In mathematics, the Menger curvature of a triple of points in n-dimensional Euclidean space Rn is the reciprocal of the radius of the circle that passes through the three points. It is named after the Austrian-American mathematician Karl Menger.
Definition Let x, y and z be three points in Rn; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let Π ⊆ Rn be the Euclidean plane spanned by x, y and z and let C ⊆ Π be the unique Euclidean circle in Π that passes through x, y and z (the circumcircle of x, y and z). Let R be the radius of C. Then the Menger curvature c(x, y, z) of x, y and z is defined by
c ( x , y , z ) = 1 R . {\displaystyle c(x,y,z)={\frac {1}{R}}.}
If the three points are collinear, R can be informally considered to be +∞, and it makes rigorous sense to define c(x, y, z) = 0. If any of the points x, y and z are coincident, again define c(x, y, z) = 0. Using the well-known formula relating the side lengths of a triangle to its area, it follows that
c ( x , y , z ) = 1 R = 4 A | x − y | | y − z | | z − x | , {\displaystyle c(x,y,z)={\frac {1}{R}}={\frac {4A}{|x-y||y-z||z-x|}},}
where A denotes the area of the triangle spanned by x, y and z. Another way of computing Menger curvature is the identity
c ( x , y , z ) = 2 sin ∠ x y z | x − z | {\displaystyle c(x,y,z)={\frac {2\sin \angle xyz}{|x-z|}}}
where ∠ x y z {\displaystyle \angle xyz} is the angle made at the y-corner of the triangle spanned by x,y,z. Menger curvature may also be defined on a general metric space. If X is a metric space and x,y, and z are distinct points, let f be an isometry from { x , y , z } {\displaystyle \{x,y,z\}} into R 2 {\displaystyle \mathbb {R} ^{2}} . Define the Menger curvature of these points to be
c X ( x , y , z ) = c ( f ( x ) , f ( y ) , f ( z ) ) . {\displaystyle c_{X}(x,y,z)=c(f(x),f(y),f(z)).}
Note that f need not be defined on all of X, just on {x,y,z}, and the value cX (x,y,z) is independent of the choice of f.
Integral Curvature Rectifiability Menger curvature can be used to give quantitative conditions for when sets in R n {\displaystyle \mathbb {R} ^{n}} may be rectifiable. For a Borel measure μ {\displaystyle \mu } on a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} define
c p ( μ ) = ∫ ∫ ∫ c ( x , y , z ) p d μ ( x ) d μ ( y ) d μ ( z ) . {\displaystyle c^{p}(\mu )=\int \int \int c(x,y,z)^{p}d\mu (x)d\mu (y)d\mu (z).}
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