Spherical cap

An example of a spherical cap in blue.

In geometry, a spherical cap, spherical dome, or spherical segment of one base 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[1]

and the curved surface area of the spherical cap is[1]

or

The relationship between and is irrelevant as long as . The red 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

.

The volume may also be found by integrating under a surface of rotation, using and factorizing.

.

Applications

Volumes of union and intersection of two intersecting spheres

The volume of the union of two intersecting spheres of radii r1 and r2 is [2]

,

where

is the sum of the volumes of the two isolated spheres, and

the sum of the volumes of the two spherical caps forming their intersection. If d < r1 + r2 is the distance between the two sphere centers, elimination of the variables h1 and h2 leads to[3][4]

 .

Surface area bounded by circles of latitude

The surface area bounded by two circles of latitude is the difference of surface areas of their respective spherical caps. For a sphere of radius r, and latitudes φ1 and φ2, the area is [5]

For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016[6]) is 2π·6371²|sin 90° sin 66.56°| = 21.04 million km², or 0.5·|sin 90° sin 66.56°| = 4.125% of the total surface area of the Earth.

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 [7]

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 [8] (1986, USSR Academ. Press) the following formulas were derived: , where ,

.

For odd

.

It is shown in [9] that, if and , then where is the integral of the standard normal distribution.

See also

References

  1. 1 2 Polyanin, Andrei D; Manzhirov, Alexander V. (2006), Handbook of Mathematics for Engineers and Scientists, CRC Press, p. 69, ISBN 9781584885023.
  2. Connolly, Michael L. (1985). "Computation of molecular volume". J. Am. Chem. Soc. 107: 1118–1124. doi:10.1021/ja00291a006.
  3. Pavani, R.; Ranghino, G. (1982). "A method to compute the volume of a molecule". Comput. Chem. 6: 133–135. doi:10.1016/0097-8485(82)80006-5.
  4. Bondi, A. (1964). "Van der Waals volumes and radii". J. Phys. Chem. 68: 441–451. doi:10.1021/j100785a001.
  5. Scott E. Donaldson, Stanley G. Siegel. "Successful Software Development". Retrieved 29 August 2016.
  6. "Obliquity of the Ecliptic (Eps Mean)". Neoprogrammics.com. Retrieved 2014-05-13.
  7. 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.
  8. Chudnov, Alexander M. (1986). "On minimax signal generation and reception algorithms (rus.)". Problems of Information Transmission. 22 (4): 49–54.
  9. Chudnov, Alexander M (1991). "Game-theoretical problems of synthesis of signal generation and reception algorithms (rus.)". Problems of Information Transmission. 27 (3): 57–65.

Additional reading

Wikimedia Commons has media related to Spherical caps.

Derivation and some additional formulas.

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