Sphericity

File:Rounding & sphericity EN.svg

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

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 cylindrical object such as a shaft, is called roundness.

Definition

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

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

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.

Ellipsoidal objects

The sphericity, $\mathrm{\Psi }$, of an oblate spheroid (similar to the shape of the planet Earth) is:

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

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

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, $V_p$

$A_s^3={\left(4\pi r^2\right)}^3=4^3{\pi }^3{r}^6=4\pi \left(4^2{\pi }^2{r}^6\right)=4\pi \cdot 3^2\left(\frac{4^2{\pi }^2}{3^2}{r}^6\right)=36\pi {\left(\frac{4\pi }{3}{r}^3\right)}^2=36\pi V_p^2$

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 $\mathrm{\Psi }$ as:

$\mathrm{\Psi }=\frac{A_s}{A_p}=\frac{{\pi }^{\frac{1}{3}}{\left(6V_p\right)}^{\frac{2}{3}}}{A_p}$

Sphericity of common objects

Name	Picture	Volume	Surface area	Sphericity
Sphere	File:Sphere wireframe 10deg 6r.svg	$\frac{4\pi }{3}{r}^3$	$4\pi r^2$	$1$
Disdyakis triacontahedron	File:Disdyakistriacontahedron.jpg	$\frac{900+720\sqrt{5}}{11}{s}^3$	$\frac{180\sqrt{179-24\sqrt{5}}}{11}{s}^2$	$\frac{{\left({(5+4\sqrt{5})}^2\frac{11\pi }{5}\right)}^{\frac{1}{3}}}{\sqrt{179-24\sqrt{5}}}\approx 0.9857$
Tricylinder	File:Tricylinder.png	$16-8\sqrt{2}{r}^3$	$48-24\sqrt{2}{r}^2$	$\frac{\sqrt[3]{36\pi +18\pi \sqrt{2}}}{6}\approx 0.9633$
Rhombic triacontahedron	File:Rhombictriacontahedron.svg	$4\sqrt{5+2\sqrt{5}}{s}^3$	$12\sqrt{5}{s}^2$	$\frac{\sqrt[6]{455625{\pi }^2+202500{\pi }^2\sqrt{5}}}{15}\approx 0.9609$
Icosahedron	File:Icosahedron.svg	$\frac{15+5\sqrt{5}}{12}{s}^3$	$5\sqrt{3}{s}^2$	$\frac{\sqrt[3]{2100\pi \sqrt{3}+900\pi \sqrt{15}}}{30}\approx 0.9393$
Bicylinder	File:Steinmetz-solid.svg	$\frac{16}{3}{r}^3$	$16r^2$	$\frac{\sqrt[3]{2\pi }}{2}\approx 0.9226$
Ideal bicone $(h=r\sqrt{2})$	File:Bicone.svg	$\frac{2\pi }{3}{r}^{2}h=\frac{2\pi \sqrt{2}}{3}{r}^3$	$2\pi r\sqrt{r^2+h^2}=2\pi \sqrt{3}{r}^2$	$\frac{\sqrt[6]{432}}{3}\approx 0.9165$
Dodecahedron	File:POV-Ray-Dodecahedron.svg	$\frac{15+\sqrt{5}}{4}{s}^3$	$3\sqrt{25+10\sqrt{5}}{s}^2$	${\left(\frac{{(15+7\sqrt{5})}^2\pi }{12{(25+10\sqrt{5})}^{\frac{3}{2}}}\right)}^{\frac{1}{3}}\approx 0.9105$
Rhombic dodecahedron	File:Rhombicdodecahedron.jpg	$\frac{16\sqrt{3}}{9}{s}^3$	$8\sqrt{2}{s}^2$	$\frac{\sqrt[6]{2592{\pi }^2}}{6}\approx 0.9047$
Ideal torus $(R=r)$	File:Torus2.png	$2{\pi }^{2}R{r}^2=2{\pi }^2{r}^3$	$4{\pi }^{2}Rr=4{\pi }^2{r}^2$	$\frac{\sqrt[3]{18{\pi }^2}}{2\pi }\approx 0.8947$
Ideal cylinder $(h=2r)$	File:Circular cylinder rh.svg	$\pi r^{2}h=2\pi r^3$	$2\pi r(r+h)=6\pi r^2$	$\frac{\sqrt[3]{18}}{3}\approx 0.8736$
Octahedron	File:Octahedron.svg	$\frac{\sqrt{2}}{3}{s}^3$	$2\sqrt{3}{s}^2$	$\frac{\sqrt[3]{3\pi \sqrt{3}}}{3}\approx 0.8456$
Hemisphere	File:Sphere symmetry group cs.png	$\frac{2\pi }{3}{r}^3$	$3\pi r^2$	$\frac{2\sqrt[3]{2}}{3}\approx 0.8399$
Cube	File:Hexahedron.svg	$s^3$	$6s^2$	$\frac{\sqrt[3]{36\pi }}{6}\approx 0.8060$
Ideal cone $(h=2r\sqrt{2})$	File:Blue-cone.png	$\frac{\pi }{3}{r}^{2}h=\frac{2\pi \sqrt{2}}{3}{r}^3$	$\pi r(r+\sqrt{r^2+h^2})=4\pi r^2$	$\frac{\sqrt[3]{4}}{2}\approx 0.7937$
Tetrahedron	File:Tetrahedron.svg	$\frac{\sqrt{2}}{12}{s}^3$	$\sqrt{3}{s}^2$	$\frac{\sqrt[3]{12\pi \sqrt{3}}}{6}\approx 0.6711$

See also

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

External links

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