Suppose two functions 
have continuous derivatives of order high enough. We say that 
and 
have a contact of order 
(
) at a point 
, if for all 
, we have 
. In geometric terms, let 
be the graphs of 
and 
respectively. Then, at the point 
,
(1) 
and 
have zeroth order contact iff 
but 
and 
are not tangent at 
;
(2) 
and 
have first order contact iff 
and 
are tangent to each other at 
;
(3) 
and 
have second order contact iff their osculating curvatures at 
are equal;
(4) 
and 
have 
#card=math&code=%28n%2B2%29&id=GzpzI)-th order contact iff the 
-th derivatives of their curvatures at 
are equal.
Note that at any inflection point 
, the curvature is zero. This means the osculating circle at 
is just a straight line. We define the evolute of a curve to be the locus of the centres of all the osculating circles.
