Formula for the "volume" of an n-simplex
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.
The
pairwise distance polynomials between n points in a real Euclidean space are Euclidean invariants that are associated via the Cayley-Menger relations.[1] These relations served multiple purposes such as generalising Heron's Formula, computing the content of a n-dimensional simplex, and ultimately determining if any real symmetric matrix is a Euclidean distance matrix in the field of Distance geometry.[2]
History
Karl Menger was a young geometry professor at the University of Vienna and Arthur Cayley was a British mathematician who specialized in algebraic geometry. Menger extended Cayley's algebraic excellence to propose a new axiom of metric spaces using the concepts of distance geometry and relation of congruence, known as the Cayley–Menger determinant. This ended up generalising one of the first discoveries in distance geometry, Heron's formula, which computes the area of a triangle given its side lengths.[3]
Definition
Let
be
points in
-dimensional Euclidean space, with
.[a] These points are the vertices of an n-dimensional simplex: a triangle when
; a tetrahedron when
, and so on. Let
be the Euclidean distances between vertices
and
. 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:[4][5]
![{\displaystyle {\begin{aligned}v_{n}^{2}&={\frac {1}{(n!)^{2}2^{n}}}{\begin{vmatrix}2d_{01}^{2}&d_{01}^{2}+d_{02}^{2}-d_{12}^{2}&\cdots &d_{01}^{2}+d_{0n}^{2}-d_{1n}^{2}\\d_{01}^{2}+d_{02}^{2}-d_{12}^{2}&2d_{02}^{2}&\cdots &d_{02}^{2}+d_{0n}^{2}-d_{2n}^{2}\\\vdots &\vdots &\ddots &\vdots \\d_{01}^{2}+d_{0n}^{2}-d_{1n}^{2}&d_{02}^{2}+d_{0n}^{2}-d_{2n}^{2}&\cdots &2d_{0n}^{2}\end{vmatrix}}\\[10pt]&={\frac {(-1)^{n+1}}{(n!)^{2}2^{n}}}{\begin{vmatrix}0&d_{01}^{2}&d_{02}^{2}&\cdots &d_{0n}^{2}&1\\d_{01}^{2}&0&d_{12}^{2}&\cdots &d_{1n}^{2}&1\\d_{02}^{2}&d_{12}^{2}&0&\cdots &d_{2n}^{2}&1\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\d_{0n}^{2}&d_{1n}^{2}&d_{2n}^{2}&\cdots &0&1\\1&1&1&\cdots &1&0\end{vmatrix}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/308890971283a91e445d7bec3f463227d944e73d)
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.[b]
Except for the final row and column of 1s, the matrix in the second form of this equation is a Euclidean distance matrix.
Special cases
2-Simplex
To reiterate, a simplex is an n-dimensional polytope and the convex hull of
points which do not lie in any
dimensional plane.[6] Therefore, a 2-simplex occurs when
and the simplex results in a triangle. Therefore, the formula for determining
of a triangle is provided below:[5]
As a result, the equation above presents the content of a 2-simplex (area of a planar triangle with side lengths
,
, and
) and it is a generalised form of Heron's Formula.[5]
3-Simplex
Similarly, a 3-simplex occurs when
and the simplex results in a tetrahedron.[6] Therefore, the formula for determining
of a tetrahedron is provided below:[5]
As a result, the equation above presents the content of a 3-simplex, which is the volume of a tetrahedron where the edge between vertices
and
has length
.[5]
Proof
Let the column vectors
be
points in
-dimensional Euclidean space. Starting with the volume formula
![{\displaystyle v_{n}={\frac {1}{n!}}\left|\det {\begin{pmatrix}A_{0}&A_{1}&\cdots &A_{n}\\1&1&\cdots &1\end{pmatrix}}\right|\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4deab76124ca414fc56e1d99a49f55d24ed2108e)
we note that the determinant is unchanged when we add an extra row and column to make an
matrix,
![{\displaystyle P={\begin{pmatrix}A_{0}&A_{1}&\cdots &A_{n}&0\\1&1&\cdots &1&0\\\|A_{0}\|^{2}&\|A_{1}\|^{2}&\cdots &\|A_{n}\|^{2}&1\end{pmatrix}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00afbde8e6c772b91ccda972c1e652bfcb6df4ec)
where
is the square of the length of the vector
. Additionally, we note that the
matrix
![{\displaystyle Q={\begin{pmatrix}-2&0&\cdots &0&0&0\\0&-2&\cdots &0&0&0\\\vdots &\vdots &\ddots &\vdots &\vdots &\vdots \\0&0&\cdots &-2&0&0\\0&0&\cdots &0&0&1\\0&0&\cdots &0&1&0\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79d9705e5875ee95e19fd9cea8740e2b241699c4)
has a determinant of
. Thus,[7]
![{\displaystyle \det {\begin{pmatrix}0&d_{01}^{2}&d_{02}^{2}&\cdots &d_{0n}^{2}&1\\d_{01}^{2}&0&d_{12}^{2}&\cdots &d_{1n}^{2}&1\\d_{02}^{2}&d_{12}^{2}&0&\cdots &d_{2n}^{2}&1\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\d_{0n}^{2}&d_{1n}^{2}&d_{2n}^{2}&\cdots &0&1\\1&1&1&\cdots &1&0\end{pmatrix}}=\det(P^{T}QP)=\det(Q)\det(P)^{2}=(-1)^{n+1}2^{n}(n!)^{2}v_{n}^{2}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03e06fd9dae5b5a5a509cb5b2a04d2bc4f185cf4)
Example
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
,
![{\displaystyle {\begin{aligned}16A^{2}&={\begin{vmatrix}2a^{2}&a^{2}+b^{2}-c^{2}\\a^{2}+b^{2}-c^{2}&2b^{2}\end{vmatrix}}\\[8pt]&=4a^{2}b^{2}-(a^{2}+b^{2}-c^{2})^{2}\\[6pt]&=(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})\\[6pt]&=(a+b+c)(a+b-c)(a-b+c)(-a+b+c)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffbea4ab48bf8b955cd7859d82caa17aa3788a3b)
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.[8]
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[9][10]
![{\displaystyle {\begin{aligned}144V^{2}={}&{\frac {1}{2}}{\begin{vmatrix}2d_{01}^{2}&d_{01}^{2}+d_{02}^{2}-d_{12}^{2}&d_{01}^{2}+d_{03}^{2}-d_{13}^{2}\\d_{01}^{2}+d_{02}^{2}-d_{12}^{2}&2d_{02}^{2}&d_{02}^{2}+d_{03}^{2}-d_{23}^{2}\\d_{01}^{2}+d_{03}^{2}-d_{13}^{2}&d_{02}^{2}+d_{03}^{2}-d_{23}^{2}&2d_{03}^{2}\end{vmatrix}}\\[8pt]={}&4d_{01}^{2}d_{02}^{2}d_{03}^{2}+(d_{01}^{2}+d_{02}^{2}-d_{12}^{2})(d_{01}^{2}+d_{03}^{2}-d_{13}^{2})(d_{02}^{2}+d_{03}^{2}-d_{23}^{2})\\[6pt]&{}-d_{01}^{2}(d_{02}^{2}+d_{03}^{2}-d_{23}^{2})^{2}-d_{02}^{2}(d_{01}^{2}+d_{03}^{2}-d_{13}^{2})^{2}-d_{03}^{2}(d_{01}^{2}+d_{02}^{2}-d_{12}^{2})^{2}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6630123298d507c70f13225ed5413cd7e0d14abc)
Finding the circumradius of a simplex
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
![{\displaystyle {\begin{vmatrix}0&r^{2}&r^{2}&r^{2}&\cdots &r^{2}&1\\r^{2}&0&d_{01}^{2}&d_{02}^{2}&\cdots &d_{0n}^{2}&1\\r^{2}&d_{01}^{2}&0&d_{12}^{2}&\cdots &d_{1n}^{2}&1\\r^{2}&d_{02}^{2}&d_{12}^{2}&0&\cdots &d_{2n}^{2}&1\\\vdots &\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\r^{2}&d_{0n}^{2}&d_{1n}^{2}&d_{2n}^{2}&\cdots &0&1\\1&1&1&1&\cdots &1&0\end{vmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea703a6f9a961bb652b07967e62791f4a9509d8)
In particular, when
, this gives the circumradius of a triangle in terms of its edge lengths.
Set Classifications
From these determinants, we also have the following classifications:
Straight
A set Λ (with at least three distinct elements) is called straight if and only if, for any three elements A, B, and C of Λ,[11]
![{\displaystyle \det {\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&1\\d(AB)^{2}&0&d(BC)^{2}&1\\d(AC)^{2}&d(BC)^{2}&0&1\\1&1&1&0\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9849b1bc1b4cdf47efae0c3576729b7a1543ad2c)
Plane
A set Π (with at least four distinct elements) is called plane if and only if, for any four elements A, B, C and D of Π,[11]
![{\displaystyle \det {\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&d(AD)^{2}&1\\d(AB)^{2}&0&d(BC)^{2}&d(BD)^{2}&1\\d(AC)^{2}&d(BC)^{2}&0&d(CD)^{2}&1\\d(AD)^{2}&d(BD)^{2}&d(CD)^{2}&0&1\\1&1&1&1&0\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74da11738b51c0d34f20c75200b136f27e5f8b44)
but not all triples of elements of Π are straight to each other;
Flat
A set Φ (with at least five distinct elements) is called flat if and only if, for any five elements A, B, C, D and E of Φ,[11]
![{\displaystyle \det {\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&d(AD)^{2}&d(AE)^{2}&1\\d(AB)^{2}&0&d(BC)^{2}&d(BD)^{2}&d(BE)^{2}&1\\d(AC)^{2}&d(BC)^{2}&0&d(CD)^{2}&d(CE)^{2}&1\\d(AD)^{2}&d(BD)^{2}&d(CD)^{2}&0&d(DE)^{2}&1\\d(AE)^{2}&d(BE)^{2}&d(CE)^{2}&d(DE)^{2}&0&1\\1&1&1&1&1&0\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30b4d827cfeca35c90453d2ad2a280eab6af98eb)
but not all quadruples of elements of Φ are plane to each other; and so on.
Menger's Theorem
Karl Menger made a further discovery after the development of the Cayley–Menger determinant, which became known as Menger's Theorem. The theorem states:
- A semimetric
is Euclidean of dimension n if and only if all Cayley-Menger determinants on
points is strictly positive, all determinants on
points vanish, and a Cayley-Menger determinant on at least one set of
points is nonnegative (in which case it is necessarily zero).[1]
In simpler terms, if every subset of
points can be isometrically embedded in an
but not generally
dimensional Euclidean space, then the semimetric is Euclidean of dimension
unless
consists of exactly
points and the Cayley–Menger determinant on those
points is strictly negative. This type of semimetric would be classified pseudo-Euclidean.[1]
Realization of a Euclidean distance matrix
Given the Cayley-Menger relations as explained above, the following section will bring forth two algorithms to decide whether a given matrix is a distance matrix corresponding to a Euclidean point set. The first algorithm will do so when given a matrix AND the dimension,
, via a geometric constraint solving algorithm. The second algorithm does so when the dimension,
, is not provided. This algorithm theoretically finds a realization of the full
Euclidean distance matrix in the smallest possible embedding dimension in quadratic time.
Theorem (d is given)
For the sake and context of the following theorem, algorithm, and example, slightly different notation will be used than before resulting in an altered formula for the volume of the
dimensional simplex below than above.
- Theorem. An
matrix
is a Euclidean Distance Matrix if and only if for all
submatrices
of
, where
,
. For
to have a realization in dimension
, if
, then
.[12]
As stated before, the purpose to this theorem comes from the following algorithm for realizing a Euclidean Distance Matrix or a Gramian Matrix.
Algorithm
- Input
- Euclidean Distance Matrix
or Gramian Matrix
.
- Output
- Pointset
![{\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
- Procedure
- If the dimension
is fixed, we can solve a system of polynomial equations, one for each inner product entry of
, where the variables are the coordinates of each point
in the desired dimension
. - Otherwise, we can solve for one point at a time.
- Solve for the coordinates of
using its distances to all previously placed points
. Thus,
is represented by at most
coordinate values, ensuring minimum dimension and complexity.
Example
Let each point
have coordinates
. To place the first three points:
- Put
at the origin, so
. - Put
on the first axis, so
. - To place
:
In order to find a realization using the above algorithm, the discriminant of the distance quadratic system must be positive, which is equivalent to
having positive volume. In general, the volume of the
dimensional simplex formed by the
vertices is given by[12]
.
In this formula above,
is the Cayley–Menger determinant. This volume being positive is equivalent to the determinant of the volume matrix being positive.
Theorem (d not given)
Let K be a positive integer and D be a n × n symmetric hollow matrix with nonnegative elements, with n ≥ 2. D is a Euclidean distance matrix with dim(D) = K if and only if there exist
and an index set I =
such that
![{\displaystyle {\begin{cases}x_{i}=0\\x_{i_{j}}(j-1)\neq 0,&{\mbox{ }}j\in I_{2,K+1}\\x_{i_{j}}(i)=0,&{\mbox{ }}j\in I_{2,K},i\in I_{j,K},\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffe6ced4784f4305dc4533efe6a1adcad59ba22)
where
realizes D, where
denotes the
component of the
vector.
The extensive proof of this theorem can be found at the following reference.[13]
Algorithm - K = edmsph(D, x)
Source:[13]
- Γ
![{\displaystyle =\bigcap _{j\in I}S^{K}(x_{j},D_{ij})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09a9e284d5d2b9ee94237ff58e42af024c483c67)
- if Γ
∅; then - return ∞
- else if Γ
![{\displaystyle x_{i}=p_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e63c4d72eeb54a3ee823b18ad834ce056fbc1a)
- else if Γ
![{\displaystyle x_{i}=p_{i}^{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/471b0650aacfc95c87d3d3f60db2264c8eadd81c)
← expand(
) - I ← I ∪ {i}
- K ← K + 1
- else
- error: dim aff(span(
)) < K - 1
- end if
end for return K
See also
Notes
- ^ An n-dimensional body can't be immersed into k-dimensional space if
- ^ The (hyper)volume of a figure does not depend on its vertices' numbering order.
References
- ^ a b c Sitharam, Meera; St. John, Audrey; Sidman, Jessica. Handbook of Geometric Constraint Systems Principles. Boca Raton, FL: CRC Press. ISBN 978-1-4987-3891-0
- ^ http://ufo2.cise.ufl.edu/index.php/Distance_Geometry Distance Geometry
- ^ Six Mathematical Gems from the History of Distance Geometry
- ^ Sommerville, D. M. Y. (1958). An Introduction to the Geometry of n Dimensions. New York: Dover Publications.
- ^ a b c d e Cayley-Menger Determinant
- ^ a b Simplex Encyclopedia of Mathematics
- ^ "Simplex Volumes and the Cayley–Menger Determinant". www.mathpages.com. Archived from the original on 16 May 2019. Retrieved 2019-06-08.
- ^ Heath, Thomas L. (1921). A History of Greek Mathematics (Vol II). Oxford University Press. pp. 321–323.
- ^ Audet, Daniel. "Déterminants sphérique et hyperbolique de Cayley–Menger" (PDF). Bulletin AMQ. LI: 45–52.
- ^ Dörrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. New York: Dover Publications. pp. 285–9.
- ^ a b c Distance Geometry Wiki Page
- ^ a b Sitharam, Meera. "Lecture 1 through 6"." Geometric Complexity CIS6930, University of Florida. Received 28 Mar.2020
- ^ a b Realizing Euclidean Distance Matrices by Sphere Intersection