An example of a spherical cap in blue (and another in red)
3D model of a 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 (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.
Volume and surface area
The
volume of the spherical cap and the area of the curved surface may be calculated using combinations of
The radius $r$ of the sphere
The radius $a$ of the base of the cap
The height $h$ of the cap
The
polar angle$\theta$ 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 $\phi$ denotes the
latitude in
geographic coordinates, then $\theta +\phi =\pi /2=90^{\circ }\,$, and $\cos \theta =\sin \phi$.
The relationship between $h$ and $r$ is relevant as long as $0\leq h\leq 2r$. For example, the red section of the illustration is also a spherical cap for which $h>r$.
The formulas using $r$ and $h$ can be rewritten to use the radius $a$ of the base of the cap instead of $r$, using the
Pythagorean theorem:
Deriving the surface area intuitively from the
spherical sector volume
Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume $V_{sec}$ 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 $V={\frac {1}{3}}bh'$, where $b$ is the infinitesimal
area of each pyramidal base (located on the surface of the sphere) and $h'$ is the height of each pyramid from its base to its apex (at the center of the sphere). Since each $h'$, in the limit, is constant and equivalent to the radius $r$ of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:
the sum of the volumes of the two spherical caps forming their intersection. If $d\leq r_{1}+r_{2}$ is the
distance between the two sphere centers, elimination of the variables $h_{1}$ and $h_{2}$ leads
to^{
[4]}^{
[5]}
The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii $r_{1}$ and $r_{2}$, separated by some distance $d$, and for which their surfaces intersect at $x=h$. That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2's cap (with height $(r_{2}-r_{1})-(d-h)$) and sphere 1's cap (with height $h$),
This formula is valid only for configurations that satisfy $0<d<r_{2}$ and $d-(r_{2}-r_{1})<h\leq r_{1}$. If sphere 2 is very large such that $r_{2}\gg r_{1}$, hence $d\gg h$ and $r_{2}\approx d$, 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 spheres
Consider two intersecting spheres of radii $r_{1}$ and $r_{2}$, with their centers separated by distance $d$. They intersect if
$|r_{1}-r_{2}|\leq d\leq r_{1}+r_{2}$
From the law of cosines, the polar angle of the spherical cap on the sphere of radius $r_{1}$ is
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 $r$, and caps with heights $h_{1}$ and $h_{2}$, the area is
$A=2\pi r|h_{1}-h_{2}|\,,$
or, using geographic coordinates with latitudes $\phi _{1}$ and $\phi _{2}$,^{
[6]}
$A=2\pi r^{2}|\sin \phi _{1}-\sin \phi _{2}|\,,$
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·6371^{2}|sin 90° − sin 66.56°| = 21.04 million km^{2}, 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.
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 $n$-dimensional volume of a hyperspherical cap of height $h$ and radius $r$ in $n$-dimensional Euclidean space is given by:^{
[8]}
It is shown in ^{
[10]} that, if $n\to \infty$ and $q{\sqrt {n}}={\text{const.}}$, then $p_{n}(q)\to 1-F({q{\sqrt {n}}})$ where $F()$ is the integral of the
standard normal distribution.
A more quantitative bound is $A/A_{n}=n^{\Theta (1)}\cdot [(2-h/r)h/r]^{n/2}$.
For large caps (that is when $(1-h/r)^{4}\cdot n=O(1)$ as $n\to \infty$), the bound simplifies to $n^{\Theta (1)}\cdot e^{-(1-h/r)^{2}n/2}$.
^{
[11]}
^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.
^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 Krauthgamer (Ed.). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 10-24.
Further reading
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.