Spherical cap

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The spherical cap is the purple section.

In geometry, a spherical cap or spherical dome is a portion of a sphere cut off by a plane. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.

Volume and surface area

If the radius of the base of the cap is , and the height of the cap is , then the volume of the spherical cap is

and the curved surface area of the spherical cap is

The relationship between and is irrelevant as long as 0 ≤ . The blue section of the illustration is also a spherical cap.

The parameters , and are not independent:

.

Substituting this into the area formula gives:

Note also that in the upper hemisphere of the diagram, , and in the lower hemisphere ; hence in either hemisphere and so an alternative expression for the volume is

.

Application

The volume of all points which are in at least one of two intersecting spheres of radii r1 and r2 is [1]

,

where

is the total of the two isolated spheres, and

the sum of the two spherical caps of the intersection. If d <r1+r2 is the distance between the two sphere centers, elimination of the variables h1 and h2 leads to[2] [3]

Generalizations

Sections of other solids

The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid.

Hyperspherical cap

Generally, the -dimensional volume of a hyperspherical cap of height and radius in -dimensional Euclidean space is given by [4]

where (the gamma function) is given by .

The formula for can be expressed in terms of the volume of the unit n-ball and the hypergeometric function or the regularized incomplete beta function as

,

and the area formula can be expressed in terms of the area of the unit n-ball as

,

where .

Earlier in [5] (1986, USSR Academ. Press) the formulas were received: , where

For odd

It is shown in [6] that if then where is the integral of the standard normal distribution.

See also

References

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  4. Li, S. (2011). "Concise Formulas for the Area and Volume of a Hyperspherical Cap". Asian J. Math. Stat. 4 (1): 66–70. doi:10.3923/ajms.2011.66.70
  5. Chudnov A.M. (1986). “On Minimax Signal Generation and Reception Algorithms”, Problems of Information Transmission, 22:4 (1986), 49–54, rus. (Mi ppi958, an: 0624.94005)
  6. Chudnov A.M. (1991). “Game-Theoretical Problems of Synthesis of Signal Generation and Reception Algorithms”, Problems of Information Transmission, 27:3 (1991), 57–65, rus. (Mi ppi570, an:0778.94001)
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External links

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