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Difference between revisions of "Affine distance"

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An invariant determined by two line elements in an [[Equi-affine plane|equi-affine plane]]. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110001.png" /> together with a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110002.png" /> passing through it is called a line element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110003.png" />. For two line elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110005.png" /> the affine distance is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110007.png" /> is the surface area of the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a0110009.png" /> is the point of intersection of the straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a01100010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011000/a01100011.png" />. The affine distance for two elements tangent to a parabola is equal to the affine arc length of this parabola (cf. [[Affine parameter|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.
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An invariant determined by two line elements in an [[Equi-affine plane|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|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.

Latest revision as of 15:22, 30 July 2014

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.

How to Cite This Entry:
Affine distance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_distance&oldid=32573
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article