The Cone Limit of the Catenoid
The Cone Limit of the Catenoid
Consider the catenoids given by the parametric equations
x=ccoshcos[u]
v
c
y=ccoshsin[u]
v
c
z=v
where is a positive parameter that you can vary.
c
When , this reduces to the equation of a circle in the - plane of radius centered at the origin.
z=v=0
x
y
c
For a given height , let . The slope of the cone is vcosh, 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.
v
R(v)=ccosh
min
c
v
c
c
0
1
c
0
c
0
R(c)