Fermat Surface

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:
:$x(s,t) =$

:$y(s,t) =$

:$z(s,t) = .$

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

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Algebraic surfaces

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