Function Repository Resource:

GyrationTensor

Source Notebook

Compute the gyration tensor and derived metrics quantifying the shape of particle distributions

Contributed by: Nicholas E. Brunk, Wolfram|Alpha Math Team

ResourceFunction["GyrationTensor"][{p₁,p₂,…}]

returns the gyration tensor of points {p₁,p₂,…}.

ResourceFunction["GyrationTensor"][{p₁,p₂,…},type]

returns the derived metric specified via type.

Details

ResourceFunction["GyrationTensor"] allows points {p₁,p₂,…}of any consistent dimensionality ℝⁿ.

Metric type availability depends upon the dimensionality of the coordinates in ℝⁿ:

Dimension n

type

any

"ResourceFunction["GyrationTensor"]", "RadiusOfGyration", "Eigenvalues"

"Acircularity", "NormalizedAcircularity"

"Asphericity", "NormalizedAsphericity", "RelativeAnisotropy"

"HyperAsphericity", "NormalizedHyperAsphericity"

Examples

Basic Examples (1)

By default, GyrationTensor returns the tensor itself:

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Scope (2)

Other metrics derived from the gyration tensor may be computed:

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Metrics may be computed in other dimensions, for example 2D and 4D, respectively:

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Properties and Relations (2)

The metrics "NormalizedAsphericity" and "RelativeAnisotropy" have the same limiting behavior in describing spherically symmetric vs. linear distributions:

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These metrics do, however, differ in non-limiting cases:

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Possible Issues (2)

Available metrics are determined by the dimensions of the input vectors, such that "Acircularity"—a 2D metric—is not computed for 3D:

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A given dimension can be omitted to check symmetry in the remaining fewer dimensions:

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Non-normalized metrics scale with the size of the distribution:

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Normalized equivalents do not and are scaled between 0 and 1:

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Neat Examples (8)

Closely approximate the radius of gyration of a given 2D region, for example, reproducing known quantities such as a hoop (unit circle):

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$coords = RandomPoint[ImplicitRegion[x^2 + y^2 == 1, {x, y}], 300]; Graphics3D[{Point[coords /. {x_, y_} :> {x, y, 0}]}]$

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The same works for a hollow, 3D unit sphere:

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The hollow sphere has a known radius of gyration R_g=R=1, which can be approximated (quite accurately, given the SpherePoints function):

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The same works for a solid, 3D unit sphere, using ImplicitRegion:

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$coords = RandomPoint[ImplicitRegion[x^2 + y^2 + z^2 <= 1, {x, y, z}], 30000]; Graphics3D[{{Opacity[0.5], Sphere[]}, Point[coords], {Thickness[0.005], Line[{{0, 0, 0}, {1, 0, 0}}]}}]$

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The solid sphere has a known radius of gyration :

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The intermediate result of a hollow spherical shell of a given thickness may also be estimated, using ImplicitRegion:

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$\[ScriptCapitalR] = ImplicitRegion[.75 <= x^2 + y^2 + z^2 <= 1, {x, y, z}]; coords = RandomPoint[\[ScriptCapitalR], 300]; Show[RegionPlot3D[\[ScriptCapitalR], PlotStyle -> Opacity[0.5], PlotPoints -> 50], Graphics3D[ Point[coords], {Thickness[0.005], Line[{{0, 0, 0}, +{1, 0, 0}}]}]]$

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The radius of gyration may be estimated:

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This result can be verified, in that it converges on the previous answer for the perfectly hollow unit sphere as thickness t→0:

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$\[ScriptCapitalR] = ImplicitRegion[.99 <= x^2 + y^2 + z^2 <= 1, {x, y, z}]; coords = RandomPoint[\[ScriptCapitalR], 300]; ResourceFunction["GyrationTensor"][coords, "RadiusOfGyration"]$

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The same works for a solid, 3D cylinder, e.g. with unit radius and length, for which a Region may be defined from a pre-existing graphics primitive:

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$\[ScriptCapitalR] = Region@Cylinder[{{0, 0, 0}, {0, 0, 1}}]; coords = RandomPoint[\[ScriptCapitalR], 30000]; Show[RegionPlot3D[\[ScriptCapitalR], PlotStyle -> Opacity[0.5], PlotPoints -> 50], Graphics3D[ Point[coords], {Thickness[0.005], Line[{{0, 0, 0}, +{1, 0, 0}}]}], AspectRatio -> 1]$

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A solid cylinder has a known radius of gyration :

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Estimate the radius of gyration of a rabbit from ExampleData:

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$\[ScriptCapitalR] = ExampleData[{"Geometry3D", "StanfordBunny"}, "Region"]; coords = RandomPoint[\[ScriptCapitalR], 30000]; Show[\[ScriptCapitalR], Graphics3D[{Point[coords]}], Boxed -> True]$

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Note that this is a hollow rabbit, so it is more relevant to the Easter chocolate shell version than a live rabbit:

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Shape metrics derived from the gyration tensor may be used to characterize the shape of a random walk. An unbiased random walk will generally have lower normalized asphericity than a biased random walk:

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