Editing Quadric (section)

=== Generalization of quadrics: quadratic sets ===
It is not reasonable to formally extend the definition of quadrics to spaces over genuine skew fields (division rings). Because one would obtain secants bearing more than 2 points of the quadric which is totally different from ''usual'' quadrics.<ref>R. [[Rafael Artzy|Artzy]]: ''The Conic <math>y=x^2</math> in Moufang Planes'', Aequat.Mathem. 6 (1971), p. 31-35</ref><ref>E. Berz: ''Kegelschnitte in Desarguesschen Ebenen'', Math. Zeitschr. 78 (1962), p. 55-8</ref><ref>external link E. Hartmann: ''Planar Circle Geometries'', p. 123</ref> The reason is the following statement.

:A [[division ring]] <math>K</math> is [[commutative ring|commutative]] if and only if any [[quadratic equation|equation]] <math>x^2+ax+b=0, \ a,b \in K</math>, has at most two solutions.

There are ''generalizations'' of quadrics: [[quadratic set]]s.<ref>Beutelspacher/Rosenbaum: p. 135</ref> A quadratic set is a set of points of a projective space with the same geometric properties as a quadric: every line intersects a quadratic set in at most two points or is contained in the set.