Sphericity

 ( Geometric Measurement ) 

Sphericity is a measure of how closely the shape of a physical object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed at which it can turn without failing. Sphericity is a specific example of a compactness measure of a shape.

Sphericity applies in three dimensions; its analogue in two dimensions, such as the cross sectional circles along a cylinder object such as a shaft, is called roundness.

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Definition Defined by Wadell in 1935, the sphericity, $\backslash Psi$, of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:

$\backslash Psi\; =\; \backslash frac\{\backslash pi^\{\backslash frac\{1\}\{3\}\}(6V\_p)^\{\backslash frac\{2\}\{3\}\}\}\{A\_p\}$

where $V\_p$ is volume of the object and $A\_p$ is the surface area. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality, any shape which is not a sphere will have sphericity less than 1.

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Ellipsoidal objects The sphericity, $\backslash Psi$, of an oblate spheroid (similar to the shape of the planet Earth) is:

$\backslash Psi\; =$

\frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p} = \frac{2\sqrt3{ab^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln{\left(\frac{a+\sqrt{a^2-b^2}}b\right)}},

where a and b are the Semi-major axis and semi-minor axis axes respectively.

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

First we need to write surface area of the sphere, $A\_s$ in terms of the volume of the object being measured, $V\_p$

$A\_\{s\}^3\; =\; \backslash left(4\; \backslash pi\; r^2\backslash right)^3\; =\; 4^3\; \backslash pi^3\; r^6\; =\; 4\; \backslash pi\; \backslash left(4^2\; \backslash pi^2\; r^6\backslash right)\; =\; 4\; \backslash pi\; \backslash cdot\; 3^2\; \backslash left(\backslash frac\{4^2\; \backslash pi^2\}\{3^2\}\; r^6\backslash right)\; =\; 36\; \backslash pi\; \backslash left(\backslash frac\{4\; \backslash pi\}\{3\}\; r^3\backslash right)^2\; =\; 36\backslash ,\backslash pi\; V\_\{p\}^2$

therefore

$A\_\{s\}\; =\; \backslash left(36\backslash ,\backslash pi\; V\_\{p\}^2\backslash right)^\{\backslash frac\{1\}\{3\}\}\; =\; 36^\{\backslash frac\{1\}\{3\}\}\; \backslash pi^\{\backslash frac\{1\}\{3\}\}\; V\_\{p\}^\{\backslash frac\{2\}\{3\}\}\; =\; 6^\{\backslash frac\{2\}\{3\}\}\; \backslash pi^\{\backslash frac\{1\}\{3\}\}\; V\_\{p\}^\{\backslash frac\{2\}\{3\}\}\; =\; \backslash pi^\{\backslash frac\{1\}\{3\}\}\; \backslash left(6V\_\{p\}\backslash right)^\{\backslash frac\{2\}\{3\}\}$

hence we define $\backslash Psi$ as:

$$

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Sphericity of common objects

Sphere		$\backslash frac\{4\backslash pi\}\{3\}\backslash ,r^3$	$4\backslash pi\backslash ,r^2$	$1$
Disdyakis triacontahedron		$\backslash frac\{900+720\backslash sqrt\{5\}\}\{11\}\backslash ,s^3$	$\backslash frac\{180\backslash sqrt\{179-24\backslash sqrt\{5\}\}\}\{11\}\backslash ,s^2$	$\backslash frac\{\backslash left(\backslash left(5+4\backslash sqrt\{5\}\backslash right)^\{2\}\backslash frac\{11\backslash pi\}\{5\}\backslash right)^\{\backslash frac\{1\}\{3\}\}\}\{\backslash sqrt\{179-24\backslash sqrt\{5\}\}\}\backslash approx0.9857$
Steinmetz solid		$16-8\backslash sqrt\{2\}\backslash ,r^3$	$48-24\backslash sqrt\{2\}\backslash ,r^2$	$\backslash frac\{\backslash sqrt3\{36\backslash pi+18\backslash pi\backslash sqrt\{2\}\}\}\{6\}\backslash approx0.9633$
Rhombic triacontahedron		$4\backslash sqrt\{5+2\backslash sqrt\{5\}\}\backslash ,s^3$	$12\backslash sqrt\{5\}\backslash ,s^2$	$\backslash frac\{\backslash sqrt6\{455625\backslash pi^\{2\}+202500\backslash pi^\{2\}\backslash sqrt\{5\}\}\}\{15\}\backslash approx0.9609$
Icosahedron		$\backslash frac\{15+5\backslash sqrt\{5\}\}\{12\}\backslash ,s^3$	$5\backslash sqrt\{3\}\backslash ,s^2$	$\backslash frac\{\backslash sqrt3\{2100\backslash pi\backslash sqrt\{3\}+900\backslash pi\backslash sqrt\{15\}\}\}\{30\}\backslash approx0.9393$
Steinmetz solid		$\backslash frac\{16\}\{3\}\backslash ,r^3$	$16\backslash ,r^2$	$\backslash frac\{\backslash sqrt3\{2\backslash pi\}\}\{2\}\backslash approx0.9226$
Ideal bicone $(h=r\backslash sqrt\{2\})$		$\backslash frac\{2\backslash pi\}\{3\}\backslash ,r^\{2\}h=\backslash frac\{2\backslash pi\backslash sqrt\{2\}\}\{3\}\backslash ,r^3$	$2\backslash pi\backslash ,r\backslash sqrt\{r^\{2\}+h^\{2\}\}=2\backslash pi\backslash sqrt\{3\}\backslash ,r^2$	$\backslash frac\{\backslash sqrt6\{432\}\}\{3\}\backslash approx0.9165$
Dodecahedron		$\backslash frac\{15+\backslash sqrt\{5\}\}\{4\}\backslash ,s^3$	$3\backslash sqrt\{25+10\backslash sqrt\{5\}\}\backslash ,\; s^2$	$\backslash left(\backslash frac\{\backslash left(15\; +\; 7\backslash sqrt\{5\}\backslash right)^2\; \backslash pi\}\{12\backslash left(25+10\backslash sqrt\{5\}\backslash right)^\{\backslash frac\{3\}\{2\}\}\}\backslash right)^\{\backslash frac\{1\}\{3\}\}\backslash approx0.9105$
Rhombic dodecahedron		$\backslash frac\{16\backslash sqrt\{3\}\}\{9\}\backslash ,s^3$	$8\backslash sqrt\{2\}\backslash ,s^2$	$\backslash frac\{\backslash sqrt6\{2592\backslash pi^2\}\}\{6\}\backslash approx0.9047$
Ideal torus $(R=r)$		$2\backslash pi^2Rr^2=2\backslash pi^2\backslash ,r^3$	$4\backslash pi^2Rr=4\backslash pi^2\backslash ,r^2$	$\backslash frac\{\backslash sqrt3\{18\backslash pi^2\}\}\{2\backslash pi\}\backslash approx0.8947$
Ideal cylinder $(h=2r)$		$\backslash pi\backslash ,r^2h=2\backslash pi\backslash ,r^3$	$2\backslash pi\backslash ,r(r+h)=6\backslash pi\backslash ,r^2$	$\backslash frac\{\backslash sqrt3\{18\}\}\{3\}\backslash approx0.8736$
Octahedron		$\backslash frac\{\backslash sqrt\{2\}\}\{3\}\backslash ,s^3$	$2\backslash sqrt\{3\}\backslash ,s^2$	$\backslash frac\{\backslash sqrt3\{3\backslash pi\backslash sqrt\{3\}\}\}\{3\}\backslash approx0.8456$
Hemisphere		$\backslash frac\{2\backslash pi\}\{3\}\backslash ,r^3$	$3\backslash pi\backslash ,r^2$	$\backslash frac\{2\backslash sqrt3\{2\}\}\{3\}\backslash approx0.8399$
Cube		$\backslash ,s^3$	$6\backslash ,s^2$	$\backslash frac\{\backslash sqrt3\{36\backslash pi\}\}\{6\}\backslash approx0.8060$
Ideal cone $(h=2r\backslash sqrt\{2\})$		$\backslash frac\{\backslash pi\}\{3\}\backslash ,r^2h=\backslash frac\{2\backslash pi\backslash sqrt\{2\}\}\{3\}\backslash ,r^3$	$\backslash pi\backslash ,r(r+\backslash sqrt\{r^2+h^2\})=4\backslash pi\backslash ,r^2$	$\backslash frac\{\backslash sqrt3\{4\}\}\{2\}\backslash approx0.7937$
Tetrahedron		$\backslash frac\{\backslash sqrt\{2\}\}\{12\}\backslash ,s^3$	$\backslash sqrt\{3\}\backslash ,s^2$	$\backslash frac\{\backslash sqrt3\{12\backslash pi\backslash sqrt\{3\}\}\}\{6\}\backslash approx0.6711$

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See also

• Equivalent spherical diameter
• Flattening
• Isoperimetric ratio
• Rounding (sediment)
• Roundness
• Willmore energy

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External links

• Grain Morphology: Roundness, Surface Features, and Sphericity of Grains

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