next surface  previous surface  2D curves  3D curves  surfaces  fractals  polyhedra 
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:
next surface  previous surface  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017