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Spherical cap

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An example of a spherical cap in blue (and another in red.)

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.

Contents

Volume and surface areaEdit

If the radius of the sphere 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

 

in which   is the polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap. (If it   denotes the latitude in geographic coordinates,  .)

The relationship between   and   is irrelevant as long as  . For example, the red section of the illustration is also a spherical cap for which  .

These formulas can be rewritten to use the radius   of the base of the cap instead of  , using the relationship:

 

so that

 

Substituting this into the formulas gives:

 
 

The volume formula may be derived by integrating under a surface of rotation and factorizing as follows.

 

ApplicationsEdit

Volumes of union and intersection of two intersecting spheresEdit

The volume of the union of two intersecting spheres of radii   and   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   is the distance between the two sphere centers, elimination of the variables   and   leads to[3][4]

 

Surface area bounded by parallel disksEdit

The curved surface area of the spherical segment bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius  , and caps with heights   and  , the area is

 

or, using geographic coordinates with latitudes   and  ,[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.

This formula can also be used to demonstrate that half the surface area of the Earth lies between latitudes 30° South and 30° North in a spherical zone which encompasses all of the Tropics.

GeneralizationsEdit

Sections of other solidsEdit

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 capEdit

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  

 .

AsymptoticsEdit

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

A more quantitive way of writing this, is in [10] where the bound   is given. For large caps (that is when   as  ), the bound simplifies to  .

See alsoEdit

ReferencesEdit

  1. ^ a b 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.
  10. ^ Anja Becker, Léo Ducas, Nicolas Gama, and Thijs Laarhoven. 2016. New directions in nearest neighbor searching with applications to lattice sieving. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms (SODA '16), Robert Kraughgamer (Ed.). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 10-24.

Further readingEdit

  • Richmond, Timothy J. (1984). "Solvent accessible surface area and excluded volume in proteins: Analytical equation for overlapping spheres and implications for the hydrophobic effect". J. Mol. Biol. 178 (1): 63–89. doi:10.1016/0022-2836(84)90231-6.
  • Lustig, Rolf (1986). "Geometry of four hard fused spheres in an arbitrary spatial configuration". Mol. Phys. 59 (2): 195–207. Bibcode:1986MolPh..59..195L. doi:10.1080/00268978600102011.
  • Gibson, K. D.; Scheraga, Harold A. (1987). "Volume of the intersection of three spheres of unequal size: a simplified formula". J. Phys. Chem. 91 (15): 4121–4122. doi:10.1021/j100299a035.
  • Gibson, K. D.; Scheraga, Harold A. (1987). "Exact calculation of the volume and surface area of fused hard-sphere molecules with unequal atomic radii". Mol. Phys. 62 (5): 1247–1265. Bibcode:1987MolPh..62.1247G. doi:10.1080/00268978700102951.
  • Petitjean, Michel (1994). "On the analytical calculation of van der Waals surfaces and volumes: some numerical aspects". Int. J. Quant. Chem. 15 (5): 507–523. doi:10.1002/jcc.540150504.
  • Grant, J. A.; Pickup, B. T. (1995). "A Gaussian description of molecular shape". J. Phys. Chem. 99 (11): 3503–3510. doi:10.1021/j100011a016.
  • Busa, Jan; Dzurina, Jozef; Hayryan, Edik; Hayryan, Shura (2005). "ARVO: A fortran package for computing the solvent accessible surface area and the excluded volume of overlapping spheres via analytic equations". Comp. Phys. Commun. 165: 59–96. Bibcode:2005CoPhC.165...59B. doi:10.1016/j.cpc.2004.08.002.

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