Cayley-Menger Determinant

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A determinant that gives the volume of a simplex in j dimensions. If S is a j-simplex in R^n with vertices v_1,...,v_(j+1) and B=(beta_(ik)) denotes the (j+1)×(j+1) matrix given by

 beta_(ik)=|v_i-v_k|_2^2,
(1)

then the content V_j is given by

 V_j^2(S)=((-1)^(j+1))/(2^j(j!)^2)det(B^^),
(2)

where B^^ is the (j+2)×(j+2) matrix obtained from B by bordering B with a top row (0,1,...,1) and a left column (0,1,...,1)^(T). Here, the vector L2-norms |v_i-v_k|_2 are the edge lengths and the determinant in (2) is the Cayley-Menger determinant (Sommerville 1958, Gritzmann and Klee 1994). The first few coefficients for j=0, 1, ... are -1, 2, -16, 288, -9216, 460800, ... (OEIS A055546).

For j=2, (2) becomes

 -16Delta^2=|0 1 1 1; 1 0 c^2 b^2; 1 c^2 0 a^2; 1 b^2 a^2 0|,
(3)

which gives the area for a plane triangle with side lengths a, b, and c, and is a form of Heron's formula.

For j=3, the content of the 3-simplex (i.e., volume of the general tetrahedron) is given by the determinant

 288V^2=|0 1 1 1 1; 1 0 d_(12)^2 d_(13)^2 d_(14)^2; 1 d_(21)^2 0 d_(23)^2 d_(24)^2; 1 d_(31)^2 d_(32)^2 0 d_(34)^2; 1 d_(41)^2 d_(42)^2 d_(43)^2 0|,
(4)

where the edge between vertices i and j has length d_(ij). Setting the left side equal to 0 (corresponding to a tetrahedron of volume 0) gives a relationship between the distances between vertices of a planar quadrilateral (Uspensky 1948, p. 256).

Buchholz (1992) gives a slightly different (and slightly less symmetrical) form of this equation.

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