Quadric surfaces are important objects in
Multivariable Calculus and Vector Analysis classes. We like them
because they are natural 3D-extensions of the so-called
conics (ellipses, parabolas, and
hyperbolas), and they provide fairly nice surfaces to use
as examples for the rest of your class.
The basic quadric surfaces are described by the following
five equations, where \(A\), \(B\), and \(C\) are constants.
Rather than memorize the equations, you should learn how
to examine cross sections to figure out what surface a
given equation represents, as I will describe below.
Using This Gallery
In this gallery you’ll find interactive pictures 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 first picture below shows a
sphere with an adjustable radius; click and drag the blue
slider to change it. You can rotate the sphere by grabbing
and dragging. You can also zoom in or out using the scroll
wheel on your mouse or the equivalent touch motion.
radius = 2.3
Gridlines:
Surface:
There are additional
controls at the bottom of the picture that
determine how the surface is shown: you can change the
material the sphere is made out of and also show or hide
the gridlines. If you get lost when moving
around, the “Reset view” button will take you
back to where you started.
The next picture shows you various cross
sections of the sphere, that is, the intersection of the
sphere with a plane parallel to one of the coordinate
planes. The picture initially shows the intersection of
the sphere of radius 2 with the three planes \(x = 1 \),
\(y = 1.4\), and \(z = -1.2\).
Click and drag the blue sliders to adjust which plane is
being used to “slice” the sphere to produce
the cross section; while you are holding the slider, the
plane in question appears. As with the previous picture,
you can (and should!) rotate and zoom to get a better view.
x = 1.0
y = 1.0
z = 1.0
A Note about Domains
Sometimes a computer can graph a surface in more than one way.
Look at the two pictures below; they both show graphs of the function
\(z = x^2+y^2\). The first one is probably more familiar, and
it’s what most of us would draw by hand. The second picture can
be useful, though, because the gridlines on the surface show you the
cross sections \(x=c\) and \(y=d\) of the surface.
In technical terms, the two pictures show graphs of the same function
but with different domains. On the left the domain is a disk,
described by
$$
0 \leq x^2 + y^2 \leq 2
$$
On the right, the domain is a square,
$$
\begin{align}
-1 \leq x \leq 1 \\
-1 \leq y \leq 1
\end{align}
$$
In this gallery I’ve drawn a lot of surfaces with square domains
to emphasize the vertical cross sections. I’ve also included buttons
below certain pictures which let you change the domain to a disk. You
might be surprised how different some of the pictures will look when
you change the domain.
In fact, that leads to a good way to gauge how well you understand
the quadric surfaces. On each page you’ll be able to adjust the
coefficients of the equation. Do this with both domains, and see if
you can tell that it affects both graphs in the same way.
Now you’re ready to look at the gallery images. Use the menu at the
top this page to choose which quadric surface you’d like to see!
In 2016, Nathan Dunfield redid
the interactive 3D graphics from scratch using three.js
instead of LiveGraphics3D
so they work on web browsers that lack Java (which by 2016 was most of them). Dunfield also updated
the styling to better support phones and smaller tablets, and switched
to MathJax for displaying the formulas.
