Cayley-Menger determinant

In mathematics , the Cayley-Menger determinant is particularly important for calculating volumes .

It was given by Cayley in 1841 and calculates the volume of triangles, tetrahedra and higher-dimensional simplices.

definition

Be a simplex with corners in -dimensional space . ${\displaystyle S}$ ${\displaystyle v_{1},\ldots ,v_{n+1}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$

Let be the matrix whose first row or column is or and whose entries for ${\displaystyle B}$ ${\displaystyle (n+2)\times (n+2)}$ ${\displaystyle (0,1,\ldots ,1)}$ ${\displaystyle (0,1,\ldots ,1)}$ ${\displaystyle 2\leq i,j\leq n+2}$

${\displaystyle b_{ij}=\|v_{i-1}-v_{j-1}\|^{2}}$

are. Then the Cayley-Menger determinant of the simplex is defined as the determinant of . ${\displaystyle B}$

Volume calculation of simplices

General formula

The volume of the simplex is calculated using the Cayley-Menger determinant ${\displaystyle S}$

${\displaystyle (\operatorname {Vol} (S))^{2}={\frac {(-1)^{n+1}}{2^{n}(n!)^{2}}}\det(B).}$

Examples

Area of ​​a triangle

general triangle

The area of a triangle with side length is calculated as the square root of ${\displaystyle a,b,c}$

${\displaystyle -{\frac {1}{16}}\det \left({\begin{array}{cccc}0&1&1&1\\1&0&c^{2}&b^{2}\\1&c^{2}&0&a^{2}\\1&b^{2}&a^{2}&0\end{array}}\right).}$

This is a reformulation of Heron's theorem .

Volume of a tetrahedron

general tetrahedron

→ Main article : Volume formula of the general tetrahedron

The volume of a tetrahedron with edge lengths is calculated as a square root ${\displaystyle d_{ij},1\leq i,j\leq 4}$

${\displaystyle {\frac {1}{288}}\det \left({\begin{array}{ccccc}0&1&1&1&1\\1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\\1&d_{21}^{2}&0&d_{23}^{2}&d_{24}^{2}\\1&d_{31}^{2}&d_{32}^{2}&0&d_{34}^{2}\\1&d_{41}^{2}&d_{42}^{2}&d_{43}^{2}&0\end{array}}\right).}$

In particular, it applies when the four points lie in one plane. ${\displaystyle \det(B)=0}$

An essentially equivalent formula had already been given by Piero della Francesca in the 15th century . In the English-speaking world, it is often referred to as Tartaglia's formula .

Characterization of Euclidean spaces

Karl Menger used the Cayley-Menger determinant to give a purely metric characterization of Euclidean spaces among metric spaces . Marcel Berger later gave a more general characterization of Riemannian manifolds of constant sectional curvature using the Cayley-Menger determinates.

Different results of the distance geometry can be proved with the help of the Cayley-Menger determinates, for example the Stewart theorem .

Symmetries

The group of linear mappings of the , which leave the Cayley-Menger determinant of a tetrahedron (as a function of the 6 edge lengths) invariant, has order 23040 and is isomorphic to the Weyl group . ${\displaystyle \mathbb {R} ^{6}}$ ${\displaystyle D_{6}}$

These symmetries are also given the Dehn invariant and thus map each tetrahedron into a tetrahedron with the same decomposition .

literature

• A. Cayley: A theorem in the geometry of position. Cambridge Mathematical Journal, II: 267-271 (1841). on-line
• L. Blumenthal: Theory and applications of distance geometry. Clarendon Press, Oxford (1953). Chapter IV.40
• M. Berger: Geometry I. Springer-Verlag, Berlin (1987). Chapter 9.7

Web links

• Cayley-Menger determinant (MathWorld)
• Simplex Volumes and the Cayley-Menger Determinant (MathPages)
• Cayley-Menger determinant in space (proof of the 3-dimensional volume formula)

Individual evidence

1. ↑ K. Menger: Studies on general metrics. Math. Ann. 100 (1928), no. 1, 75-163. doi : 10.1007 / BF01448840
2. ↑ M. Berger: Une caractérisation purement métrique des variétés riemanniennes à courbure constante. EB Christoffel (Aachen / Monschau, 1979), pp. 480–492, Birkhäuser, Basel-Boston, Mass., 1981.
3. ↑ D. Michelucci, p Foufou: Using Cayley-Menger Determinants for Geometric Constraint Solving. ACM Symposium on Solid Modeling and Applications (2004) online (pdf)
4. ↑ Philip P. Boalch: Regge and Okamoto symmetries, Comm. Math. Phys. 276 (2007), no. 1, 117-130.
5. ↑ J. Roberts .: Classical 6j-symbols and the tetrahedron. Geom. & Top. 3 (1999)