Affine normal

A straight line defined in an affine-invariant manner at each point of a hypersurface in an affine space with the aid of the third-order differential neighbourhood of this hypersurface; it is essential that the principal quadratic form of the hypersurface does not degenerate. The affine normal at a point $M$ of a plane curve coincides with the diameter of the parabola that has third-order contact with the curve at $M$. If tangent hyper-quadrics are employed, a similar interpretation can be given to the affine normal to a hypersurface. In particular, the affine normal of a hyper-quadric coincides with its diameter.