Feature Column from the AMS

Quadric Surfaces

Let S be the surface which is the graph of
z = f(x,y) = ax² + 2bxy + ay².
Equivalently,
f(x,y) = (x,y)A(x,y)^t,
where A is the matrix

The character of S depends on the eigenvalues of the matrix A:: If both eigenvalues are positive, S is bowl-shaped, opening up.; If both eigenvalues are negative, S is bowl-shaped, opening down.; If one eigenvalue is positive and the other is negative, S issaddle-shaped.

Here is a picture of S for a particular choice of a andb, together with the eigenvalues of the corresponding matrix A.This picture is a frame from a movie, showing how S changes as Avaries. Clicking on the image will bring up the movie.

- Steven Weintraub