使用 tikzpicture 环境在圆内的三维空间中绘制图形场

Question 1

我稍微改变了函数的定义，使其仅存在于圆中x^2 + y^2 <= 4。我还改变了阴影，因为我认为这样形状会更清晰一些。容量问题可以通过分别更改x和y（此图中的t和r）的采样数来解决。非常感谢@marmot 提出使用极坐标的想法。

\addplot3[surf,mesh/ordering=y varies,shader=faceted,samples=72,samples y=20,variable=t,variable y=r,domain=0:360,domain y=0:2,line join=round] ({r*cos(t)},{r*sin(t)},{r*r*r*sin(t)*cos(t)*cos(t)-r*sin(t)});

我使用r和t表示 ρ 和 θ 只是为了便于阅读。圆也可以用极坐标绘制，使用

\addplot3[samples=30,smooth,variable=t,domain=0:360] ({2*cos(t)},{2*sin(t)},2);

我还添加了axis on top冗余选项opacity=0.9，但这并没有提供适当的 3D 效果。我认为这应该是另一个问题。使用键可以更改视图view={}{}，我发现view={110}{25}效果非常好。

完整代码：

\documentclass[tikz,margin=2mm]{standalone}
\usepackage{amssymb}

\usepackage{pgfplots}
\usetikzlibrary{arrows.meta, patterns}
\pgfplotsset{compat=1.8}
\pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
\pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}
\usepackage{amsmath}

\usepackage{mathtools}

\begin{document}

\begin{tikzpicture}[scale=1.5]
        \begin{axis}[
            legend pos=outer north east,
            axis lines = center,
            label style={font=\tiny},
            legend style={font=\tiny},
            xticklabel style = {font=\tiny},
            yticklabel style = {font=\tiny},
            zticklabel style = {font=\tiny},
            xlabel = $x$,
            xmin=-4,
            ymin=-4,
            zmin=-4,
            xmax=4,
            ymax=4,
            zmax=4,
            ylabel = $y$,
            zlabel = $z$,
            domain=-3:3,
            domain y=-3:3,
            view={110}{25},
            axis on top,
        ]
            % Field
%            \addplot3[opacity=0.9,surf,mesh/ordering=y varies,shader=interp,samples=60] {y*x*x-y};
            \addplot3[surf,mesh/ordering=y varies,shader=faceted,samples=72,samples y=20,variable=t,variable y=r,domain=0:360,domain y=0:2,line join=round] ({r*cos(t)},{r*sin(t)},{r*r*r*sin(t)*cos(t)*cos(t)-r*sin(t)});
            \addlegendentry{$f(x,y)$}
            % This planes should be circles...
            \addplot3[samples=30,smooth,variable=t,domain=0:360] ({2*cos(t)},{2*sin(t)},2);
            \addplot3[samples=30,smooth,variable=t,domain=0:360] ({2*cos(t)},{2*sin(t)},-2);
            % Points:
            \addplot3[soldot,black] coordinates {(sqrt(3),1,2)} node[above left] {\tiny $(\sqrt 3,1,2)$};
            \addplot3[soldot,black] coordinates {(-sqrt(3),1,2)} node[above right] {\tiny $(-\sqrt 3,1,2)$};
            \addplot3[soldot,black] coordinates {(sqrt(3),-1,-2)} node[below left] {\tiny $(\sqrt 3,-1,-2)$};
            \addplot3[soldot,black] coordinates {(-sqrt(3),-1,-2)} node[below right] {\tiny $(-\sqrt 3,-1,-2)$};
    \end{axis}
\end{tikzpicture}

\end{document}

当然结果是：

免责声明：我没有解决点的位置及其标签放置问题，我将把这个问题留给你:)

编辑
根据要求，当从正 z 轴查看并且轴限值较小时，点标签会被剪裁。使用clip=false可防止出现这种情况。

代码：

\documentclass[tikz,margin=2mm]{standalone}
\usepackage{amssymb}

\usepackage{pgfplots}
\usetikzlibrary{arrows.meta, patterns}
\pgfplotsset{compat=1.8}
\pgfplotsset{soldot/.style={color=black,only marks,mark=*}}
\pgfplotsset{holdot/.style={color=red,fill=white,very thick,only marks,mark=*}}
\usepackage{amsmath}

\usepackage{mathtools}

\begin{document}

\begin{tikzpicture}[scale=1.5]
        \begin{axis}[
            legend pos=outer north east,
            axis lines = center,
            label style={font=\tiny},
            legend style={font=\tiny},
            xticklabel style = {font=\tiny},
            yticklabel style = {font=\tiny},
            zticklabel style = {font=\tiny},
            xlabel = $x$,
%            xmin=-4,
%            ymin=-4,
            xmin=-2.5,
            ymin=-2.5,
            zmin=-4,
%            xmax=4,
%            ymax=4,
            xmax=2.5,
            ymax=2.5,
            zmax=4,
            ylabel = $y$,
            zlabel = $z$,
            domain=-3:3,
            domain y=-3:3,
%            view={110}{25},
            view={0}{90},
            axis on top,
            clip=false,
        ]
            % Field
%            \addplot3[opacity=0.9,surf,mesh/ordering=y varies,shader=interp,samples=60] {y*x*x-y};
            \addplot3[surf,mesh/ordering=y varies,shader=faceted,samples=72,samples y=20,variable=t,variable y=r,domain=0:360,domain y=0:2,line join=round] ({r*cos(t)},{r*sin(t)},{r*r*r*sin(t)*cos(t)*cos(t)-r*sin(t)});
            \addlegendentry{$f(x,y)$}
            % This planes should be circles...
            \addplot3[samples=30,smooth,variable=t,domain=0:360] ({2*cos(t)},{2*sin(t)},2);
            \addplot3[samples=30,smooth,variable=t,domain=0:360] ({2*cos(t)},{2*sin(t)},-2);
            % Points:
            \addplot3[soldot,black] coordinates {(sqrt(3),1,2)} node[above right] {\tiny $(\sqrt 3,1,2)$};
            \addplot3[soldot,black] coordinates {(-sqrt(3),1,2)} node[above left] {\tiny $(-\sqrt 3,1,2)$};
            \addplot3[soldot,black] coordinates {(sqrt(3),-1,-2)} node[below right] {\tiny $(\sqrt 3,-1,-2)$};
            \addplot3[soldot,black] coordinates {(-sqrt(3),-1,-2)} node[below left] {\tiny $(-\sqrt 3,-1,-2)$};
    \end{axis}
\end{tikzpicture}

\end{document}

结果：

Answer