Affine distance

An invariant determined by two line elements in an equi-affine plane. A point $M$ together with a straight line $m$ passing through it is called a line element $(M,m)$. For two line elements $(M,m)$ and $(N,n)$ the affine distance is $2f^{1/3}$, where $f$ is the surface area of the triangle $MNP$ and $P$ is the point of intersection of the straight lines $m$ and $n$. The affine distance for two elements tangent to a parabola is equal to the affine arc length of this parabola (cf. Affine parameter). In the three-dimensional equi-affine space the affine distance may also be defined in terms of elements consisting of pairwise incident points, straight lines and planes.