Torispherical Dome

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TorisphericalDome

A torispherical dome is the surface obtained from the intersection of a spherical cap with a tangent torus, as illustrated above. The radius of the sphere R is called the "crown radius," and the radius a of the torus is called the "knuckle radius." Torispherical domes are used to construct pressure vessels.

Let c be the distance from the center of the torus to the center of the torus tube, let a<c be the radius of the torus tube, and let h be the height from the base of the dome to the top. Then the radius of the base is given by a+c<R. In addition, by elementary geometry, a torispherical dome satisfies

c^2+(R-h)^2=(R-a)^2,

(1)

h=R-sqrt((a+c-R)(a-c-R)).

(2)

The transition from sphere to torus occurs at the critical radius

[画像: r=c[1+(R/a-1)^(-1)], ]

(3)

so the dome has equation

[画像: {x^2+y^2+z^2=R^2 for sqrt(x^2+y^2)<r; (c-sqrt(x^2+y^2))^2=a^2-[z-(R-h)]^2 for r<sqrt(x^2+y^2)<a+c, ]

(4)

where

R-h=sqrt((a+c-R)(a-c-R)).