Quadric Surfaces Quadric Surfaces
Let S be the surface which is the graph of z = f(x,y) = ax^{2} + 2bxy + ay^{2}.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 bowlshaped, opening up.
 If both eigenvalues are negative, S is bowlshaped, opening down.
 If one eigenvalue is positive and the other is negative, S issaddleshaped.
Here is a picture of S for a particular choice of a and b, 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

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