at a given point of a curve
The circle that has contact of order with at (see Osculation). If the curvature of at is zero, then the osculating circle degenerates into a straight line. The radius of the osculating circle is called the radius of curvature of at , and its centre the centre of curvature (see Fig.). If is the plane curve given by an equation , then the radius of the osculating circle is given by
If is the spatial curve given by equations
then the radius of the osculating circle is given by
(where the primes denote differentiation with respect to ).
|[a1]||R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 39|
|[a2]||D.J. Struik, "Lectures on classical differential geometry" , Dover, reprint (1988) pp. 14|
Osculating circle. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Osculating_circle&oldid=14753