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Interactive Gallery of Quadric Surfaces

Jonathan Rogness

Jonathan Rogness is in the Mathematics Department at the University of Minnesota.

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.

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

Paraboloid Equation

Cone Equation

Hyperboloid (1Sh) Equation

Hyperboloid (2Sh) Equation

Ellipsoid Equation

It can be tough to memorize what the graph of each equation is. A better approach is to use cross sections to figure out what surface a given equation represents.



About this Gallery -- read before you go on.


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*x+y*y. The picture on the left is probably more familiar, and it's what most of us would (attempt to) 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 <= x*x + y*y< = 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 that let you change the domain to a disk. You might be surprised how different some of the pictures 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.

Go to the next page  to open the Gallery of Quadric Surfaces


Software Specifications

The mathlets in the gallery will run on any JavaTM-enabled browser. (Click the button at the right if you need to upgrade your Java plug-in.) Make sure Javascript is enabled in your browser if you want to use the domain and gridline controls below the mathlets. Some browsers have bugs in their Javascript implementation that prevent these controls from changing the surface in the pre-existing mathlet; if any errors are detected, the gallery will open a new window with the desired mathlet.


The interactive pictures in the gallery are created with LiveGraphics3D, a Java applet written by Martin Kraus. Some of the design elements were adapted from the freely available code at css/edge.

Published August, 2005
© Copyright 2004 by Jonathan Rogness

Jonathan Rogness, "Interactive Gallery of Quadric Surfaces," Loci (September 2005)


Journal of Online Mathematics and its Applications