Definitions
Nearby Words

# 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(xyz) of x, y and z is defined by

$c \left(x, y, z\right) = frac1\left\{R\right\}.$

If the three points are collinear, R can be informally considered to be +∞, and it makes rigorous sense to define c(xyz) = 0. If any of the points x, y and z are coincident, again define c(xyz) = 0.

Using the well-known formula relating the side lengths of a triangle to its area, it follows that

$c \left(x, y, z\right) = frac1\left\{R\right\} = frac\left\{4 A\right\}$
>,

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