Sphericity

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, [cite journal |first=Hakon |last=Wadell |title=Volume, Shape and Roundness of Quartz Particles |journal=Journal of Geology |volume=43 |year=1935 |pages=250–280 ] 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

Ellipsoidal Objects

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{2sqrt [3] {ab^2{a+frac{b^2}{sqrt{a^2-b^2ln{(frac{a+sqrt{a^2-b^2b)

"(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 ight)^3 = 4^3 pi^3 r^6 = 4 pi left(4^2 pi^2 r^6 ight) = 4 pi cdot 3^2 left(frac{4^2 pi^2}{3^2} r^6 ight) = 36 pi left(frac{4 pi}{3} r^3 ight)^2 = 36,pi V_{p}^2

therefore

:A_{s} = left(36,pi V_{p}^2 ight)^{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} ight)^{frac{2}{3

hence we define Psi as:

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

Sphericity of common objects

phericity 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. 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 the statistical test used to evaluate sphericity.

References

See also

*Rounding (sediment)
*Sphericity scale

External links

* [http://www.howround.com/ How round is your circle?]
* [http://people.uncw.edu/dockal/gly312/grains/grains.htm Grain Morphology: Roundness, Surface Features, and Sphericity of Grains]


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  • Sphericity — Sphe*ric i*ty, n. [Cf. F. sph[ e]ricit[ e].] The quality or state of being spherial; roundness; as, the sphericity of the planets, or of a drop of water. [1913 Webster] …   The Collaborative International Dictionary of English

  • sphericity — [sfi ris′ə tē] n. [ML sphaericitas] the state of being spherical; round form; roundness …   English World dictionary

  • sphericity — sferiškumas statusas T sritis fizika atitikmenys: angl. spherical shape; sphericity vok. Kugelform, f; Kugelförmigkeit, f; Kugelgestalt, f rus. сферичность, f; шарообразность, f pranc. sphéricité, f …   Fizikos terminų žodynas

  • Sphericity scale — The sphericity scale is a measure used by geologists and geographers which measures the sphericity of a stone or particle. A geographer can tell from a stone s sphericity if significant erosion has taken place in its location (such as a river).… …   Wikipedia

  • sphericity — noun see sphere I …   New Collegiate Dictionary

  • sphericity — /sfi ris i tee/, n., pl. sphericities. a spherical state or form. [1615 25; < NL sphericitas, equiv. to LL spheric(us) (see SPHERICS1) + itas ITY] * * * …   Universalium

  • sphericity — noun a) The quality of being spherical, being a sphere. b) The ratio of the surface area of a given particle to the surface area of a sphere with the same volume. Syn: sphericality, sphericalness, sphereness, orbicularity …   Wiktionary

  • sphericity — sphe·ric·i·ty || sfɪ rɪsÉ™tɪ n. roundness, state of being a sphere …   English contemporary dictionary

  • sphericity — n. Rotundity, roundness, globosity, globularity …   New dictionary of synonyms

  • sphericity — sphe·ric·i·ty …   English syllables

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