Let O be a point on a curve C in the plane
where the osculating circle to C at O exists. Let T and N be the unit tangent and normal vectors to C at O, respectively. Let k be the curvature of
C
at O.
(N is oriented so that O + 1/k N is the center of the osculating circle to C at O.)
For any r > 0, define
Cr to be the circle with radius r centered at O,
P = O + r T, the point
at the top of Cr,
Q to be the
intersection of C and Cr, and
R to be the
point on the line through P and Q such that OR is parallel
with N
Then, as r decreases to
0,
R
converges to the point R0 = O + 4 N.