An undulary, also called an elliptic catenary, is the Delaunay curve traced by a focus of a noncircular ellipse as the ellipse rolls without slipping along a line. Its height above the rolling line is a single-valued periodic function of position along that line. Revolving the undulary about that line gives an unduloid (Delaunay 1841, Bendito et al. 2014).
For an ellipse with semiaxes 
,
let 
and 
.
A parametrization of the focal roulette is
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is real and the rolling line
is the 
-axis. The distance 
varies between 
and 
. The resulting unduloid
has mean curvature of magnitude 
.
In the circular limit 
, the roulette
becomes a line parallel to the rolling line,
and the surface of revolution is a circular
cylinder.
