Barrel

A barrel is a solid of revolution with parallel circular top and bottom faces sharing a common axis, and with 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 vector space is a barrel if it is absorbing, closed, balanced, and convex (Taylor and Lay 1980, p. 111).

Kepler's analysis of the diagonal-rod method then used to measure barrels in the wine trade led to Kepler's wine barrel problem, a problem in optimization theory concerning the volume inferred from a measurement through the bunghole (Kepler 1615, Cardil 2012, Kepler 2018).

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)