Fermat cubic

3D model of Fermat cubic (real points)

In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by

x^{3}+y^{3}+z^{3}=1.\

Methods of algebraic geometry provide the following parameterization of Fermat's cubic:

x(s,t)={3t-{1 \over 3}(s^{2}+st+t^{2})^{2} \over t(s^{2}+st+t^{2})-3}

y(s,t)={3s+3t+{1 \over 3}(s^{2}+st+t^{2})^{2} \over t(s^{2}+st+t^{2})-3}

z(s,t)={-3-(s^{2}+st+t^{2})(s+t) \over t(s^{2}+st+t^{2})-3}.

In projective space the Fermat cubic is given by

w^{3}+x^{3}+y^{3}+z^{3}=0.

The 27 lines lying on the Fermat cubic are easy to describe explicitly: they are the 9 lines of the form (w : aw : y : by) where a and b are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates.

Real points of Fermat cubic surface.

References

Ness, Linda (1978), "Curvature on the Fermat cubic", Duke Mathematical Journal, 45 (4): 797–807, doi:10.1215/s0012-7094-78-04537-4, ISSN 0012-7094, MR 0518106
Elkies, Noam. "Complete cubic parameterization of the Fermat cubic surface".