The elliptic catenary - profile curve of an unduloid - is given by the equations in $xy$-plane

$x(t) = \int\limits_0^t \sqrt{a^2 - c^2 \cos^2 \tau} d\tau + c \sin t \cdot \frac{a - c \cos t}{\sqrt{a^2 - c^2 \cos^2 t}},$

$y(t) = \sqrt{a^2 - c^2} \frac{a - c \cos t}{\sqrt{a^2 - c^2 \cos^2 t}},$

where $a$ and $c$ are the major semi-axis and half of the focal distance of the rolling ellipse.

Lines on the model show the typical behaviour of geodesic lines on surfaces of revolution and CMC (constant mean curvature) surfaces.

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