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CROSSCAP
Surface studied by Steiner???
Other name: mitre. 
Cartesian parametrization #1:
with .
Cartesian parametrization #2:
so that ,
Cartesian parametrization #3:, obtained for . Cartesian parametrization #4:,
obtained
for
.

The crosscap is the image of the quotient sphere with
antipodal points identified (i.e. the real
projective plane), by the map: .
The crosscap is one of the simplest immersions
of the real projective plane
into .
It has only one segment line of selfintersection that ends with two cuspidal points (here O and (0, 0, a)) (compare to the Roman surface and the Boy surface, which are two other immersions of the projective plane). 

The figure opposite illustrates the fact that the crosscap is a model of the projective plane: 
Start with a holed sphere (homeomorphic to the disk), and stick edge to edge a with a and b with b, to form the segment line of selfintersection 
Another construction, starting from a disk with its edge twisted to a figureeight, with already a selfintersection. 

The crosscap also has interesting geometrical properties. In particular, it is a reunion of a family of ellipses, in three different ways:
First family (cf. parametrization
#2: the sections by the planes containing Oz
with polar angle q are the ellipses,
with secondary vertices (0, 0, a) and ,
and constant majoraxis equal to 2a, and with equation:



If the previous ellipses are replaced by circles, we
get a circled surface with cylindrical
parametrization: .
This latter surface, homeomorphic to the previous one, is the image by inversion of the Plücker conoid of order 1, the lines of the conoid becoming the circles of this crosscap. 

Second family (cf. parametrization
#3): the sections by the planes containing Oy
are the ellipses:;
their majoraxis is constant equal to a, and their minoraxis oscillates
between 0 and a (the zero case corresponding to the selfintersection
segment). This proves that the crosscap is a special case of sine
torus.
In blue, the locus of the vertices. 


Third family (cf. parametrization #4): the sections by the planes are the ellipses: , with principal vertex (0,0,a). 




Here is a polyhedral version of the crosscap.
Be careful, this is not a true polyhedron: the double central edge is common to 4 faces. 
The crosscap must not be mistaken for the pseudo crosscap:
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© Robert FERRÉOL 2017