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| [[File:Spherical Cap.svg|thumb|200px|The spherical cap is the purple section.]]
| | Let me first start with introducing average joe. My name is Mia Salas. Playing crochet is something my significant other doesn't fancy but I. Her husband and her decided to reside in Michigan but she have got to move one day or the opposite. I work as a database administrator. Check out his website here: http://euroseonet.hol.es/ |
| 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}]}
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| {\,\,}_{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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Let me first start with introducing average joe. My name is Mia Salas. Playing crochet is something my significant other doesn't fancy but I. Her husband and her decided to reside in Michigan but she have got to move one day or the opposite. I work as a database administrator. Check out his website here: http://euroseonet.hol.es/