A surface such as the Möbius strip or Klein bottle (Gray 1997, pp. 322-323) on which there exists a closed path such that the directrix is reversed when moved around this path. The real projective plane is also a nonorientable surface, as are the Boy surface, cross-cap, and Roman surface, all of which are homeomorphic to the real projective plane (Pinkall 1986).
There is a general method for constructing nonorientable surfaces which proceeds as follows (Banchoff 1984, Pinkall 1986). Choose three homogeneous polynomials of positive even degree and consider the map
 |
(1)
|
Then restricting 
,
,
and 
to the surface of a sphere by writing
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
and restricting 
to 
and 
to ![[0,pi/2]](/images/equations/NonorientableSurface/Inline16.svg)
defines a map of the real projective plane
to 
.
In three dimensions, there is no unbounded nonorientable surface which does not intersect itself (Kuiper 1961, Pinkall 1986).
." Comment. Math. Helv. 35, 85-92, 1961.Pinkall,
U. "Models of the Real Projective Plane." Ch. 6 in