# Quadrics

The following pictures were generated with the Ray tracer Povray. I wrote
the files in 2002 for a Math 21a course at Harvard. Povray has since then stayed stable
(while quite a few other 3D programs have been abandoned or purposely shelved
by companies). Povray is nice because it describes polynomials effectively.
The ellipsoid is poly {2, <1,0,0,0,1,0,0,1,0,-1>
The paraboloid is poly {2, <1,0,0,0,0,0,-1,1,0,-1>.
The hyperboloid is poly {2, <1,0,0,0,-1,0,0,1,0,-1>
It uses the exact format x.B.x + A.x -b = 0 as we defined quadratic
manifolds. Now for a symmetric 3x3 matrix, there are 6 numbers to give
and for the matrix A, there are 3 numbers and for b an other number. That is
why the format poly{2,...} has 10 arguments. Now you know why mathematicians
love this so much. By the way, the program can also handle higher order
polynomials. There are just much more arguments to define the polynomials.
The pictures illustrate how important it is to look at the traces, the
intersections of the surface with the coordinate planes.
Click onto the picture to see it large (2400x1800 pixels).