Compute ellipsoid hypervolume

Computes the geometric volume (area in 2D, volume in 3D, hypervolume in higher dimensions) of a \(p\)-dimensional ellipsoid defined by its semi-axis lengths.

For semi-axes \(a_1, \dots, a_p\), the volume is: $$ V_p = \frac{\pi^{p/2}}{\Gamma(p/2 + 1)} \prod_{i=1}^{p} a_i $$ where \(\pi^{p/2} / \Gamma(p/2 + 1)\) is the volume of the unit \(p\)-dimensional ball.

In probabilistic niche models, semi-axis lengths are typically derived from covariance eigenvalues and a chi-square cutoff.

Usage

ellipsoid_volume(n_dimensions, semi_axes_lengths)

Arguments

n_dimensions

Integer. Number of dimensions \(p\).

semi_axes_lengths

Numeric vector of length n_dimensions containing the ellipsoid semi-axis lengths.

Value

Numeric. Geometric volume (or hypervolume) of the ellipsoid.

See also

build_ellipsoid, ellipsoid_calculator

Examples

range_df <- data.frame(bio_1 = c(22, 28),
                       bio_12 = c(1000, 3500))
ell <- nicheR::build_ellipsoid(range = range_df)
#> Starting: building ellipsoidal niche from ranges...
#> Step: computing covariance matrix...
#> Step: computing additional ellipsoidal niche metrics...
#> Done: created ellipsoidal niche.

ell$volume
#> [1] 12056.31

# or recalculate
nicheR::ellipsoid_volume(n_dimensions = ell$dimensions, semi_axes_lengths = ell$semi_axes_lengths)
#> [1] 12056.31