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 (forming a great circle), so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.
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 calculus
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]
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.
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 (8.12 million sq mi), 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.
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.
Generally, the -dimensional volume of a hyperspherical cap of height and radius in -dimensional Euclidean space is given by:[8]
where (the gamma function) is given by .
^Connolly, Michael L. (1985). "Computation of molecular volume". Journal of the American Chemical Society. 107 (5): 1118–1124. doi:10.1021/ja00291a006.
^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.
^Bondi, A. (1964). "Van der Waals volumes and radii". The Journal of Physical Chemistry. 68 (3): 441–451. doi:10.1021/j100785a001.
^Becker, Anja; Ducas, Léo; Gama, Nicolas; Laarhoven, Thijs (10 January 2016). Krauthgamer, Robert (ed.). New directions in nearest neighbor searching with applications to lattice sieving. Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '16), Arlington, Virginia. Philadelphia: Society for Industrial and Applied Mathematics. pp. 10–24. ISBN978-1-61197-433-1.
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. PMID6548264.
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.