6

I would like to draw the body D defined by x^2+y^2<=9, z<=16-x^2-y^2 and z>=0.

We have to graph x^2+y^2=9, z=16-x^2-y^2 and z=0 with less opacity, and then fill (or use patterns) the intersections to create the body D defined by the previous expressions:

What I want

However, I am not able to complete the task. Here is a MWE:

\documentclass{article}
\usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}

\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\begin{document}

\begin{center}
    \begin{tikzpicture}
        \begin{axis} [
                axis on top,
                axis lines=center,
                xlabel=$x$,
                ylabel=$y$,
                zlabel=$z$,
                xmin=-5,
                ymin=-5,
                zmin=-1,
                xmax=5,
                ymax=5,
                zmax=17,
                xtick={-4,-3,0,3,4},
                xticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ytick={-4,-3,0,3,4},
                yticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ztick={0,16},
                zticklabels={$0$,$16$},
                ticklabel style={font=\tiny},
                legend pos=outer north east,
                legend style={at={(1.25,.81)},anchor=north east},
                legend cell align={left},
                view={135}{25}
            ]
            \addplot3[orange,opacity=.5,samples=51,samples y=21,variable=t,variable y=r,domain=0:360,domain y=0:2*pi,restrict z to domain=0:16] ({r*cos(t)},{r*sin(t)},{16-r*r});   % z=16-x^2-y^2
            \addlegendentry{\(16-x^2-y^2\)}
            \addplot3[green,opacity=.5,samples=51,samples y=21,variable=t,variable y=r,domain=0:360,domain y=0:2*pi,restrict z to domain=0:16] ({3*cos(t)},{3*sin(t)},{0});   % x^2+y^2=9
            \addlegendentry{\(x^2+y^2=9\)}
            \fill[cyan,opacity=.5] (3,3,0) -- (3,-3,0) -- (-3,-3,0) 
        -- (-3,3,0);    % z=0
            \addlegendentry{\(z=0\)}    % This label is not showing because we used 'fill', not 'addplot3'
            % Here should be the code of the intersection of the three surfaces...
        \end{axis}
    \end{tikzpicture}
\end{center}

\end{document}

What I have done

EDIT. Many thanks to marmot who helped me partially complete what I want:

\documentclass{article}
\usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}

\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\begin{document}

\begin{center}
    \begin{tikzpicture}
        \begin{axis} [
                axis on top,
                axis lines=center,
                xlabel=$x$,
                ylabel=$y$,
                zlabel=$z$,
                xmin=-5,
                ymin=-5,
                zmin=-1,
                xmax=5,
                ymax=5,
                zmax=17,
                xtick={-4,-3,0,3,4},
                xticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ytick={-4,-3,0,3,4},
                yticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ztick={0,16},
                zticklabels={$0$,$16$},
                ticklabel style={font=\tiny},
                legend pos=outer north east,
                legend style={at={(1.45,.85)},anchor=north east},
                legend cell align={left},
                view={135}{25}
            ]
            \addplot3[opacity=.5,surf,shader=interp,colormap={orange}{color=(orange!50) color=(orange!50)},variable=t,variable y=r,domain=0:360,domain y=0:2*pi,restrict z to domain=0:16] ({r*cos(t)},{r*sin(t)},{16-r*r});  % z=16-x^2-y^2
            \addlegendentry{\(z=16-x^2-y^2\)}
            \addplot3[opacity=.5,surf,shader=interp,domain=0:360,domain y=0:16,colormap={green}{color=(green!50) color=(green!50)}] ({3*cos(x)},{3*sin(x)},{y}); % x^2+y^2=9
            \addlegendentry{\(x^2+y^2=9\)}
            %\fill[cyan,opacity=.5] (4,4,0) -- (4,-4,0) -- (-4,-4,0) -- (-4,4,0);    % z=0
            \addplot3[fill=cyan,opacity=.5,domain=-4:4, domain y=-4:4] (x,y,0);
            \addlegendentry{\(z=0\)}
        \end{axis}
    \end{tikzpicture}
\end{center}

\end{document}

New MWE

However, we need to:

  • Fix the output of z=0 (I do not want to use fill because then we cannot add a legend since it is not a plot).
  • Fill the solid D using patterns: Solid using patterns or filling it: Solid filling it What requires less programming time.

Some links of interest:

Thanks!!

5
+50

How about

\documentclass{article}
\usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}

\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\begin{document}

\begin{center}
    \begin{tikzpicture}
        \begin{axis} [
                axis on top,
                axis lines=center,
                xlabel=$x$,
                ylabel=$y$,
                zlabel=$z$,
                xmin=-5,
                ymin=-5,
                zmin=-1,
                xmax=5,
                ymax=5,
                zmax=17,
                xtick={-4,-3,0,3,4},
                xticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ytick={-4,-3,0,3,4},
                yticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ztick={0,16},
                zticklabels={$0$,$16$},
                ticklabel style={font=\tiny},
                legend pos=outer north east,
                legend style={at={(1.45,.85)},anchor=north east},
                legend cell align={left},
                view={135}{25}
            ]
            \addplot3[opacity=.5,surf,shader=interp,colormap={orange}{color=(orange!50) color=(orange!50)},variable=t,variable y=r,domain=0:360,domain y=0:2*pi,restrict z to domain=0:16] ({r*cos(t)},{r*sin(t)},{16-r*r});  % z=16-x^2-y^2
            \addlegendentry{\(z=16-x^2-y^2\)}
            \addplot3[opacity=.5,surf,shader=interp,domain=0:360,domain y=0:16,colormap={green}{color=(green!50) color=(green!50)}] ({3*cos(x)},{3*sin(x)},{y}); % x^2+y^2=9
            \addlegendentry{\(x^2+y^2=9\)}
            %\fill[cyan,opacity=.5] (4,4,0) -- (4,-4,0) -- (-4,-4,0) -- (-4,4,0);    % z=0
            \addplot3[surf,shader=interp,opacity=.5,domain=-4:4, domain y=-4:4,
            colormap={dull}{color=(cyan) color=(cyan)},opacity=0.5] {0};
            \addlegendentry{\(z=0\)}
            \addplot3[mesh,color=gray,ultra thin,domain=0:360,domain y=0:4,z
            buffer=sort,samples=31,samples y=41] 
            ({min(y,3)*cos(x)},{min(y,3)*sin(x)},{16-y*y});
        \end{axis}
    \end{tikzpicture}
\end{center}

\end{document}

enter image description here

? It might make a nice cage for nasty ducks. ;-)

This shows the surface. I do not know what you want to do with the original plots.

\documentclass{article}
\usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}

\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\begin{document}

\begin{center}
    \begin{tikzpicture}
        \begin{axis} [
                axis on top,
                axis lines=center,
                xlabel=$x$,
                ylabel=$y$,
                zlabel=$z$,
                xmin=-5,
                ymin=-5,
                zmin=-1,
                xmax=5,
                ymax=5,
                zmax=17,
                xtick={-4,-3,0,3,4},
                xticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ytick={-4,-3,0,3,4},
                yticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ztick={0,16},
                zticklabels={$0$,$16$},
                ticklabel style={font=\tiny},
                legend pos=outer north east,
                legend style={at={(1.25,.81)},anchor=north east},
                legend cell align={left},
                view={135}{25}
            ]
%             \addplot3[orange,opacity=.5,samples=51,samples y=21,variable=t,variable y=r,domain=0:360,domain y=0:2*pi,restrict z to domain=0:16] ({r*cos(t)},{r*sin(t)},{16-r*r});   % z=16-x^2-y^2
%             \addlegendentry{\(16-x^2-y^2\)}
%             \addplot3[green,opacity=.5,samples=51,samples y=21,variable=t,variable y=r,domain=0:360,domain y=0:2*pi,restrict z to domain=0:16] ({3*cos(t)},{3*sin(t)},{0});   % x^2+y^2=9
%             \addlegendentry{\(x^2+y^2=9\)}
%             \fill[cyan,opacity=.5] (3,3,0) -- (3,-3,0) -- (-3,-3,0) 
%         -- (-3,3,0);    % z=0
%             %\addlegendentry{\(z=0\)}    % This label is not showing because we used 'fill', not 'addplot3'
            % Here should be the code of the intersection of the three surfaces...
            \addplot3[surf,shader=interp,domain=0:360,domain y=0:7,z buffer=sort] 
            ({3*cos(x)},{3*sin(x)},{y});
            \addplot3[data cs=polar,surf,shader=interp,domain=0:360,domain y=0:3] {16-y*y};
        \end{axis}
    \end{tikzpicture}
\end{center}
\end{document}

enter image description here

This looks like a poisonous mushroom to me:

\documentclass{article}
\usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}

\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\begin{document}

\begin{center}
    \begin{tikzpicture}
        \begin{axis} [
                axis on top,
                axis lines=center,
                xlabel=$x$,
                ylabel=$y$,
                zlabel=$z$,
                xmin=-5,
                ymin=-5,
                zmin=-1,
                xmax=5,
                ymax=5,
                zmax=17,
                xtick={-4,-3,0,3,4},
                xticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ytick={-4,-3,0,3,4},
                yticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ztick={0,16},
                zticklabels={$0$,$16$},
                ticklabel style={font=\tiny},
                legend pos=outer north east,
                legend style={at={(1.25,.81)},anchor=north east},
                legend cell align={left},
                view={135}{25}
            ]
%             \addplot3[orange,opacity=.5,samples=51,samples y=21,variable=t,variable y=r,domain=0:360,domain y=0:2*pi,restrict z to domain=0:16] ({r*cos(t)},{r*sin(t)},{16-r*r});   % z=16-x^2-y^2
%             \addlegendentry{\(16-x^2-y^2\)}
%             \addplot3[green,opacity=.5,samples=51,samples y=21,variable=t,variable y=r,domain=0:360,domain y=0:2*pi,restrict z to domain=0:16] ({3*cos(t)},{3*sin(t)},{0});   % x^2+y^2=9
%             \addlegendentry{\(x^2+y^2=9\)}
%             \fill[cyan,opacity=.5] (3,3,0) -- (3,-3,0) -- (-3,-3,0) 
%         -- (-3,3,0);    % z=0
%             %\addlegendentry{\(z=0\)}    % This label is not showing because we used 'fill', not 'addplot3'
            % Here should be the code of the intersection of the three surfaces...
            \addplot3[surf,shader=interp,domain=0:360,domain y=0:7,z
            buffer=sort,colormap=
        {greenblack}{color=(green!50!black) color=(green)}] 
            ({3*cos(x)},{3*sin(x)},{y});
            \addplot3[data cs=polar,surf,shader=interp,domain=0:360,domain y=0:3,color=orange] {16-y*y};
        \end{axis}
    \end{tikzpicture}
\end{center}
\end{document}

enter image description here

You can also draw it in one stretch.

\documentclass{article}
\usepackage[a4paper,margin=1in,footskip=0.25in]{geometry}
\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\begin{document}

\begin{center}
    \begin{tikzpicture}
        \begin{axis} [
                axis on top,
                axis lines=center,
                xlabel=$x$,
                ylabel=$y$,
                zlabel=$z$,
                xmin=-5,
                ymin=-5,
                zmin=-1,
                xmax=5,
                ymax=5,
                zmax=17,
                xtick={-4,-3,0,3,4},
                xticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ytick={-4,-3,0,3,4},
                yticklabels={$-4$,$-3$,$0$,$3$,$4$},
                ztick={0,16},
                zticklabels={$0$,$16$},
                ticklabel style={font=\tiny},
                legend pos=outer north east,
                legend style={at={(1.25,.81)},anchor=north east},
                legend cell align={left},
                view={135}{25}
            ]
            % Here should be the code of the intersection of the three surfaces...
            \addplot3[surf,shader=interp,domain=0:360,domain y=0:4,z
            buffer=sort,colormap=
        {greenorange}{color=(green!80!black) color=(orange)},samples=31,samples
        y=41] 
            ({min(y,3)*cos(x)},{min(y,3)*sin(x)},{16-y*y});
        \end{axis}
    \end{tikzpicture}
\end{center}
\end{document}

enter image description here

  • 1
    It looks very nice! I would like to draw the cylinder in green, the upside-down paraboloid in orange and the plane in cyan. Then we need to show the body, probably using intersections and patterns or fill, I do not which is the best. – manooooh Feb 25 at 15:43
  • @manooooh green-orange .... – user121799 Feb 25 at 15:55
  • It is perfect!! How does ({min(y,3)*cos(x)},{min(y,3)*sin(x)},{16-y*y}) work? Have you used any change of variables? – manooooh Feb 25 at 23:07
  • 1
    @manooooh No, this is just a trick to be able to draw the thing in one stretch. If you move along the z axis from 16 to lower values, the radius of the shape will increas until it hits its maximum value of 3. – user121799 Feb 25 at 23:44
  • 1
    @manooooh This depends only on one variable. Why do you expect a mesh? You only draw a circle in the xy plane. Where is the mesh supposed to be? R U looking 4 \addplot3[mesh,color=gray,ultra thin,domain=0:360,,domain y=0:3, samples=31,samples y=31] ({y*cos(x)},{y*sin(x)},{0});? – user121799 Feb 26 at 1:44

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