Orientation Tensor

2D orientation tensor characterizes distribution of axial angles using the Eigenvalue method (Watson 1966, Scheidegger 1965).

Usage

ortensor2d(x, w = NULL, norm = FALSE)

Arguments

x

numeric. Axial angular data (in degrees).

w

(optional) Weights. A vector of positive numbers and of the same length as x.

norm

logical. Whether the tensor should be normalized.

Value

2x2 matrix

Details

The moment of inertia can be minimized by calculating the Cartesian coordinates of the orientation data, and calculating their covariance matrix. This yields $$I = x \cdot x^\intercal$$ where \(x\) is the Cartesian vector of the orientations. Orientation tensor \(T\) and the inertia tensor \(I\) are related by $$I = E - T$$ where \(E\) denotes the unit matrix, so that $$T = \frac{1}{n} \sum_{i=i}^{n} x_i \cdot x_i^\intercal$$

References

Watson, G. S. (1966). The Statistics of Orientation Data. The Journal of Geology, 74(5), 786–797.

Scheidegger, A. E. (1964). The tectonic stress and tectonic motion direction in Europe and Western Asia as calculated from earthquake fault plane solutions. Bulletin of the Seismological Society of America, 54(5A), 1519–1528. doi:10.1785/BSSA05405A1519

Bachmann, F., Hielscher, R., Jupp, P. E., Pantleon, W., Schaeben, H., & Wegert, E. (2010). Inferential statistics of electron backscatter diffraction data from within individual crystalline grains. Journal of Applied Crystallography, 43(6), 1338–1355. https://doi.org/10.1107/S002188981003027X

See also

ot_eigen2d()

Examples

test <- rvm(100, mean = 0, k = 10)
ortensor2d(test)
#>            [,1]       [,2]
#> [1,] 0.90068830 0.01410628
#> [2,] 0.01410628 0.09931170