Geometric Version of Shapiro Conjecture 6
Set p:=d+1-k and m:=k.  

Then H is a p-plane, and
G(t ) is the m-plane osculating the rational normal curve g(t)   =   (1, t, t 2, ..., t m+p-1 ) .

    The geometric condition on the p-plane H
G(s )     H   {0}
states that H meets the m-plane osculating the rational normal curve at the point g(s).

Geometric Version of Shapiro Conjecture
    Each of the (finitely many) p-planes meeting mp different m-planes osculating the rational normal curve in m+p space at real points is real.

    Consider the case when m=p=2 and work in projective 3-space.
This asserts that only real lines meet 4 lines tangent to the rational normal curve at real points.
We now give a geometric demonstration of this case of the Shapiro conjecture.