Barrel

A barrel solid of revolution composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical about the midplane.

The term also has a technical meaning in functional analysis. In particular, a subset of a topological linear space is a barrel if it is absorbing, closed, and absolutely convex (Taylor and Lay 1980, p. 111). (A subset of a topological linear space is absorbing if for each there is an such that is in if for each such that . A subset of a topological linear space is absolutely convex if for each and in , is in if .)

When buying supplies for his second wedding, the great astronomer Johannes Kepler became unhappy about the inexact methods used by the merchants to estimate the liquid contents of a wine barrel. Kepler therefore investigated the properties of nearly 100 solids of revolution generated by rotation of conic sections about non-principal axes (Kepler, MacDonnell, Shechter, Tikhomirov 1991).

For sides consisting of an arc of an ellipse, the equation of the side is given by

(1)

with . Solving for gives

(2)

so the sides have equation

(3)

Using the equation for a solid of revolution then gives

			(4)
			(5)

For sides consisting of a parabolic segment, the equation of the side is given by

(6)

with . Solving for gives

(7)

so the sides have equation

(8)

Using the equation for a solid of revolution then gives

			(9)
			(10)