   spath/split at intersections={P}{LB},
   spath/split at intersections={EB}{LB},
   spath/split at intersections={ET}{LT},
 % Each path is now a collection of components; to work
 % with them individually we split them into a list of
 % separate paths which is stored in a macro
   spath/get components of={P}\Pcpts,
   spath/get components of={EB}\Bcpts,
   spath/get components of={ET}\Tcpts,
}
% axes x,y
\draw[axes]  (30:4)  -- (210:4) node [left]  {$x$};
\draw[axes]  (150:4) -- (330:4) node [right] {$y$};
\draw[thick] (0,0)   -- (0,2.25);
% paraboloid, bottom, back
\draw[paraboloid back,
      spath/use=\getComponentOf\Pcpts{2},
      spath/use={\getComponentOf\Bcpts{2},weld},
];
% paraboloid, bottom, front
\draw[paraboloid front,
      spath/use=\getComponentOf\Pcpts{2},
      spath/use={\getComponentOf\Bcpts{1},reverse,weld}
];
% horizontal plane
\draw[plane] (210:3) ++ (0,2.25) ++ (330:3.5) --++ (30:6) --++ (150:7) --++ (210:6) -- cycle;
% paraboloid, top, back
\draw[paraboloid back,
      spath/use=\getComponentOf\Pcpts{1},
      spath/use={\getComponentOf\Bcpts{2},reverse,weld},
      spath/use={\getComponentOf\Pcpts{3},weld},
      spath/use={\getComponentOf\Tcpts{2},weld}
];
% axis z
\draw[axes] (0,2.25) -- (0,6) node [above] {$z$};
% paraboloid, top, front
\draw[paraboloid front,
      spath/use=\getComponentOf\Pcpts{1},
      spath/use={\getComponentOf\Bcpts{1},weld},
      spath/use={\getComponentOf\Pcpts{3},weld},
      spath/use={\getComponentOf\Tcpts{1},reverse,weld}
];
\end{tikzpicture}
\end{document}

The output is the same as above.