Calculus II - Quadric Surfaces

Section 12.4 : Quadric Surfaces

4. Sketch the following quadric surface.

\[{y^2} = 4{x^2} + 16{z^2}\]

Show Solution

This is a cone that is centered on the $y$-axis and because the coefficients of the $x$ and $z$ terms are different the cross sections of the surface 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 ${y^2} = 4{x^2} + 16{z^2}$ drawn with the traditional axes. It is an elliptic cone centered on the y-axis with its point at the origin, opening out in both the positive and negative y directions.$

$A three dimensional sketch of the same elliptic cone ${y^2} = 4{x^2} + 16{z^2}$, this time drawn with “boxed” axes to make the surface easier to visualize. It is centered on the y-axis with its point at the origin and opens out in both directions along that axis.$