Menger curvature

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

$c(x,y,z)={\frac {1}{R}}.$

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

$c(x,y,z)={\frac {1}{R}}={\frac {4A}{|x-y||y-z||z-x|}},$

where A denotes the area of the triangle spanned by x, y and z.

See also

• Menger-Melnikov curvature of a measure

External links

• Leymarie, F. (2003). "Notes on Menger Curvature". Retrieved 2007-11-19. {{cite web}}: Unknown parameter |month= ignored (help)

References

• Tolsa, Xavier (2000). "Principal values for the Cauchy integral and rectifiability". Proc. Amer. Math. Soc. 128 (7): 2111–2119. doi:10.1090/S0002-9939-00-05264-3.