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
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
where A denotes the area of the triangle spanned by x, y and z.
Another way of computing Menger curvature is the identity
where 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 into . Define the Menger curvature of these points to be
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.
Read more about this topic: Menger Curvature
Famous quotes containing the word definition:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)