Английская Википедия:Fermat cubic

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Файл:3D model of Fermat cubic.stl

3D model of Fermat cubic (real points)

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

<math> x^3 + y^3 + z^3 = 1. \ </math>

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

<math> x(s,t) = {3 t - {1\over 3} (s^2 + s t + t^2)^2 \over t (s^2 + s t + t^2) - 3} </math>

<math> 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} </math>

<math> z(s,t) = {-3 - (s^2 + s t + t^2) (s + t) \over t (s^2 + s t + t^2) - 3}. </math>

In projective space the Fermat cubic is given by

<math>w^3+x^3+y^3+z^3=0.</math>

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.

Файл:FermatCubicSurface.PNG

Real points of Fermat cubic surface.

References

• Шаблон:Citation
• Шаблон:Cite web

Шаблон:Algebraic-geometry-stub