is the radius of the circle. But if in place of a circle we trace any arbitrary curve on our plane, the curvature will vary from point to point along the curve. The curvature of the curve at given point
is then defined by the a certain circle which is tangent to the curve at the point
. As a matter of fact there exist an indefinite number of circles of varying radii lying tangent to the curve at
; but among these circles one stands out prominently in that it is, so to speak, more perfectly tangent than all the others. Whereas the tangent circles intersect the curve in two points coinciding at
, this privileged circle intersects it in three such points. It is called the osculating circle (osculare meaning to kiss in Latin). The curvature of the curve at