Sphericity is a measure of how closely the shape of an object resembles that of a perfect
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
. For example, the sphericity of the
balls inside a
ball bearing
A ball bearing is a type of rolling-element bearing that uses balls to maintain the separation between the bearing races.
The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. It achieves this ...
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
The compactness measure of a shape is a numerical quantity representing the degree to which a shape is compact. The meaning of "compact" here is not related to the topological notion of compact space.
Properties
Various compactness measures are us ...
. Defined by Wadell in 1935,
the sphericity,
, of a particle is the ratio of the
surface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
of a sphere with the same volume as the given particle to the surface area of the particle:
:
where
is volume of the particle and
is the surface area of the particle. The sphericity of a sphere is
unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; a ...
by definition and, by the
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
, any particle which is not a sphere will have sphericity less than 1.
Sphericity applies in
three dimensions
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
; its analogue in
two dimensions, such as the
cross sectional circles along a
cylindrical
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an in ...
object such as a
shaft
Shaft may refer to:
Rotating machine elements
* Shaft (mechanical engineering), a rotating machine element used to transmit power
* Line shaft, a power transmission system
* Drive shaft, a shaft for transferring torque
* Axle, a shaft around whi ...
, is called
roundness
Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle. Roundness applies in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft or a cylindr ...
.
Ellipsoidal objects
The sphericity,
, of an
oblate spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ci ...
(similar to the shape of the planet
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
) is:
:
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 particle divided by the actual surface area of the particle.
First we need to write surface area of the sphere,
in terms of the volume of the particle,
:
therefore
:
hence we define
as:
:
Sphericity of common objects
See also
*
Equivalent spherical diameter
*
Flattening
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
*
Index of sphericity
*
Isoperimetric ratio
*
Rounding (sediment)
Roundness is the degree of smoothing due to abrasion of sedimentary particles. It is expressed as the ratio of the average radius of curvature of the edges or corners to the radius of curvature of the maximum inscribed sphere.
Measure of roundn ...
*
Roundness
Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle. Roundness applies in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft or a cylindr ...
*
Willmore energy
In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is ...
References
External links
{{Wiktionary, sphericity
Grain Morphology: Roundness, Surface Features, and Sphericity of Grains
Geometric measurement
Spheres
Metrology