Cayley–Menger determinant

In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a -dimensional simplex in terms of the squares of all of the distances between pairs of its vertices. The determinant is named after Arthur Cayley and Karl Menger.

DefinitionEdit

Let   be   points in  -dimensional Euclidean space, often with  . These points are the vertices of an n-dimensional simplex: a triangle when  ; a tetrahedron when  , and so on. Let   be the distances between   and  , for  . The content, i.e. the n-dimensional volume of this simplex, denoted by  , can be expressed as a function of determinants of certain matrices, as follows:[1]

 

This is the Cayley–Menger determinant. For   it is a symmetric polynomial in the  's and is thus invariant under permutation of these quantities. This fails for  , but it is always invariant under permutation of the vertices.

A proof of the second equation can be found.[2] From the second equation, the first can be derived by elementary row and column operations:

 then exchange the first and last column, gaining a  , and multiply each of its   inner rows by  .

Generalization to hyperbolic and spherical geometryEdit

There are spherical and hyperbolic generalizations.[3] A proof can be found here [4].

In a spherical space of dimension   and constant curvature  , any   points satisfy

 

where  , and   is the spherical distance between points  .

In a hyperbolic space of dimension   and constant curvature  , any   points satisfy

 

where  , and   is the hyperbolic distance between points  .

ExampleEdit

In the case of  , we have that   is the area of a triangle and thus we will denote this by  . By the Cayley–Menger determinant, where the triangle has side lengths  ,   and  ,

 

The result in the third line is due to the Fibonacci identity. The final line can be rewritten to obtain Heron's formula for the area of a triangle given three sides, which was known to Archimedes prior.[5]

In the case of  , the quantity   gives the volume of a tetrahedron, which we will denote by  . For distances between   and   given by  , the Cayley–Menger determinant gives[6][7]

 

Finding the circumradius of a simplexEdit

Given a nondegenerate n-simplex, it has a circumscribed n-sphere, with radius  . Then the (n+1)-simplex made of the vertices of the n-simplex and the center of the n-sphere is degenerate. Thus, we have

 

In particular, when  , this gives the circumradius of a triangle in terms of its edge lengths.

See alsoEdit

ReferencesEdit

  1. ^ Sommerville, D. M. Y. (1958). An Introduction to the Geometry of n Dimensions. New York: Dover Publications.
  2. ^ "Simplex Volumes and the Cayley-Menger Determinant". www.mathpages.com. Archived from the original on 16 May 2019. Retrieved 2019-06-08.
  3. ^ Blumenthal, L. M.; Gillam, B. E. (1943). "Distribution of Points in n-Space". The American Mathematical Monthly. 50 (3): 181. doi:10.2307/2302400. JSTOR 2302400.
  4. ^ Tao, Terrence (2019-05-25). "The spherical Cayley-Menger determinant and the radius of the Earth". What's new. Retrieved 2019-06-10.
  5. ^ Heath, Thomas L. (1921). A History of Greek Mathematics (Vol II). Oxford University Press. pp. 321–323.
  6. ^ Audet, Daniel. "Déterminants sphérique et hyperbolique de Cayley–Menger" (PDF). Bulletin AMQ. LI: 45–52.
  7. ^ Dörrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. New York: Dover Publications. pp. 285–9.