Catenoid

Catenoid.

A catenoid is the smallest surface formed by the rotation of a chain line .

y=a\,\operatorname {ch} \,{\frac {x}{a}}

{\ displaystyle y = a \, \ operatorname {ch} \, {\ frac {x} {a}}}

{\ displaystyle y = a \, \ operatorname {ch} \, {\ frac {x} {a}}}

around axis

O X

{\ displaystyle OX}

Ox

.

History

The catenoid was first described by Euler in 1744 . The word catenoid is derived from lat. catena means chain and Greek éidos - kind .

The catenoid can also be set parametrically:

u\in \mathbb {R} ,\quad v\in \left[0;2\pi \right),\qquad {\begin{cases}x=\operatorname {ch} (u)\,\cos(v)\\y=\operatorname {ch} (u)\,\sin(v)\\z=u\end{cases}},

{\ displaystyle u \ in \ mathbb {R}, \ quad v \ in \ left [0; 2 \ pi \ right), \ qquad {\ begin {cases} x = \ operatorname {ch} (u) \, \ cos (v) \\ y = \ operatorname {ch} (u) \, \ sin (v) \\ z = u \ end {cases}},}

u\in \mathbb {R} ,\quad v\in \left[0;2\pi \right),\qquad {\begin{cases}x=\operatorname {ch} (u)\,\cos(v)\\y=\operatorname {ch} (u)\,\sin(v)\\z=u\end{cases}},

Where $\operatorname {ch}$ ${\ displaystyle \ operatorname {ch}}$ $\operatorname {ch}$ - hyperbolic cosine .

Soap film in the form of a catenoid.

It is a minimal surface .
- In particular, a soap film takes on the shape of a catenoid, stretched over two close wire circles whose planes are perpendicular to the line connecting their centers.
A not too large portion of the catenoid can be converted isometrically (without compression and tension) into a portion of the helicoid .
The total curvature is equal to $-$ $four$ $\cdot$ $π$ {\ displaystyle -4 \ cdot \ pi} .
- Total minimum surface in $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ $\mathbb {R} ^{3}$ with total curvature $-4\cdot \pi$ ${\ displaystyle -4 \ cdot \ pi}$ $-4\cdot \pi$ is either a catenoid or an Enneper surface .

Catenoid // Great Soviet Encyclopedia : [in 30 vol.] / Ch. ed. A.M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia, 1969-1978.