Steiner Trihedral Pairs
We have seen that there exist configurations of tritangent planes
which are parallel (i.e., intersect in a line at infinity).
Let us call two tritangent planes disjoint if they intersect in a line off the surface
(this includes the case of parallel planes
in our affine view of the Clebsch cubic, for instance).
In general, the following is true:
Every pair of disjoint tritangent planes π1,π2
can be completed to a configuration of 6 tritangent planes
π1,π2,π3; γ1,γ2,γ3
with the property that
π1,π2,π3 are three pairwise disjoint tritangent planes
as are γ1,γ2,γ3.
The additional property holds that
πi intersects γj in a line of the surface.
Such a configuration of six planes is called Steiner trihedral pair.
Thus, there are nine lines involved in the definition of a Steiner trihedral pair.
The picture above shows a Steiner trihedral pair.
The planes π1,π2,π3 are the tritangent planes that we know from previous examples (shown in orange).
The planes γ1,γ2,γ3 are shown as pink triangles.
Each γj is the plane determined by three parallel lines in the πi.
The nine lines involved in the trihedral pair are the nine yellow lines that lie in the
tritangent planes (this holds true for both sets of tritangent planes).
It is known that there are exactly 120 trihedral pairs associated with each non-degenerate cubic surface.
The picture above shows just one of them.
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On 4 Jun 2017, 11:23.