Sphericity

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, \Psi , 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 V_p is volume of the particle and A_p is the surface area of the particle

Contents

Ellipsoidal objects

See also: Earth radius

The sphericity, \Psi , 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%2B\frac{b^2}{\sqrt{a^2-b^2}}\ln{(\frac{a%2B\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, A_s in terms of the volume of the particle, V_p

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 \Psi 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 \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) \,s^3 6\,s^2

\left( \frac{\pi}{6} \right)^{\frac{1}{3}} \approx 0.806
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 \frac{1}{4} \left(15 %2B 7\sqrt{5}\right)\, s^3 3 \sqrt{25 %2B 10\sqrt{5}}\, s^2

\left( \frac{\left(15 %2B 7\sqrt{5}\right)^2 \pi}{12\left(25%2B10\sqrt{5}\right)^{\frac{3}{2}}} \right)^{\frac{1}{3}} \approx 0.910
icosahedron \frac{5}{12}\left(3%2B\sqrt{5}\right)\, s^3 5\sqrt{3}\,s^2 \left( \frac{ \left(3 %2B \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 %2B \sqrt{r^2 %2B 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 %2B 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