#math/calculus
A surface is hard to sketch all at once, but every plane cut produces a familiar curve. The `section method` reconstructs the three-dimensional shape from several of these two-dimensional slices.
For an equation $F(x,y,z)=0$:
1. note symmetries;
2. set $x$, $y$, or $z$ equal to convenient constants;
3. draw the resulting plane curves in their actual planes;
4. interpolate smoothly between them.
![[tikz-08136dc583b0.svg]]
%% tikz-source
```latex
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[>=Stealth,scale=1.0,line cap=round]
\draw[->,black!45] (0,0)--(4.3,-1.7) node[right]{$x$};
\draw[->,black!45] (0,0)--(-3.3,-1.7) node[left]{$y$};
\draw[->,black!45] (0,0)--(0,5.0) node[above]{$z$};
\foreach \h/\a in {.55/.70,1.5/1.18,2.7/1.58,4.1/1.95}{
\draw[blue!65!black,thick] (0,\h) ellipse [x radius={1.55*\a},y radius={.48*\a}];
\draw[dashed,black!28] (-2.5,\h)--(2.5,\h);
}
\draw[teal!70!black,very thick] (0,.05) .. controls (1.3,1.1) and (2.1,3.2) .. (3.05,4.1);
\draw[teal!70!black,very thick] (0,.05) .. controls (-1.3,1.1) and (-2.1,3.2) .. (-3.05,4.1);
\node[align=left] at (5.4,3.3)
{$z=x^2+y^2$\\$z=c$: circles\\$x=0,y=0$: parabolas};
\end{tikzpicture}
\end{document}
```
%%
For $z=x^2+y^2$:
- $z=c$ gives circles $x^2+y^2=c$;
- $x=0$ gives $z=y^2$;
- $y=0$ gives $z=x^2$.
The sections determine an upward elliptic paraboloid. If a variable is absent, every slice perpendicular to that axis is identical: $x^2+y^2=25$ is a circular cylinder parallel to the $z$ axis.
Sections lead directly to the catalog of [[14D1 - Quadric Surfaces|quadric surfaces]].