Spherical cap

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The spherical cap is the purple section.

In geometry, a spherical cap or spherical dome 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[edit]

If the radius of the base of the cap is a, and the height of the cap is h, then the volume of the spherical cap is

V = \frac{\pi h}{6} (3a^2 + h^2),

and the curved surface area of the spherical cap is

A = 2 \pi r h.

The relationship between h and r is irrelevant as long as 0 ≤ h2r. The blue section of the illustration is also a spherical cap.

The parameters a, h and r are not independent:

r^2 = (r-h)^2 + a^2 = r^2 +h^2 -2rh +a^2,
r = \frac {a^2 + h^2}{2h}.

Substituting this into the area formula gives:

A = 2 \pi \frac{(a^2 + h^2)}{2h} h = \pi (a^2 + h^2).

Note also that in the upper hemisphere of the diagram, \scriptstyle h = r - \sqrt{r^2 - a^2}, and in the lower hemisphere \scriptstyle h = r + \sqrt{r^2 - a^2}; hence in either hemisphere \scriptstyle a = \sqrt{h(2r-h)} and so an alternative expression for the volume is

V = \frac {\pi h^2}{3} (3r-h).

Application[edit]

The volume of all points which are in at least one of two intersecting spheres of radii r1 and r2 is [1]

 V = V^{(1)}-V^{(2)},

where

V^{(1)} = \frac{4\pi}{3}r_1^3 +\frac{4\pi}{3}r_2^3

is the total of the two isolated spheres, and

V^{(2)} = \frac{\pi h_1^2}{3}(3r_1-h_1)+\frac{\pi h_2^2}{3}(3r_2-h_2)

the sum of the two spherical caps of the intersection. If d <r1+r2 is the distance between the two sphere centers, elimination of the variables h1 and h2 leads to[2] [3]

V^{(2)} = \frac{\pi}{12d}(r_1+r_2-d)^2[d^2+2d(r_1+r_2)-3(r_1-r_2)^2].

Generalizations[edit]

Sections of other solids[edit]

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[edit]

Generally, the n-dimensional volume of a hyperspherical cap of height h and radius r in n-dimensional Euclidean space is given by [4]

V = \frac{\pi ^ {\frac{n-1}{2}}\, r^{n}}{\,\Gamma \left ( \frac{n+1}{2} \right )} \int\limits_{0}^{\arccos\left(\frac{r-h}{r}\right)}\sin^n (t) \,\mathrm{d}t

where \Gamma (the gamma function) is given by  \Gamma(z) = \int_0^\infty  t^{z-1} \mathrm{e}^{-t}\,\mathrm{d}t .

The formula for V can be expressed in terms of the volume of the unit n-ball C_{n}={\scriptstyle \pi^{n/2}/\Gamma[1+\frac{n}{2}]} and the hypergeometric function {}_{2}F_{1} or the regularized incomplete beta function I_x(a,b)as

V = C_{n} \, r^{n} \left( \frac{1}{2}\, - \,\frac{r-h}{r}  \,\frac{\Gamma[1+\frac{n}{2}]}{\sqrt{\pi}\,\Gamma[\frac{n+1}{2}]}
{\,\,}_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1-n}{2};\tfrac{3}{2};\left(\tfrac{r-h}{r}\right)^{2}\right)\right)
=\frac{1}{2}C_{n} \, r^n I_{(2rh-h^2)/r^2} \left(\frac{n+1}{2}, \frac{1}{2} \right) ,

and the area formula A can be expressed in terms of the area of the unit n-ball A_{n}={\scriptstyle 2\pi^{n/2}/\Gamma[\frac{n}{2}]} as

A =\frac{1}{2}A_{n} \, r^{n-1} I_{(2rh-h^2)/r^2} \left(\frac{n-1}{2}, \frac{1}{2} \right) ,

where \scriptstyle 0\le h\le r .

Earlier in [5] (1986, USSR Academ. Press) the formulas were received:  A=A_n p_ { n-2 } (q), V=V_n p_n (q) , where   q= 1-h/r (0 \le q \le 1 ),     p_n (q) =(1-G_n(q)/G_n(1))/2,

  G _n(q)= \int \limits_{0}^{q} (1-t^2) ^ { (n-1) /2 } dt.

For odd    n=2k+1:

   G_n(q) = \sum_{i=0}^k (-1) ^i \binom k i \frac {q^{2i+1}}  {2i+1}.

It is shown in [6] that if     n  \to \infty   then   p_n (q) \to   1- F(q \sqrt {n})   where  F() is the integral of the standard normal distribution.

See also[edit]

References[edit]

  1. ^ Connolly, Michael L. (1985). "Computation of molecular volume". J. Am. Chem. Soc: 1118–1124. doi:10.1021/ja00291a006. 
  2. ^ Pavani, R.; Ranghino, G. (1982). "A method to compute the volume of a molecule". Comput. Chem. doi:10.1016/0097-8485(82)80006-5. 
  3. ^ Bondi, A. (1964). "van der Waals volumes and radii". J. Phys. Chem. (68): 441–451. doi:10.1021/j100785a001. 
  4. ^ 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
  5. ^ Chudnov A.M. (1986). “On Minimax Signal Generation and Reception Algorithms”, Problems of Information Transmission, 22:4 (1986), 49–54, rus. (Mi ppi958, an: 0624.94005)
  6. ^ Chudnov A.M. (1991). “Game-Theoretical Problems of Synthesis of Signal Generation and Reception Algorithms”, Problems of Information Transmission, 27:3 (1991), 57–65, rus. (Mi ppi570, an:0778.94001)
  • 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. Molec. 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. 
  • 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. .

External links[edit]