Editing Curvature (section)

===Osculating circle===
[[File:Osculating.svg|alt=|right|250x250px]]
Historically, the curvature of a differentiable curve was defined through the [[osculating circle]], which is the circle that best approximates the curve at a point. More precisely, given a point {{mvar|P}} on a curve, every other point {{mvar|Q}} of the curve defines a circle (or sometimes a line) passing through {{mvar|Q}} and [[tangent (geometry)|tangent]] to the curve at {{mvar|P}}. The osculating circle is the [[limit (mathematics)|limit]], if it exists, of this circle when {{mvar|Q}} tends to {{mvar|P}}. Then the ''center'' and the ''radius of curvature'' of the curve at {{mvar|P}} are the center and the radius of the osculating circle. The curvature is the [[multiplicative inverse|reciprocal]] of radius of curvature. That is, the curvature is 
: <math> \kappa = \frac{1}{R},</math>
where {{mvar|R}} is the radius of curvature<ref>{{harvnb|Kline|1998|page=458}}</ref> (the whole circle has this curvature, it can be read as turn {{math|2π}} over the length {{math|2π}}{{mvar|R}}).

This definition is difficult to manipulate and to express in formulas. Therefore, other equivalent definitions have been introduced.