Pseudocrosscap

A surface constructed by placing a family of figure-eight curves into $\Bbb{R}^3$ such that the first and last curves reduce to points. The surface has parametric equations

$\displaystyle x(u,v)$	$\textstyle =$	$\displaystyle (1-u^2)\sin v$
$\displaystyle y(u,v)$	$\textstyle =$	$\displaystyle (1-u^2)\sin(2v)$
$\displaystyle z(u,v)$	$\textstyle =$	$\displaystyle u.$

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 247-248, 1993.