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.