Contour diagrams and cross-sections [9.2]
Anh Dong, Step rice terraces in
Xã Cao Phạ, Yen Bai, Vietnam.
The lines on a contour/topographic map are also called "level curves" because all the points on a line are at the same height (same $z$ value). Here the terrace edges appear as visual, literal, and cartographical "level curves" for the landscape.
- $x$- and $y$-traces.
- Using GeoGebra to plot $f(x,y)$ as a 3-d surface.
- An $x=k$-trace as the intersection of the plane $x=k$ with the surface $f(x,y)$.
- Contours, or level-curves as the "$z=k$ traces" of a surface.
Contour diagrams
The lines on a contour/topographic map are called "level curves" because all the points on a labelled line are at the same height (same $z$ value).
In this sense one of the level curves of a surface consists of all the points on a surface that share the same $z$ value, let's say $z=2$.
But $z=2$ is also the equation for a plane that's parallel to the $x$-$y$ plane, intersecting the $z$ axis at $z=2$. An equation for the level curve for $f(x,y)=2$ is obtained by looking at the cross-section of the surface with the plane $z=2$. (You can sometimes solve for $y$ in terms of $x$...) This is a horizontal cross section.
You can also explore the surface with vertical cross sections. We just called these "cross sections" in Lab 03: In Geogebra, graph your surface by defining $f(x,y)$, and then graph a plane such as $y=3$. The intersection is the same as the graph of $f(x,3)$ vs $x$.