11. Quadric Surfaces

We are about to learn how to do derivatives and integrals on surfaces. We need a good collection of surfaces to learn with. There are many fascinating surfaces, and many engineering applications take place on complicated surfaces (studying the aerodynamics of a new airplane design, for example). But a collection of surfaces known as quadric surfaces are a great set to learn calculus with.

A quadric surface is a set of points represented by equations where some of the $\large x$, $\large y$, or $\large z$ variables are squared. They can look cool:

quadric1

We investigate these surfaces using traces. A trace is the intersection of a vertical or horizontal plane with a surface:

trace1

In the picture above, we see the intersection of the quadric surface with the plane $\large x=1$. The intersection is a hyperbola (this surface is called a hyperboloid as a result). Traces help us get an idea of what the surface will look like. Here's a video with an example:

The following image displays the 6 quadric surfaces (the constants get in the way here; what matters is which variables are squared and which are positive/negative):

quadricMap

(Image from https://www.wizeprep.com/online-courses/23904/chapter/6/core/1/6)

By changing which variables are squared and which are positive/negative, we can have versions of each of these surfaces aligned along the $\large x$ or $\large y$ axis instead of the $\large z$ axis.

Tomorrow's class:

Learn the shapes and names for the six quadric surfaces (or seven, if you want to count a sphere as unique rather than a special case of an ellipsoid).
Practice sketching traces and quadric surfaces.