Plotting multivariable functions with dependent variables in domain

Question

Trying to plot a hyperboloid with domain x^2+y^2 <= 50. For now I have set the domain to [-5,5], but this yields ugly plots, see below. Any advice on how to get around this? I tried setting the domain in terms of x and y (and x domain in terms of y and vice versa), but to no avail.

Ideally I would like to apply the same solution to the gradient vector field, should there be one.

\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}
\xdefinecolor{lightgrey}{RGB}{220,220,220}
\xdefinecolor{goldenrod}{RGB}{255,223,66}
\xdefinecolor{newblue}{RGB}{57,106,177}
\xdefinecolor{newred}{RGB}{204,37,41}
\xdefinecolor{newgreen}{RGB}{132,186,91}
\xdefinecolor{newpurple}{RGB}{144,103,167}
\begin{tikzpicture}[scale=1.2]
\begin{axis}[axis equal, view={0}{90}]
\addplot3[surf,shader=interp,opacity=0.5,domain=-5:5]
{(1/100)(50-x^2-y^2)};
\addplot [domain=0:2*pi,samples=50]({cos(deg(x))},{sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({2*cos(deg(x))},{2*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({3*cos(deg(x))},{3*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({4*cos(deg(x))},{4*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({5*cos(deg(x))},{5*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({6*cos(deg(x))},{6*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({7*cos(deg(x))},{7*sin(deg(x))});
\addplot[newpurple,double=newpurple,->] plot coordinates {
    (0,0)
    (-1,3)
};
\addplot[newblue,double=newblue,->] plot coordinates {
    (0,0)
    (7,-1)
};
\addplot[newred,double=newred,->] plot coordinates {
    (7,-1)
    (-1,3)
};
\addplot+[newred,double=newred] plot coordinates {
    (1,2)
};
\addplot[newgreen,double=newgreen] plot coordinates {
    (-3,-6)
    (3,6)
};
\addplot3[blue,/pgfplots/quiver,
    quiver/u=-x/50,
    quiver/v=-y/50,
    quiver/scale arrows=0.1,
    -stealth,samples=10] {1};          
\end{axis}
\end{tikzpicture}
\end{document}

Answer 1

You can use data cs=polar to provide the equations in polar coordinates. Also, you can use \foreach loops to simplify the code:

\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}


\begin{tikzpicture}[scale=1.2]
\begin{axis}[axis equal image, view={0}{90}]
\addplot3 [
    surf,
    shader=interp,
    opacity=0.5,
    domain=0:360,
    y domain=0:7,
    samples=100, samples y=7,
    data cs=polar
] {(50-y^2)};

\foreach \i in {1,...,7}{
    \addplot [domain=0:360,samples=100, data cs=polar] {\i};
}

\draw [purple, ultra thick, ->] (axis cs:0,0) -- (axis cs:-1,3);
\draw [blue, ultra thick, ->] (axis cs:0,0) -- (axis cs:7,-1);
\draw [red, ultra thick, ->] (axis cs:7,-1) -- (axis cs:-1,3);

\addplot3[blue,/pgfplots/quiver,
    quiver/u=-cos(x),
    quiver/v=-sin(x),
    quiver/scale arrows=0.1,
    -stealth,samples=10,
    domain=0:360,
    y domain=0:sqrt(50),
    data cs=polar] {1};          
\end{axis}
\end{tikzpicture}
\end{document}

---

If you want to stick with cartesian coordinates for specifying the function, you can limit the range of the color map using point meta min=0, and cover the unwanted bits of the surface plot using \fill [white] (rel axis cs:0,0) rectangle (rel axis cs:1,1) (axis cs:0,0) circle [radius=7]; (this requires \pgfplotsset{compat=1.4} or higher).

\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.9}
\begin{document}
\xdefinecolor{lightgrey}{RGB}{220,220,220}
\xdefinecolor{goldenrod}{RGB}{255,223,66}
\xdefinecolor{newblue}{RGB}{57,106,177}
\xdefinecolor{newred}{RGB}{204,37,41}
\xdefinecolor{newgreen}{RGB}{132,186,91}
\xdefinecolor{newpurple}{RGB}{144,103,167}
\begin{tikzpicture}[scale=1.2]
\begin{axis}[axis equal, view={0}{90}]
\addplot3[surf,shader=interp,opacity=0.5,domain=-7:7, point meta min=1]
{(50-x^2-y^2)};
\addplot [domain=0:2*pi,samples=50]({cos(deg(x))},{sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({2*cos(deg(x))},{2*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({3*cos(deg(x))},{3*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({4*cos(deg(x))},{4*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({5*cos(deg(x))},{5*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({6*cos(deg(x))},{6*sin(deg(x))});
\addplot [domain=0:2*pi,samples=50]({7*cos(deg(x))},{7*sin(deg(x))});
\addplot[newpurple,double=newpurple,->] plot coordinates {
    (0,0)
    (-1,3)
};
\addplot[newblue,double=newblue,->] plot coordinates {
    (0,0)
    (7,-1)
};
\addplot[newred,double=newred,->] plot coordinates {
    (7,-1)
    (-1,3)
};
\addplot+[newred,double=newred] plot coordinates {
    (1,2)
};
\addplot[newgreen,double=newgreen] plot coordinates {
    (-3,-6)
    (3,6)
};
\addplot3[blue,/pgfplots/quiver,
    quiver/u=-x/50,
    quiver/v=-y/50,
    quiver/scale arrows=0.1,
    -stealth,samples=10] {1};

\fill [white] (rel axis cs:0,0) rectangle (rel axis cs:1,1) (axis cs:0,0) circle [radius=7];         
\end{axis}
\end{tikzpicture}
\end{document}