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Sphericity

 
(sfə′ris·əd·ē)

(science and technology) The degree to which a shape approaches that of a sphere.


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Sphericity

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Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and rounding (horizontal).

Sphericity is a measure of how spherical (round) an object is. As such, it is a specific example of a compactness measure of a shape. Defined by Wadell in 1935,[1] the sphericity, Ψ, of a particle is the ratio of the surface area of a sphere (with the same volume as the given particle) to the surface area of the particle:

\Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}

where Vp is volume of the particle and Ap is the surface area of the particle

Contents

Ellipsoidal objects

The sphericity, Ψ, of an oblate spheroid (similar to the shape of the planet Earth) is defined as such:

\Psi = 
\frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p} = 
\frac{2\sqrt[3]{ab^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln{(\frac{a+\sqrt{a^2-b^2}}b)}}

(where a, b are the semi-major, semi-minor axes, respectively.

Derivation

Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.

First we need to write surface area of the sphere, As in terms of the volume of the particle, Vp

A_{s}^3 = \left(4 \pi r^2\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left(4^2 \pi^2 r^6\right) = 4 \pi \cdot 3^2 \left(\frac{4^2 \pi^2}{3^2} r^6\right) = 36 \pi \left(\frac{4 \pi}{3} r^3\right)^2 = 36\,\pi V_{p}^2

therefore

A_{s} = \left(36\,\pi V_{p}^2\right)^{\frac{1}{3}} = 36^{\frac{1}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = 6^{\frac{2}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}}

hence we define Ψ as:


\Psi = \frac{A_s}{A_p} = \frac{ \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}} }{A_{p}}

Sphericity of common objects

Name Picture Volume Area Sphericity
Platonic Solids
tetrahedron Tetrahedron \frac{\sqrt{2}}{12}\,s^3 \sqrt{3}\,s^2 \left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}} \approx 0.671
cube (hexahedron) Hexahedron (cube) \,s^3 6\,s^2

\left(
\frac{\pi}{6}
\right)^{\frac{1}{3}} \approx 0.806

octahedron Octahedron  \frac{1}{3} \sqrt{2}\, s^3  2 \sqrt{3}\, s^2

\left(
\frac{\pi}{3\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.846

dodecahedron Dodecahedron  \frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3  3 \sqrt{25 + 10\sqrt{5}}\, s^2

\left(
\frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}}
\right)^{\frac{1}{3}} \approx 0.910

icosahedron Icosahedron \frac{5}{12}\left(3+\sqrt{5}\right)\, s^3 5\sqrt{3}\,s^2 \left(
\frac{ \left(3 + \sqrt{5} \right)^2 \pi}{60\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.939
Round Shapes
ideal cone
(h=2\sqrt{2}r)
\frac{1}{3} \pi\, r^2 h

= \frac{2\sqrt{2}}{3} \pi\, r^3

\pi\, r (r + \sqrt{r^2 + h^2})

= 4 \pi\, r^2

\left(
\frac{1}{2}
\right)^{\frac{1}{3}} \approx 0.794
hemisphere
(half sphere)
\frac{2}{3} \pi\, r^3 3 \pi\, r^2

\left(
\frac{16}{27}
\right)^{\frac{1}{3}} \approx 0.840

ideal cylinder
(h=2\,r)
\pi r^2 h = 2 \pi\,r^3 2 \pi r ( r + h ) = 6 \pi\,r^2

\left(
\frac{2}{3}
\right)^{\frac{1}{3}} \approx 0.874

ideal torus
(R = r)
2 \pi^2 R r^2 = 2 \pi^2 \,r^3 4 \pi^2 R r = 4 \pi^2\,r^2

\left(
\frac{9}{4 \pi}
\right)^{\frac{1}{3}} \approx 0.894

sphere \frac{4}{3} \pi r^3 4 \pi\,r^2


1\,

Sphericity in statistics

In statistical analyses, sphericity relates to the equality of the variances of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. A sufficient (but not necessary) condition for sphericity is that the variances of the sets of scores are equal and the covariances of the sets of scores are equal. This is an assumption of an ANOVA with a repeated measures factor, where violations of this assumption can invalidate the analysis conclusions. Mauchly's sphericity test is one of the statistical tests used to evaluate sphericity.

See also

References

  1. ^ Wadell, Hakon (1935). "Volume, Shape and Roundness of Quartz Particles". Journal of Geology 43 (3): 250–280. doi:10.1086/624298. 

External links


 
 

 

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McGraw-Hill Science & Technology Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved.  Read more
Wikipedia on Answers.com. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Sphericity Read more

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