The Cubic Surface Galery
The 'Cubic Surfaces with Singularities' Galery
Cubic surfaces can have singularities, like ordinary double points. In the 'Cubic Surfaces with Singularities' Galery all types of cubic surfaces (in reference to their singularities) are presented.
The 'Famous Cubic Surfaces' Galery
The Clebsch Diagonal
Surface is one of the most famous cubic
surfaces because of its
symmetry. Its equation is given by
x^3+y^3+z^3+w^3 - (x+y+z+w)^3. It is now easy to see, that
this cubic is invariant under permutations on 5 letters (by setting
t:=-(x+y+z+w)). This is the reason for the symmetry of the set of
the corresponding six points in the plane. From this picture, one
sees immediately, that there are ten Eckardt Points.
See the 'Famous Cubic Surfaces' Galery for
more pictures and movies of this and other famous cubics.
The 'Cubics With Double Points' Galery
This is a very symmetric cubic surface. Equation: x^3+y^3+z^3+w^3 - (x+y+0.5z+0.5w)^3 - (0.5z+0.5w)^3 In this equation, it is not so easy to see the fact, that three double points exist. But look at the corresponding set of six points in the plane: (200,600), (400,600), (200,400), (400,400), (300,500), (300,400). From this configuration, one sees imediately, that the surface has three double points (because there are three sets of three points, which are on a line).
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© Oliver Labs, The Cubic Surface Homepage, Algebraic Geometry Group, University of Mainz, Germany |
