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Sphericity

Measure of how closely a shape resembles a sphere

Measure of how closely a shape resembles a sphere

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, \Psi , of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:

:\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, \Psi , of an oblate spheroid (similar to the shape of the planet Earth) is:

:\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{\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 \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

NamePictureVolumeSurface areaSphericity
Sphere[[File:Sphere wireframe 10deg 6r.svg50px]]\frac{4\pi}{3}\,r^34\pi\,r^21
Disdyakis triacontahedron[[File:Disdyakistriacontahedron.jpg50px]]\frac{900+720\sqrt{5}}{11}\,s^3\frac{180\sqrt{179-24\sqrt{5}}}{11}\,s^2\frac{\left(\left(5+4\sqrt{5}\right)^{2}\frac{11\pi}{5}\right)^{\frac{1}{3}}}{\sqrt{179-24\sqrt{5}}}\approx0.9857
Tricylinder[[File:Tricylinder.png50px]]16-8\sqrt{2}\,r^348-24\sqrt{2}\,r^2\frac{\sqrt[3]{36\pi+18\pi\sqrt{2}}}{6}\approx0.9633
Rhombic triacontahedron[[File:Rhombictriacontahedron.svg50px]]4\sqrt{5+2\sqrt{5}}\,s^312\sqrt{5}\,s^2\frac{\sqrt[6]{455625\pi^{2}+202500\pi^{2}\sqrt{5}}}{15}\approx0.9609
Icosahedron[[File:Icosahedron.svg50px]]\frac{15+5\sqrt{5}}{12}\,s^35\sqrt{3}\,s^2\frac{\sqrt[3]{2100\pi\sqrt{3}+900\pi\sqrt{15}}}{30}\approx0.9393
Bicylinder[[File:Steinmetz-solid.svg50px]]\frac{16}{3}\,r^316\,r^2\frac{\sqrt[3]{2\pi}}{2}\approx0.9226
Ideal bicone
(h=r\sqrt{2})[[File:Bicone.svg50px]]\frac{2\pi}{3}\,r^{2}h=\frac{2\pi\sqrt{2}}{3}\,r^32\pi\,r\sqrt{r^{2}+h^{2}}=2\pi\sqrt{3}\,r^2\frac{\sqrt[6]{432}}{3}\approx0.9165
Dodecahedron[[File:POV-Ray-Dodecahedron.svg50px]]\frac{15+\sqrt{5}}{4}\,s^33\sqrt{25+10\sqrt{5}}\, s^2\frac{\sqrt[6]{2080+928\sqrt{5}}\sqrt[3]{9\pi}\sqrt{5}}{30}\approx0.9105
Rhombic dodecahedron[[File:Rhombicdodecahedron.jpg50px]]\frac{16\sqrt{3}}{9}\,s^38\sqrt{2}\,s^2\frac{\sqrt[6]{2592\pi^2}}{6}\approx0.9047
Ideal torus(R=r)[[File:Torus2.png50px]]2\pi^2Rr^2=2\pi^2\,r^34\pi^2Rr=4\pi^2\,r^2\frac{\sqrt[3]{18\pi^2}}{2\pi}\approx0.8947
Ideal cylinder(h=2r)[[File:Circular cylinder rh.svg50px]]\pi\,r^2h=2\pi\,r^32\pi\,r(r+h)=6\pi\,r^2\frac{\sqrt[3]{18}}{3}\approx0.8736
Octahedron[[File:Octahedron.svg50px]]\frac{\sqrt{2}}{3}\,s^32\sqrt{3}\,s^2\frac{\sqrt[3]{3\pi\sqrt{3}}}{3}\approx0.8456
Hemisphere[[File:Sphere symmetry group cs.svg50px]]\frac{2\pi}{3}\,r^33\pi\,r^2\frac{2\sqrt[3]{2}}{3}\approx0.8399
Cube[[File:Hexahedron.svg50px]]\,s^36\,s^2\frac{\sqrt[3]{36\pi}}{6}\approx0.8060
Ideal cone(h=2r\sqrt{2})[[File:Blue-cone.png50px]]\frac{\pi}{3}\,r^2h=\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}\approx0.7937
Tetrahedron[[File:Tetrahedron.svg50px]]\frac{\sqrt{2}}{12}\,s^3\sqrt{3}\,s^2\frac{\sqrt[3]{12\pi\sqrt{3}}}{6}\approx0.6711

References

References

  1. Wadell, Hakon. (1935). "Volume, Shape, and Roundness of Quartz Particles". The Journal of Geology.
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