#math/calculus
Before differentiating surfaces, we need a small vocabulary of curves. A `conic section` is obtained by intersecting a double cone with a plane.
![[Pasted image 20260928224439.png|350]]
The same curves can be defined without the cone:
- An `ellipse` is the set of points $P$ for which $|PF_1|+|PF_2|$ is constant.^[$F_{1},F_{2}$ are called the `foci`, the plural form of focus.]
- A `hyperbola` is the set for which $\big||PF_1|-|PF_2|\big|$ is constant.^[I.e. the difference between distances to $F_{1}$ and $F_{2}$ is the same.]
- A `parabola` is the set for which the distance to a focus equals the distance to a `directrix`.
- A `circle` is the special ellipse whose two foci coincide.
![[tikz-9e30543a6e4f.svg]]
%% tikz-source
```latex
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[>=Stealth,scale=.82,line cap=round,font=\small]
\begin{scope}
\draw[->,black!45] (-2.2,0)--(2.2,0) node[right]{$x$};
\draw[->,black!45] (0,-1.65)--(0,1.65) node[above]{$y$};
\draw[blue!70!black,very thick] (0,0) ellipse (1.75 and 1.05);
\fill[red!75!black] (-1.4,0) circle (2pt) node[below]{$F_1$};
\fill[red!75!black] (1.4,0) circle (2pt) node[below]{$F_2$};
\node at (0,-2.05) {ellipse: sum fixed};
\end{scope}
\begin{scope}[xshift=6cm]
\draw[->,black!45] (-2.2,0)--(2.2,0) node[right]{$x$};
\draw[->,black!45] (0,-1.65)--(0,1.65) node[above]{$y$};
\draw[dashed,black!35] (-2,-1.25)--(2,1.25);
\draw[dashed,black!35] (-2,1.25)--(2,-1.25);
\draw[blue!70!black,very thick,domain=-1.25:1.25,samples=70]
plot ({cosh(\x)},{.7*sinh(\x)});
\draw[blue!70!black,very thick,domain=-1.25:1.25,samples=70]
plot ({-cosh(\x)},{.7*sinh(\x)});
\fill[red!75!black] (-1.22,0) circle (2pt) node[below]{$F_1$};
\fill[red!75!black] (1.22,0) circle (2pt) node[below]{$F_2$};
\node at (0,-2.05) {hyperbola: difference fixed};
\end{scope}
\begin{scope}[xshift=12cm]
\draw[->,black!45] (-2.2,0)--(2.2,0) node[right]{$x$};
\draw[->,black!45] (0,-1.65)--(0,1.65) node[above]{$y$};
\draw[blue!70!black,very thick,domain=-1.7:1.7,samples=60]
plot ({\x},{.5*\x*\x-.5});
\draw[red!70!black,dashed,thick] (-2,-1)--(2,-1)
node[right]{directrix};
\fill[red!75!black] (0,0) circle (2pt) node[right]{$F$};
\node at (0,-2.05) {parabola: equal distances};
\end{scope}
\end{tikzpicture}
\end{document}
```
%%
These are geometric definitions: translating or rotating the coordinate axes changes the equation, not the curve. [[14A2 - Standard Forms and Eccentricity|Standard forms]] turn that geometry into formulas; later, [[14D2 - Cones, Cylinders, and Degenerate Quadrics|slicing surfaces]] returns us to the three-dimensional origin of the conics.