Fermat cubic



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


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


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

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 &minus;1, and their 18 conjugates under permutations of coordinates.


 * Real points of Fermat cubic surface.