Cayley–Menger determinant

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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.

Definition[edit]

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 geometry[edit]

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 .

Example[edit]

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 simplex[edit]

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 also[edit]

References[edit]

  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.