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).
ray traced cone
ray traced ellipsoid
ray traced one-sheeted hyperboloid
ray traced paraboloid
ray traced cylinder
ray traced hyperbolic paraboloid
ray traced two sheeted hyperboloid
ray traced sphere