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Quadric Surfaces

 Quadric Surfaces

Let S be the surface which is the graph of
z = f(x,y) = ax2 + 2bxy + ay2.

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.

This frame and movie are © 1998 Juha Haataja and the Center for ScientificComputing, Finland.

- Steven Weintraub