Spherical cap

In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single 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.

An example of a spherical cap in blue (and another in red.)
3D model of a spherical cap.

Volume and surface areaEdit

The volume of the spherical cap and the area of the curved surface may be calculated using combinations of

  • The radius   of the sphere
  • The radius   of the base of the cap
  • The height   of the cap
  • 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
Using   and   Using   and   Using   and  
Volume   [1]    
Area  [1]    

If   denotes the latitude in geographic coordinates, then  .

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

The formulas using   and   can be rewritten to use the radius   of the base of the cap instead of  , using the Pythagorean theorem:

 

so that

 

Substituting this into the formulas gives:

 
 

Deriving the surface area intuitively from the spherical sector volumeEdit

Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume   of the spherical sector, by an intuitive argument,[2] as

 

The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume formula of  , where   is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and   is the height of each pyramid from its base to its apex (at the center of the sphere). Since each  , in the limit, is constant and equivalent to the radius   of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:

 

Deriving the volume and surface area using calculusEdit

 
Rotating the green area creates a spherical cap with height   and sphere radius  .

The volume and area formulas may be derived by examining the rotation of the function

 

for  , using the formulas the surface of the rotation for the area and the solid of the revolution for the volume. The area is

 

The derivative of   is

 

and hence

 

The formula for the area is therefore

 

The volume is

 

ApplicationsEdit

Volumes of union and intersection of two intersecting spheresEdit

The volume of the union of two intersecting spheres of radii   and   is [3]

 

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[4][5]

 

Volume of a spherical cap with a curved baseEdit

The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii   and  , separated by some distance  , and for which their surfaces intersect at  . That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2's cap (with height  ) and sphere 1's cap (with height  ),

 

This formula is valid only for configurations that satisfy   and  . If sphere 2 is very large such that  , hence   and  , which is the case for a spherical cap with a base that has a negligible curvature, the above equation is equal to the volume of a spherical cap with a flat base, as expected.

Areas of intersecting spheresEdit

Consider two intersecting spheres of radii   and  , with their centers separated by distance  . They intersect if

 

From the law of cosines, the polar angle of the spherical cap on the sphere of radius   is

 

Using this, the surface area of the spherical cap on the sphere of radius   is

 

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  ,[6]

 

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[7]) 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:[citation needed]   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 quantitative bound is  . For large caps (that is when   as  ), the bound simplifies to  . [10]

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. ^ Shekhtman, Zor. "Unizor - Geometry3D - Spherical Sectors". YouTube. Zor Shekhtman. Retrieved 31 Dec 2018.
  3. ^ Connolly, Michael L. (1985). "Computation of molecular volume". Journal of the American Chemical Society. 107 (5): 1118–1124. doi:10.1021/ja00291a006.
  4. ^ Pavani, R.; Ranghino, G. (1982). "A method to compute the volume of a molecule". Computers & Chemistry. 6 (3): 133–135. doi:10.1016/0097-8485(82)80006-5.
  5. ^ Bondi, A. (1964). "Van der Waals volumes and radii". The Journal of Physical Chemistry. 68 (3): 441–451. doi:10.1021/j100785a001.
  6. ^ Scott E. Donaldson, Stanley G. Siegel (2001). Successful Software Development. ISBN 9780130868268. Retrieved 29 August 2016.
  7. ^ "Obliquity of the Ecliptic (Eps Mean)". Neoprogrammics.com. Retrieved 2014-05-13.
  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". Journal of Molecular Biology. 178 (1): 63–89. doi:10.1016/0022-2836(84)90231-6. PMID 6548264.
  • Lustig, Rolf (1986). "Geometry of four hard fused spheres in an arbitrary spatial configuration". Molecular Physics. 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". The Journal of Physical Chemistry. 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". Molecular Physics. 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". Journal of Computational Chemistry. 15 (5): 507–523. doi:10.1002/jcc.540150504.
  • Grant, J. A.; Pickup, B. T. (1995). "A Gaussian description of molecular shape". The Journal of Physical Chemistry. 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". Computer Physics Communications. 165 (1): 59–96. Bibcode:2005CoPhC.165...59B. doi:10.1016/j.cpc.2004.08.002.

External linksEdit