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)
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with . Solving for gives
(2)
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so the sides have equation
(3)
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Using the equation for a solid of revolution then gives
(4)
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(5)
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For sides consisting of a parabolic segment, the equation of the side is given by
(6)
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with . Solving for gives
(7)
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so the sides have equation
(8)
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Using the equation for a solid of revolution then gives
(9)
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(10)
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