In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
,
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, and
trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional
volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
, of a
-dimensional
simplex in terms of the squares of all of the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
s between pairs of its vertices. The determinant is named after
Arthur Cayley and
Karl Menger
Karl Menger (; January 13, 1902 – October 5, 1985) was an Austrian-born American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebra over a field, algebras and the dimension theory of low-r ...
.
The
pairwise distance polynomials between points in a real
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
are Euclidean invariants that are associated via the Cayley-Menger relations.
[Sitharam, Meera; St. John, Audrey; Sidman, Jessica. ''Handbook of Geometric Constraint Systems Principles''. Boca Raton, FL: CRC Press. ] These relations served multiple purposes such as generalising Heron's Formula, as well as computing the content of a -dimensional simplex, and ultimately determining if any real
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
is a Euclidean distance matrix for some points in the field of
distance geometry.
History
Karl Menger
Karl Menger (; January 13, 1902 – October 5, 1985) was an Austrian-born American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebra over a field, algebras and the dimension theory of low-r ...
was a young geometry professor at the
University of Vienna
The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...
and
Arthur Cayley was a British mathematician who specialized in algebraic geometry. Menger extended Cayley's algebraic results to propose a new axiom of metric spaces using the concepts of distance geometry up to congruence equivalence, 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.
Six Mathematical Gems from the History of Distance Geometry
'
Definition
Let
be
points in
-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, with
. These points are the vertices of an -dimensional simplex: a triangle when
; a
tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
when
, and so on. Let
be the
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
s between vertices
and
. The content, i.e. the -dimensional volume of this simplex, denoted by
, can be expressed as a function of
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
s of certain matrices, as follows:
[Cayley-Menger Determinant](_blank)
' ''
:
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.
Except for the final row and column of 1s, the matrix in the second form of this equation is a
Euclidean distance matrix.
Compare this to the usual formula for the oriented
volume of a simplex, namely
times the determinant of the matrix composed of the edge vectors
. Unlike the Cayley-Menger determinant, the latter matrix changes with rotation of the simplex, though not with translation; regardless, its determinant and the resulting volume do not change.
Special cases
2-Simplex
To reiterate, a simplex is an -dimensional polytope and the
convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of
points which do not lie in any
dimensional plane.
[Simplex ''Encyclopedia of Mathematics''](_blank)
/ref> Therefore, a 2-simplex occurs when and the simplex results in a triangle. Therefore, the formula for determining of a triangle is provided below:
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.
3-Simplex
Similarly, a 3-simplex occurs when and the simplex results in a tetrahedron. Therefore, the formula for determining of a tetrahedron is provided below:
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 .
Proof
Let the column vectors be points in -dimensional Euclidean space. Starting with the volume formula
we note that the determinant is unchanged when we add an extra row and column to make an matrix,
where is the square of the length of the vector . Additionally, we note that the matrix
has a determinant of . Thus,
Example
In the case of , we have that is the area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of a triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
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.
In the case of , the quantity gives the volume of a tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, which we will denote by . For distances between and given by , the Cayley–Menger determinant gives
Finding the circumradius of a simplex
Given a nondegenerate -simplex, it has a circumscribed -sphere, with radius . Then the -simplex made of the vertices of the -simplex and the center of the -sphere is degenerate. Thus, we have
Using
,
, and
,
this is equivalent to
It is a quadratic function of .
For example, 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 (with at least three distinct elements) is called straight if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
, for any three elements , , and of ,[Distance Geometry Wiki Page](_blank)
/ref>
::
Plane
A set (with at least four distinct elements) is called plane if and only if, for any four elements , , and of ,
::
but not all triples of elements of are straight to each other;
Flat
A set (with at least five distinct elements) is called flat if and only if, for any five elements , , , and of ,
::
but not all quadruples of elements of 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:
:: For a finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
is a finite set with five elements. Th ...
of points , a semi-metric can be obtained from a Euclidean metric of dimension n if and only if every Cayley-Menger determinant on points is strictly positive, every determinant on points vanishes, and a Cayley-Menger determinant on at least one set of points is nonnegative (in which case it is necessarily zero).
In simpler terms, if every subset of points can be isometrically embedded in an -dimensional, but not generally -dimensional Euclidean space, then the semi-metric is Euclidean of dimension unless consists of exactly points and the Cayley–Menger determinant on those points is strictly negative. This type of semi-metric would be classified as ''pseudo-Euclidean''.
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
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 .[Sitharam, Meera. "Lecture 1 through 6"." Geometric Complexity CIS6930, University of Florida. Received 28 Mar.2020]
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
; 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
.
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 be a positive integer and be a symmetric hollow matrix with nonnegative elements, with . is a Euclidean distance matrix with if and only if there exist and an index set such that
where realizes , where denotes the component of the vector.
The extensive proof of this theorem can be found at the following reference.[Realizing Euclidean Distance Matrices by Sphere Intersection](_blank)
/ref>
Algorithm - K = edmsph(''D'', ''x'')
Source:
: Γ
: if Γ ∅; then
:: return ∞
: else if Γ
::
: else if Γ
::
:: ← expand()
::
::
: else
:: error:
: end if
end for
return K
See also
* Distance Geometry''
''
*Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
''
''
* Euclidean Distance Matrix''
''
* Simplex''
''
* Heron's Formula
Notes
References
{{DEFAULTSORT:Cayley-Menger determinant
Determinants