Cylindrical equation:  (case of the tower) or , i.e.  (case of the funnel). Cartesian parametrization:  (case of the tower).Première forme quadratique fondamentale : . First fundamental quadratic form:  . Second fundamental quadratic form:  . In the case k = 1, cartesian parametrization the coordinate lines of which are the asymptotic lines:  (view opposite).

The tower with constant pressure is the surface of revolution obtained by rotating a logarithmic curve around its asymptote.
Its name comes from the fact that, if this surface is filled with a homogeneous material, then the pressure applied on any horizontal section by the upper part is constant.
 

Derivation of the equation: The pressure at the altitude z is equal to ; assuming P constant and differentiating, we get the differential equation: which immediately gives the result, with . Remark: if we take the variation of g due to the altitude into account, , we get the surface , with , represented opposite. Much as the first one has a straight line as asymptote, this one has an asymptote cylinder. Note that this tower, extended to infinity, would have an infinite mass, but finite weight...

See also Gabriel's horn.
 
 

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© Robert FERRÉOL  2017