The Fermat Surface x^3+y^3+z^3+1=0
The picture of the surface below shows the 3 real lines in red (out of
27 total on the cubic surface) and in magenta shows the intersection
of a plane containing y+z=0 with the surface. Moving the cursor into the box
will start the surface rotating in space. By clicking and dragging along a
line the picture will rotate about an axis perpendicular to the line,
and will continue rotating when the mouse button is released.
If you point at the magenta dot a small square will form around it
and by clicking on the dot and dragging one changes the plane, and hence the
magenta conic of intersection. The conic goes through the point on the surface
with the same (y,z) coordinate as the magenta point. Best results are obtained
it the magenta dot remains in the plane x=2 (and y and z are positive).
In other positions one sees the problem with parameterizing by rational
functions, since errors are magnified if the denominator of the rational
function is close to zero.
Pressing shift and the mouse button allows Zooming in or out.