Examining Cross Sections

Most students are a little daunted at the thought of memorizing all the different equations which represent quadric surfaces.  Some of them -- such as the two hyperboloids -- are very tricky to tell apart.  A better approach is to learn the method of examing cross sections to determine what the graph of a given equation looks like.

For example, consider the following equation of a sphere of radius 2:

x2 + y2 + z2= 4

To examine a cross section, we choose a value for one of the three variables, say z=0. Using that value, we take a look at the resulting new equation:

x2 + y2 + 02 = 4,   or   x2 + y2 = 4

You should recognize this as a circle of radius 2, centered at the point (x,y)=(0,0). In other words, the intersection of the plane z=0 and the sphere x2 + y2 + z2 = 4 is a circle of radius 2.  A more intuitive way to think about this is that you take a big meat cleaver and chop the sphere where z=0. If you take one of the resulting half-spheres, what does the edge look like?  A circle of radius two.

Look at the picture on the left to see an illustration of this cross section. You can see where the plane z=0 slices through the sphere. (In the spirit of an interactive gallery, you can click and drag on the picture to rotate it.)

Using this method, you can figure out what any quadric surface looks like.  You choose values for x, y, or z, and see what the resulting cross sections look like.  When you put them all together, you have a picture of the surface.

[an error occurred while processing this directive]