Vault
Let a vault consist of two equal half-cylinders of radius r which intersect at right angles so that the lines of their intersections (the "groins") terminate in the polyhedron vertices of a square. Two vaults placed bottom-to-top form a Steinmetz solid on two cylinders.
Solving the equations
simultaneously gives
One quarter of the vault can therefore be described by the parametric equations
The surface area of the vault is therefore given by
| [画像: A=4intl(z)rdtheta, ] |
(8)
|
where l(z) is the length of a cross section at height z and theta is the angle a point on the center of this line makes with the origin. But z=rsintheta, so
| dz=rcosthetadtheta=rsqrt(1-sin^2theta)dtheta=sqrt(r^2-z^2)dtheta, |
(9)
|
and
| l(z)=2sqrt(r^2-x^2) |
(10)
|
The volume of the vault is
The geometric centroid is
| z^_=3/8r. |
(15)
|
See also
Cylinder, Spherical Cap, Steinmetz Solid, Torispherical DomeExplore with Wolfram|Alpha
More things to try:
References
Lines, L. Solid Geometry, with Chapters on Space-Lattices, Sphere-Packs, and Crystals. New York: Dover, pp. 112-113, 1965.Moore, M. "Symmetrical Intersections of Right Circular Cylinders." Math. Gaz. 58, 181-185, 1974.Referenced on Wolfram|Alpha
VaultCite this as:
Weisstein, Eric W. "Vault." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Vault.html