Exact coordinate formulas for higher-dimensional solid angles

Derive exact elementary expressions for the solid angle of an arbitrary higher-dimensional simplex in terms of the coordinates of its defining vectors.

Background

The paper defines solid angles of simplices through spherical measures and also represents them as integrals over the positive orthant. Although formulas are available in three dimensions, the authors state that exact elementary coordinate formulas are not known in higher dimensions. Such formulas would provide a direct computational and analytic description of higher-dimensional solid angles beyond the integral representations used in the paper.