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Difference between revisions of "Affine normal"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011060/a0110601.png" /> of a plane curve coincides with the diameter of the parabola that has third-order contact with the curve at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011060/a0110602.png" />. 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.
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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.

Latest revision as of 13:23, 29 April 2014

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.

How to Cite This Entry:
Affine normal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_normal&oldid=31969
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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