Calculation of the volume of simplex in barycentric coordinates in a multidimensional Euclidean space

Stepanova M.A. Calculation of the volume of simplex in barycentric coordinates in a multidimensional Euclidean space. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2025, vol. 25, no. 5, pp. 996–998 (in Russian). doi: 10.17586/2226-1494-2025-25-5-996-998

The paper describes three ways of calculating the k-dimensional volume of the k-dimensional simplex in the n-dimensional Euclidean space (n ≥ k) in the canonical barycentric coordinate system. The first method is to calculate for the n-dimensional simplex using the determinant of the barycentric matrix, the columns of which are the barycentric coordinates of the simplex vertices. The second method is to calculate the volume for k-dimensional simplex using the Cayley–Menger determinant through the lengths of the simplex edges which can be found from the barycentric coordinates of the vertices. The third method is to compute using a Gram determinant for a system of vectors constructed from the vertices of a given simplex in a (n + 1)-dimensional Euclidean space.