Torispherical Dome

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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)

so

 h=R-sqrt((a+c-R)(a-c-R)).
(2)

The transition from sphere to torus occurs at the critical radius

 r=c[1+(R/a-1)^(-1)],
(3)

so the dome has equation

 {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)).
(5)

The torispherical dome has volume

V=pi/6[3a^2cpi+4R^3-2sqrt((a-c-R)(a+c-R))×(2a^2+c^2+2aR+2R^2)+6a^2csin^(-1)(c/(a-R))]
(6)
=pi/3[2hR^2-(2a^2+c^2+2aR)(R-h)+3a^2csin^(-1)((R-h)/(R-a))].
(7)

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