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 elliptic cone, and hyperboloids of one and two sheets.
One way to analyze surfaces is through cross sections, which 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.
Rather than include a long section in this page where we work out the cross sections of all the quadric surfaces, we instead send you to an interactive gallery of quadric surfaces, where you will learn how to analyze these surfaces and understand their properties.
