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 S of a topological linear space X is absorbing if for each x in X there is an r>0 such that ax is in X if for each a such that |a|>=r. A subset S of a topological linear space is absolutely convex if for each x and y in S, ax+by is in S if |a|+|b|<=1.)
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
with x(0)=r_1. Solving for a gives
so the sides have equation
Using the equation for a solid of revolution then gives
![画像:piint_0^h[x(z)]^2dz](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Barrel/Inline20.svg){kind=link}
For sides consisting of a parabolic segment, the equation of the side is given by
| x(z)=r_2+a(z-1/2h)^2 |
(6)
|
with x(0)=r_1. Solving for a gives
| [画像: a=(4(r_1-r_2))/(h^2), ] |
(7)
|
so the sides have equation
Using the equation for a solid of revolution then gives
![画像:piint_0^h[x(z)]^2dz](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Barrel/Inline28.svg){kind=link}
See also
Cylinder, Hyperboloid, Solid of RevolutionExplore with Wolfram|Alpha
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References
Harris, J. W. and Stocker, H. "Barrel." §4.10.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 112, 1998.Kepler, J. Nova Stereometria Doliorum Vinariorum/New Solid Geometry of Wine Barrels. (Ed. and Trans. E. Knobloch). Paris, France: Les Belles Lettres, 2018.MacDonnell, J. "The Mathematician's Quest for Superlatives from Geometrical and Calculus Considerations." http://www.faculty.fairfield.edu/jmac/ther/superlatives.htm.Shechter, B.-S. "Kepler's Wine Barrel Problem in a Dynamic Geometry Environment." http://users.math.uoc.gr/~ictm2/Proceedings/pap420.pdf.Taylor, A. E. and Lay, D. C. Introduction to Functional Analysis, 2nd ed. New York: Wiley, 1980.Tikhomirov, V. M. "New Solid Geometry of Wine Barrels." Stories About Maxima and Minima. Providence, RI: Amer. Math. Soc., 1991.Referenced on Wolfram|Alpha
BarrelCite this as:
Weisstein, Eric W. "Barrel." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Barrel.html