sphericity

{{Short description|Measure of how closely a shape resembles a sphere}}

File:Rounding & sphericity EN.svg (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,{{cite journal |first=Hakon |last=Wadell |title=Volume, Shape, and Roundness of Quartz Particles |journal=The Journal of Geology |volume=43 |year=1935 |pages=250–280 |doi=10.1086/624298 |issue=3 |bibcode=1935JG.....43..250W }} 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

class="wikitable"
Name !Picture !Volume !Surface area !Sphericity
Sphere \|50px \| $\frac{4\pi}{3}\,r^3$ \| $4\pi\,r^2$ \|data-sort-value=1\| $1$
Disdyakis triacontahedron \|50px \| $\frac{900+720\sqrt{5}}{11}\,s^3$ \| $\frac{180\sqrt{179-24\sqrt{5}}}{11}\,s^2$ \|data-sort-value=0.9857\| $\frac{\left(\left(5+4\sqrt{5}\right)^{2}\frac{11\pi}{5}\right)^{\frac{1}{3}}}{\sqrt{179-24\sqrt{5}}}\approx0.9857$
Tricylinder \|50px \| $16-8\sqrt{2}\,r^3$ \| $48-24\sqrt{2}\,r^2$ \|data-sort-value=0.9633\| $\frac{\sqrt[3]{36\pi+18\pi\sqrt{2}}}{6}\approx0.9633$
Rhombic triacontahedron \|50px \| $4\sqrt{5+2\sqrt{5}}\,s^3$ \| $12\sqrt{5}\,s^2$ \|data-sort-value=0.9609\| $\frac{\sqrt[6]{455625\pi^{2}+202500\pi^{2}\sqrt{5}}}{15}\approx0.9609$
Icosahedron \|50px \| $\frac{15+5\sqrt{5}}{12}\,s^3$ \| $5\sqrt{3}\,s^2$ \|data-sort-value=0.9393\| $\frac{\sqrt[3]{2100\pi\sqrt{3}+900\pi\sqrt{15}}}{30}\approx0.9393$
Bicylinder \|50px \| $\frac{16}{3}\,r^3$ \| $16\,r^2$ \|data-sort-value=0.9226\| $\frac{\sqrt[3]{2\pi}}{2}\approx0.9226$
Ideal bicone $(h=r\sqrt{2})$ \|50px \| $\frac{2\pi}{3}\,r^{2}h=\frac{2\pi\sqrt{2}}{3}\,r^3$ \| $2\pi\,r\sqrt{r^{2}+h^{2}}=2\pi\sqrt{3}\,r^2$ \|data-sort-value=0.9165\| $\frac{\sqrt[6]{432}}{3}\approx0.9165$
Dodecahedron \|50px \| $\frac{15+\sqrt{5}}{4}\,s^3$ \| $3\sqrt{25+10\sqrt{5}}\, s^2$ \|data-sort-value=0.9105\| $\left(\frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}}\right)^{\frac{1}{3}}\approx0.9105$
Rhombic dodecahedron \|50px \| $\frac{16\sqrt{3}}{9}\,s^3$ \| $8\sqrt{2}\,s^2$ \|data-sort-value=0.9047\| $\frac{\sqrt[6]{2592\pi^2}}{6}\approx0.9047$
Ideal torus $(R=r)$ \|50px \| $2\pi^2Rr^2=2\pi^2\,r^3$ \| $4\pi^2Rr=4\pi^2\,r^2$ \|data-sort-value=0.8947\| $\frac{\sqrt[3]{18\pi^2}}{2\pi}\approx0.8947$
Ideal cylinder $(h=2r)$ \|50px \| $\pi\,r^2h=2\pi\,r^3$ \| $2\pi\,r(r+h)=6\pi\,r^2$ \|data-sort-value=0.8736\| $\frac{\sqrt[3]{18}}{3}\approx0.8736$
Octahedron \|50px \| $\frac{\sqrt{2}}{3}\,s^3$ \| $2\sqrt{3}\,s^2$ \|data-sort-value=0.8456\| $\frac{\sqrt[3]{3\pi\sqrt{3}}}{3}\approx0.8456$
Hemisphere \|50px \| $\frac{2\pi}{3}\,r^3$ \| $3\pi\,r^2$ \|data-sort-value=0.8299\| $\frac{2\sqrt[3]{2}}{3}\approx0.8399$
Cube \|50px \| $\,s^3$ \| $6\,s^2$ \|data-sort-value=0.8060\| $\frac{\sqrt[3]{36\pi}}{6}\approx0.8060$
Ideal cone $(h=2r\sqrt{2})$ \|50px \| $\frac{\pi}{3}\,r^2h=\frac{2\pi\sqrt{2}}{3}\,r^3$ \| $\pi\,r(r+\sqrt{r^2+h^2})=4\pi\,r^2$ \|data-sort-value=0.7937\| $\frac{\sqrt[3]{4}}{2}\approx0.7937$
Tetrahedron \|50px \| $\frac{\sqrt{2}}{12}\,s^3$ \| $\sqrt{3}\,s^2$ \|data-sort-value=0.6711\| $\frac{\sqrt[3]{12\pi\sqrt{3}}}{6}\approx0.6711$

class="wikitable"

Name

!Picture

!Volume

!Surface area

!Sphericity

sphericity

Definition

Ellipsoidal objects

Derivation

Sphericity of common objects

See also

References

External links