Cayley-Menger Determinant
A determinant that gives the volume of a simplex in
dimensions. If
is a
-simplex in
with vertices
and
denotes the
matrix given by
|
(1)
|
then the content
is given by
|
(2)
|
where
is the
matrix obtained from
by bordering
with a top row
and
a left column
.
Here, the vector L2-norms
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
, 1, ... are
, 2,
, 288,
, 460800, ...
(OEIS A055546).
For
, (2) becomes
![]() |
(3)
|
which gives the area for a plane triangle with side lengths
,
, and
, and is a form
of Heron's formula.
For
, the content of the 3-simplex (i.e.,
volume of the general tetrahedron) is given by the
determinant
![]() |
(4)
|
where the edge between vertices
and
has length
. 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.


determinants




