The Cone Limit of the Catenoid

Consider the catenoids given by the parametric equations
,
,
,
where is a positive parameter that you can vary.
When , this reduces to the equation of a circle in the - plane of radius centered at the origin.
For a given height , let . The slope of the cone is , where is the value that minimizes . This cone is unique; its horizontal slices are circles with radii that are the greatest lower bound of the radii of the horizontal slices of the catenoids at the same height; the catenoid intersects the interior of any larger cone.

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Reference
[1] E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Boca Raton: Chapman and Hall/CRC, 2006.
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