View source for Affine distance

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.