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The self-intersection of a one-sided Surface. It can be described as a circular Hole which, when entered, exits from its opposite point (from a topological viewpoint, both singular points on the cross-cap are equivalent). The cross-cap has a segment of double points which terminates at two ``Pinch Points'' known as Whitney Singularities.

The cross-cap can be generated using the general method for Nonorientable Surfaces using the polynomial function

{\bf f}(x,y,z)=(xz, yz, {\textstyle{1\over 2}}(z^2-x^2))
\end{displaymath} (1)

(Pinkall 1986). Transforming to Spherical Coordinates gives
$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\cos u\sin(2v)$ (2)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sin u\sin(2v)$ (3)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(\cos^2 v-\cos^2 u\sin^2 v)$ (4)

for $u\in [0, 2\pi)$ and $v\in[0,\pi/2]$. To make the equations slightly simpler, all three equations are normally multiplied by a factor of 2 to clear the arbitrary scaling constant. Three views of the cross-cap generated using this equation are shown above. Note that the middle one looks suspiciously like Maeder's Owl Minimal Surface.

Another representation is

{\bf f}(x,y,z)=(yz, 2xy, x^2-y^2),
\end{displaymath} (5)

(Gray 1993), giving parametric equations
$\displaystyle x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sin u\sin(2v)$ (6)
$\displaystyle y$ $\textstyle =$ $\displaystyle \sin(2u)\sin^2 v$ (7)
$\displaystyle z$ $\textstyle =$ $\displaystyle \cos(2u)\sin^2 v,$ (8)

(Geometry Center) where, for aesthetic reasons, the $y$- and $z$-coordinates have been multiplied by 2 to produce a squashed, but topologically equivalent, surface. Nordstrand gives the implicit equation
\end{displaymath} (9)

which can be solved for $z$ to yield
z={-2x^2\pm\sqrt{(y^2+2x^2)(1-4x^2-y^2)}\over 4x^2+y^2}.
\end{displaymath} (10)


Taking the inversion of a cross-cap such that (0, 0, $-1/2$) is sent to $\infty$ gives a Cylindroid, shown above (Pinkall 1986).

The cross-cap is one of the three possible Surfaces obtained by sewing a Möbius Strip to the edge of a Disk. The other two are the Boy Surface and Roman Surface.

See also Boy Surface, Möbius Strip, Nonorientable Surface, Projective Plane, Roman Surface


Fischer, G. (Ed.). Plate 107 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 108, 1986.

Geometry Center. ``The Crosscap.''

Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 64, 1986.

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© 1996-9 Eric W. Weisstein