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| [[File:Spherical Cap.svg|thumb|200px|The spherical cap is the purple section.]]
| | I'm Yoshiko Oquendo. Bookkeeping has been his working day occupation for a while. I've always cherished residing in Idaho. Playing croquet is something I will by no means give up.<br><br>My page: [http://Www.Redtrance.com/profile.php?u=CoWaller http://Www.Redtrance.com/profile.php?u=CoWaller] |
| In [[geometry]], a '''spherical cap''' or '''spherical dome''' is a portion of a [[sphere]] cut off by a [[Plane (mathematics)|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 ''[[Sphere#Hemisphere|hemisphere]]''.
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| ==Volume and surface area==
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| If the radius of the base of the cap is <math>a</math>, and the height of the cap is <math>h</math>, then the [[volume]] of the spherical cap is
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| :<math>V = \frac{\pi h}{6} (3a^2 + h^2),</math>
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| and the curved surface [[area]] of the spherical cap is
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| :<math>A = 2 \pi r h.</math>
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| The relationship between <math>h</math> and <math>r</math> is irrelevant as long as <math>h > 0</math> and <math>h < 2r</math>. The blue section of the illustration is also a spherical cap.
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| The parameters <math>a</math>, <math>h</math> and <math>r</math> are not independent:
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| :<math>r^2 = (r-h)^2 + a^2 = r^2 +h^2 -2rh +a^2,</math>
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| :<math>r = \frac {a^2 + h^2}{2h}</math>.
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| Substituting this into the area formula gives:
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| :<math>A = 2 \pi \frac{(a^2 + h^2)}{2h} h = \pi (a^2 + h^2).</math>
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| Note also that in the upper hemisphere of the diagram, <math>\scriptstyle h = r - \sqrt{r^2 - a^2}</math>, and in the lower hemisphere <math>\scriptstyle h = r + \sqrt{r^2 - a^2}</math>; hence in either hemisphere <math>\scriptstyle a = \sqrt{h(2r-h)}</math> and so an alternative expression for the volume is
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| :<math>V = \frac {\pi h^2}{3} (3r-h)</math>.
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| == Application ==
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| The volume of all points which are in at least one of two intersecting spheres
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| of radii {{math|r<sub>1</sub>}} and {{math|r<sub>2</sub>}} is
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| <ref>{{cite journal|first1=Michael L.|last1=Connolly|year=1985|doi=10.1021/ja00291a006|title=Computation of molecular volume|journal=J. Am. Chem. Soc|pages=1118–1124}}</ref>
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| :<math> V = V^{(1)}-V^{(2)}</math>,
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| where
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| :<math>V^{(1)} = \frac{4\pi}{3}r_1^3 +\frac{4\pi}{3}r_2^3</math>
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| is the total of the two isolated spheres, and
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| :<math>V^{(2)} = \frac{\pi h_1^2}{3}(3r_1-h_1)+\frac{\pi h_2^2}{3}(3r_2-h_2)</math>
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| the sum of the two spherical caps of the intersection. If {{math|d <r<sub>1</sub>+r<sub>2</sub>}} is the
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| distance between the two sphere centers, elimination of the variables {{math|h<sub>1</sub>}} and {{math|h<sub>2</sub>}} leads
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| to<ref>{{cite journal|doi=10.1016/0097-8485(82)80006-5|year=1982|title=A method to compute the volume of a molecule|journal=Comput. Chem.|first1=R.|last1=Pavani|first2=G.|last2=Ranghino}}</ref>
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| <ref>{{cite journal|first1=A.|last1=Bondi|doi=10.1021/j100785a001|year=1964|title=van der Waals volumes and radii|journal=J. Phys. Chem.|number=68|pages=441–451}}</ref>
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| :<math>V^{(2)} = \frac{\pi}{12d}(r_1+r_2-d)^2[d^2+2d(r_1+r_2)-3(r_1-r_2)^2].</math>
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| ==Generalizations==
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| ===Sections of other solids===
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| The '''spheroidal dome''' is obtained by sectioning off a portion of a [[spheroid]] so that the resulting dome is [[circular symmetry|circularly symmetric]] (having an axis of rotation), and likewise the ellipsoidal dome is derived from the [[ellipsoid]].
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| ===Hyperspherical cap===
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| Generally, the <math>n</math>-dimensional volume of a hyperspherical cap of height <math>h</math> and radius <math>r</math> in <math>n</math>-dimensional Euclidean space is given by <ref>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</ref>
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| :<math>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</math>
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| where <math>\Gamma</math> (the [[gamma function]]) is given by <math> \Gamma(z) = \int_0^\infty t^{z-1} \mathrm{e}^{-t}\,\mathrm{d}t </math>.
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| The formula for <math>V</math> can be expressed in terms of the volume of the unit [[n-ball]] <math>C_{n}={\scriptstyle \pi^{n/2}/\Gamma[1+\frac{n}{2}]}</math> and the [[hypergeometric function]] <math>{}_{2}F_{1}</math> or the [[regularized incomplete beta function]] <math>I_x(a,b)</math>as
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| :<math>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)
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| =\frac{1}{2}C_{n} \, r^n I_{(2rh-h^2)/r^2} \left(\frac{n+1}{2}, \frac{1}{2} \right)</math> ,
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| and the area formula <math>A</math> can be expressed in terms of the area of the unit [[n-ball]] <math>A_{n}={\scriptstyle 2\pi^{n/2}/\Gamma[\frac{n}{2}]}</math> as
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| :<math>A =\frac{1}{2}A_{n} \, r^{n-1} I_{(2rh-h^2)/r^2} \left(\frac{n-1}{2}, \frac{1}{2} \right)</math> ,
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| where <math>\scriptstyle 0\le h\le r </math>.
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| ==See also==
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| * [[Circular segment]] — the analogous 2D object.
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| * [[Dome (mathematics)]]
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| * [[Solid angle]] — contains formula for n-sphere caps
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| * [[Spherical segment]]
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| * [[Spherical sector]]
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| * [[Spherical wedge]]
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| == References ==
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| {{reflist}}
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| * {{cite journal|first1= Timothy J. | last1=Richmond
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| |title=Solvent accessible surface area and excluded volume in proteins: Analytical equation for overlapping spheres and implications for the hydrophobic effect
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| |journal=J. Molec. Biol.
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| |year=1984 | doi=10.1016/0022-2836(84)90231-6
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| |volume=178 | number=1
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| |pages=63–89 }}
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| * {{cite journal| first1=Rolf | last1=Lustig
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| |title=Geometry of four hard fused spheres in an arbitrary spatial configuration
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| |journal= Mol. Phys.
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| |year=1986
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| |volume=59 | number=2 | pages=195–207 |bibcode=1986MolPh..59..195L
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| |doi= 10.1080/00268978600102011}}
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| * {{cite journal | first1=K. D. | last1=Gibson
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| |first2=Harold A. |last2=Scheraga
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| |title=Volume of the intersection of three spheres of unequal size: a simplified formula
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| |year=1987 | journal= J. Phys. Chem.
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| |volume=91 | number =15 | pages =4121–4122
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| }}
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| *{{cite journal | first1=K. D. | last1=Gibson
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| |first2=Harold A. | last2=Scheraga
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| |title=Exact calculation of the volume and surface area of fused hard-sphere molecules with unequal atomic radii
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| |year=1987 | journal=Mol. Phys.
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| |volume=62 | number=5 | pages=1247–1265 | bibcode=1987MolPh..62.1247G
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| |doi=10.1080/00268978700102951}}
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| *{{ cite journal | first1=Michel | last1=Petitjean
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| |title=On the analytical calculation of van der Waals surfaces and volumes: some numerical aspects
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| |journal=Int. J. Quant. Chem.
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| |year=1994 | volume=15 | number=5 | pages=507–523
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| }}
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| * {{cite journal | first1=J. A. | last1=Grant
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| |first2=B. T. | last2=Pickup
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| |title=A Gaussian description of molecular shape
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| |journal=J. Phys. Chem.
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| |year=1995 | volume=99 | number= 11
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| |doi=10.1021/j100011a016 |pages=3503–3510}}
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| * {{cite journal | first1= Jan | last1=Busa | first2=Jozef | last2=Dzurina
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| |first3=Edik | last3=Hayryan | first4=Shura | last4=Hayryan
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| |title=ARVO: A fortran package for computing the solvent accessible surface area and the excluded volume of overlapping spheres via analytic equations
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| |journal= Comp. Phys. Commun. |bibcode=2005CoPhC.165...59B
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| |year=2005 | volume=165 | pages=59–96 | doi=10.1016/j.cpc.2004.08.002
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| }}
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| * {{cite journal |last=Li |first=S. |title=Concise Formulas for the Area and Volume of a Hyperspherical Cap |journal=Asian J. Math. Stat. |volume=4 |number=1|pages=66–70 |year=2011 |doi=10.3923/ajms.2011.66.70}}.
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| ==External links==
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| {{Commons category|Spherical caps}}
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| * {{MathWorld |id=SphericalCap |title=Spherical cap}}, derivation and some additional formulas
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| * [http://formularium.org/?go=81 Online calculator for spherical cap volume and area]
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| * [http://mathforum.org/dr.math/faq/formulas/faq.sphere.html#spherecap Summary of spherical formulas]
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| [[Category:Spheres]]
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