Math Insight

Quadric surfaces

The graphs of functions of two variables are examples of surfaces. More generally, a set of points $(x,y,z)$ that satisfy an equation relating all three variables is often a surface. A simple example is the unit sphere, the set of points which satisfy the equation $x^2+y^2+z^2=1$.

One special class of equations are a set of equations which involve $x$, $y$, $z$, $x^2$, $y^2$, and $z^2$. The graphs of these equations are surfaces known as quadric surfaces. Quadric surfaces are often used as example surfaces since they are relatively simple. There are six different quadric surfaces: the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the double cone, and hyperboloids of one sheet and two sheets.

Quadric surfaces are natural 3D-extensions of the so-called conics (ellipses, parabolas, and hyperbolas), and they provide examples of fairly nice surfaces to use as examples in multivariable calculus

The basic quadric surfaces are described by the following equations, where $A$, $B$, and $C$ are constants. \begin{gather*} z= Ax^2+By^2 \qquad\qquad\qquad z^2=Ax^2+By^2\\ \frac{x^2}{A^2}+\frac{y^2}{B^2} - \frac{z^2}{C^2} = 1 \qquad\qquad -\frac{x^2}{A^2}-\frac{y^2}{B^2} + \frac{z^2}{C^2} = 1\\ \frac{x^2}{A^2}+\frac{y^2}{B^2} + \frac{z^2}{C^2} = 1 \end{gather*}

Rather than memorize the equations, you should learn how to examine cross sections to figure out what surface a given equation represents. Cross sections are cuts through the surface for a given fixed valued of $x$, $y$,or $z$. For example, consider the quadric surface given by the equation $$z=4x^2+9 y^2.$$ To check the horizontal cross-sections, we'd choose values for $z$, such as $z=36$. In this case, $$36=4x^2+9 y^2,$$ or $$\frac{x^2}{9}+\frac{y^2}{4} =1.$$ We see that the cross section in the plane $z=36$ is an ellipse which stretches 3 units in the positive and negative $x$-direction, and 2 units in the positive and negative $y$-direction.

To help you understand the quadric surfaces, we've incorporated the content of the Interactive Gallery of Quadric Surfaces into Math Insight. In these gallery pages, you'll find interactive applets of the quadric surfaces. You can see what the cross sections look like, and also see how various coefficients can affect how they look.

The following quadric surfaces are included in the gallery.