TikZ - 绘制椭圆抛物面

Question

我想使用 TikZ 绘制函数 f(x,y)=2x^2+3y^2-2xy-4x+7y 的图形。

我尝试了以下操作：

\documentclass{article}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
  \begin{axis}[axis lines=middle,xmin=-10,ymin=-12,xmax=10,ymax=10,zmin=-30,zmax=490,ticks=none]
    \addplot3 [surf,samples=25,data cs=polar, domain=0:360, y domain=0:10] {2*y^2*cos(x)*cos(x)+3*y^2*sin(x)*sin(x)-2*y^2*sin(x)*cos(x)-4*y*cos(x)+7*y*sin(x)};
\end{axis}
\end{tikzpicture}
\end{document}

这让我

但我想获得一些“更干净”的东西，例如：

我的尝试是基于这个答案问题，但由于我的函数不对称，所以效果不太好。谢谢你的帮助！

Answer 1

\documentclass{article}
\usepackage{amsmath} % only for the blablabla before the plot
\usepackage{pgfplots}
\pgfplotsset{compat=1.15}
\usetikzlibrary{calc}

\begin{document}
Your function can be plotted symmetrically in the coordinate system defined by
the principal axes of the ellipse. Your function is minimized at $x=x_0=19/10$ and 
$y=y_0=9/5$ with $f(x_0,y_0)=-10.1=:z_0$. So the first step is a translation
\[ x'~=~x-x_0\;,\quad y'~=~y-y_0\quad\text{and}\quad
z'~=~z-z_0\;.\]
The second step is to rotate to the principal axes, which can be achieved by
diagronalizing the Hesse matrix of $f$,
\[
 H~=~\begin{pmatrix} 4 &  -2\\ -2 &  6\end{pmatrix}\;.
\]
This yields a rotation angle of $31.8^\circ$. The perhaps simplest way to
produce the figure is thus to plot the function in the principal axes system and
draw the unrotated axes by hand.
\begin{center}
\begin{tikzpicture}
\begin{axis}
[%view={135}{45},%colormap/blackwhite, 
axis equal,
width=12cm,
axis lines=center, axis on top,
axis line style={draw=none},
ticks=none,
set layers=default,
domain=0:1.50,
samples=20, % this was 200, but I changed it to 20 because of my slow PC
z buffer=sort,
]
\addplot3 [surf,shader=interp,opacity=0.8,
domain y=0:180] ({1.62*cos(y)*sqrt(x)},{sin(y)*sqrt(x)},{x});
\addplot3 [surf,shader=interp,opacity=0.8,
domain y=-180:0,on layer=axis foreground] ({1.62*cos(y)*sqrt(x)},{sin(y)*sqrt(x)},{x});
\coordinate (O) at (axis cs: 0,0); 
\def\AxLen{2.5}
\coordinate (oriX) at (axis cs: {\AxLen*cos(-31.8)},{\AxLen*sin(-31.8)});
\coordinate (oriY) at (axis cs: {-\AxLen*sin(-31.8)},{\AxLen*cos(-31.8)});
\coordinate (oriZ) at (axis cs: 0,0,\AxLen*1);
\coordinate (realO) at ($-{19/10}*(oriX)-{9/5}*(oriY)$);
\end{axis}
\draw[-latex] (O) -- (oriX) node[right]{$x'$};
\draw[-latex] (O) -- (oriY) node[below]{$y'$};
\draw[-latex] (O) -- (oriZ) node[right]{$z'$};
\end{tikzpicture}
\end{center}
\end{document}

编辑：小布局，并替换rotated为unrotated。