#math/calculus
The section method reveals that second-degree equations in three variables repeatedly produce the same few shapes. A `quadric surface` is the zero set of a quadratic polynomial in $x,y,z$. After suitable shifts and rotations, the main nondegenerate forms are:
| Surface | Standard form |
|---|---|
| Ellipsoid | $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ |
| Elliptic paraboloid | $z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ |
| Hyperbolic paraboloid | $z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$ |
| One-sheet hyperboloid | $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ |
| Two-sheet hyperboloid | $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ |
| Elliptic cone | $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$ |
Signs tell the family; sections tell the shape. For the saddle $z=x^2-y^2$, $y=0$ opens upward while $x=0$ opens downward.
![[tikz-d1ed0e611053.svg]]
%% tikz-source
```latex
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[>=Stealth,scale=.95,line cap=round]
\draw[->,black!45] (-4.2,0)--(4.4,0) node[right]{$x$};
\draw[->,black!45] (0,-3.0)--(0,3.2) node[above]{$y$};
\foreach \c in {.45,1.1,2.0}{
\draw[blue!65!black,thick,domain=-2.7:-sqrt(\c),samples=55]
plot ({\x},{sqrt(max(0,\x*\x-\c))});
\draw[blue!65!black,thick,domain=sqrt(\c):2.7,samples=55]
plot ({\x},{sqrt(max(0,\x*\x-\c))});
\draw[blue!65!black,thick,domain=-2.7:-sqrt(\c),samples=55]
plot ({\x},{-sqrt(max(0,\x*\x-\c))});
\draw[blue!65!black,thick,domain=sqrt(\c):2.7,samples=55]
plot ({\x},{-sqrt(max(0,\x*\x-\c))});
\draw[red!65!black,thick,domain=-2.7:-sqrt(\c),samples=55]
plot ({sqrt(max(0,\x*\x-\c))},{\x});
\draw[red!65!black,thick,domain=sqrt(\c):2.7,samples=55]
plot ({sqrt(max(0,\x*\x-\c))},{\x});
\draw[red!65!black,thick,domain=-2.7:-sqrt(\c),samples=55]
plot ({-sqrt(max(0,\x*\x-\c))},{\x});
\draw[red!65!black,thick,domain=sqrt(\c):2.7,samples=55]
plot ({-sqrt(max(0,\x*\x-\c))},{\x});
}
\draw[black!60,thick] (-2.8,-2.8)--(2.8,2.8);
\draw[black!60,thick] (-2.8,2.8)--(2.8,-2.8);
\node[blue!65!black] at (2.35,1.0) {$z>0$};
\node[red!65!black] at (1.0,2.35) {$z<0$};
\node[align=left] at (6.5,-.4)
{level curves of $z=x^2-y^2$\\the sign flips across $y=\pm x$};
\end{tikzpicture}
\end{document}
```
%%
The saddle foreshadows [[16C2 - Hessian and the Second Derivative Test|indefinite curvature]]: different directions can bend in opposite ways.