Calculus III - Quadric Surfaces

Section 12.4 : Quadric Surfaces

1. Sketch the following quadric surface.

\[\frac{{{y^2}}}{9} + {z^2} = 1\]

Show Solution

This is a cylinder that is centered on the $x$-axis. The cross sections of the cylinder will be ellipses.

Make sure that you can “translate” the equations given in the notes to the other coordinate axes. Once you know what they look like when centered on one of the coordinates axes then a simple and predictable variable change will center them on the other coordinate axes.

Here are a couple of sketches of the region. We’ve given them with the more traditional axes as well as “boxed” axes to help visualize the surface.

$A three dimensional sketch of the surface $\frac{{{y^2}}}{9} + {z^2} = 1$ drawn with the traditional axes. It is an elliptic cylinder lying along the x-axis, with an elliptical cross section that is 3 units across in the y direction and 1 unit in the z direction.$

$A three dimensional sketch of the same elliptic cylinder $\frac{{{y^2}}}{9} + {z^2} = 1$, this time drawn with “boxed” axes to make the surface easier to visualize. It lies along the x-axis with an elliptical cross section that is wider in the y direction than it is tall in the z direction.$