Interactive Gallery of Quadric Surfaces

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 examples of fairly nice surfaces to use as examples for the rest of your class.

The basic quadric surfaces are described by the following equations, where A, B, and C are constants.

Paraboloid EquationCone Equation
Hyperboloid (1Sh) EquationHyperboloid (2Sh) Equation
Ellipsoid Equation

Rather than memorize the equations, you should learn how to examine cross sections to figure out what surface a given equation represents.


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.  These pictures all include "sliders," like the following two examples.  The picture on the left shows a sphere with an adjustable radius; click and drag the blue dot to change it.  The picture on the right shows you the various cross sections of the sphere.  Click and drag the blue dots to adjust the chosen values for x, y, and z.  You can also rotate the pictures and zoom in or out.

Note: because of a bug in some browsers (particularly older versions of Netscape, Mozilla or Firefox), the images below sometimes fail to display properly when you first load the page; if you see technical error messages instead of pictures, click the "Reload" button on your browser.
Rotate: Click and Drag
Zoom: Shift + Click and Drag Up or Down
Restore Image: Home key
Click and Drag the Blue Dots to Adjust Values

Once you've gotten the hang of these two demos, you're ready to look at the gallery images.  Use the menu at upper left portion of this page to choose which quadric surface you'd like to see.

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 = x2+y2. The one on the left is probably more familiar, and it's what most of us would draw by hand. The picture on the right can be useful, though, because the gridlines on the surface show you the cross sections x=c and y=c of the surface.

Paraboloid Paraboloid

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 ≤ x2+y2 ≤ 2

On the right, the domain is a square,

-1 ≤ x ≤ 1
-1 ≤ y ≤ 1

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.

Technical and Legal Details


Valid HTML 4.01!

This gallery appeared in the 2005 volume of Journal of Online Mathematics, an electronic journal of the Mathematical Association of America.

This entire site was built with LiveGraphics3D, which in turn is made for use with Mathematica; if you are interested in the specifics of how the images were constructed, let me know. Some of the design elements were adapted from the freely available code at css/edge.

This material is Copyright 2004 by Jonathan Rogness, and licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License.  You can find more information on this license at  Please let me know if you make use of this site, or if you have any suggestions.  Thanks!

Jonathan Rogness

The views and opinions expressed in this site are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota.

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