Normalized solid angle in R^3 from a generator matrix

Convenience wrapper around solid_angle_3d that accepts the three generators as the columns of a \(3 \times 3\) matrix instead of three separate vectors.

Usage

solid_angle_3d_det(V)

Arguments

V

A \(3 \times 3\) numeric matrix whose columns are the cone generators. Need not be unit-normalized; columns are rescaled internally.

Value

A single numeric value in \([0, 1]\), the normalized solid angle of the cone spanned by the columns of V.

Details

Equivalent to solid_angle_3d(V[, 1], V[, 2], V[, 3]) after normalize_vectors has been applied to V. Use this form when the cone is already represented as a matrix, for example in batch loops over a list of cones.

References

Van Oosterom, A., & Strackee, J. (1983). The solid angle of a plane triangle. IEEE Transactions on Biomedical Engineering, 30(2), 125-126. doi:10.1109/TBME.1983.325207

See also

solid_angle_3d for the three-vector form; compute_solid_angle for the general dispatcher; normalize_vectors for the column-normalization helper.

Examples

V <- cbind(c(1, 0, 0), c(0, 1, 0), c(0, 0, 1))
solid_angle_3d_det(V)   # 0.125
#> [1] 0.125