Notes on Menger Curvature

F. Leymarie

January 2000 - updated September 2003

Update (Sept. 2003):

The number K(x_1 , x_2 , x_3) = 1/ R , where R is the radius of the circumcircle through the three points x_1 , x_2 , x_3 , is called the Menger curvature of the triple [3].

The derivation of the formula below, then relies on using the well known relation between the area of the triangle, A, through these three points and R:

K=1/R = 4 A / abc , where a, b and c are the length of the sides of the triangle.

Note that Heron's formula gives us:

16 A^2 = (a+b+c) (b+c-a) (a+c-b) (a+b-c)

and this directly give the formula below. An alternative formula is provided by the Cayley-Menger determinant [1], which avoids directly computing the side lengths (as square roots): 16A^2=| a^2(a^2-b^2-c^2)+b^2(b^2-a^2-c^2)+c^2(c^2-a^2-b^2)|.

Let f be a regular curve of class C² in a Euclidean space, Eⁿ. Let x₁,x₂,x₃ be distinct points of f. Then, define [1, vol.1, p.273]:

$$K(x_{1},x_{2},x_{3})=\frac{\sqrt{(x_{1}x_{2}+x_{2}x_{3}+x_{3... ...}x_{1})}}{x_{1}x_{2}\cdot x_{2}x_{3}\cdot x_{3}x_{1}}\, \, \, ,$$

where the (non-negative) number K is called Menger's curvature. As x₂ and x₃ approach x₁ on f, K(x₁,x₂,x₃) tends towards the curvature of f at x₁. Also, K=0 if and only if x₁, x₂ and x₃ are collinear .

In the complex domain, for , this notion is called the Menger-Melnikov curvature [2]:

$$c(z_{1},z_{2},z_{3})^{2}=\sum _{\sigma }\frac{1}{(z_{\sigma ... ...igma (3)})\overline{(z_{\sigma (2)}-z_{\sigma (3)})}}\, \, \, ,$$

where the sum is taken over all permutations of of .This identity is transformed for 1-sets in Eⁿ to:

$$c(x_{1},x_{2},x_{3})^{2}=\sum _{\sigma }\frac{(x_{\sigma (1)... ...t\vert x_{\sigma (2)}-x_{\sigma (3)}\right\vert ^{2}}\, \, \, ,$$

or equivalently, after some manipulations:

$$c(x_{1},x_{2},x_{3})^{2}=4\, \left\{ \frac{\left\vert x_{1}-... ...2}\, \left\vert x_{1}-x_{2}\right\vert ^{2}}\right\} \, \, \, ,$$

which, by Schwartz inequality, can be shown to always be non-negative.

 

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