,
with 
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VERONESE SURFACE
| Giuseppe Veronese (1854-1917): Italian mathematician.
See also Wikipedia, BERGER, page 47. |
| Cartesian parametrization: ,
with . |
The Veronese surface is the image of the quotient
of the 2-dimensional sphere by the antipodal relation (in other words,
the real projective plane),
by the map: 
.
Since this function is injective, the Veronese surface
is a surface (i.e. a 2-manifold) without singular points embedded in 
(since it is included in the hyperplane 
of 
) and
homeomorphic to the real projective plane.
The "projection" 
defines a homeomorphism of the Veronese surface onto its image, which is
therefore an embedding of the real projective plane in 
.
However, all the "projections" of this type of the surface
in 
, called
Steiner surfaces, have singular
points:
For example, the "projection" 
maps the Veronese surface onto the Roman
surface, and the projection 
maps it onto the cross-cap.
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© Robert FERRÉOL
2017