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Osculating circle

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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

Figure: o070540a

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 ).


Comments

References

[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
How to Cite This Entry:
Osculating circle. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Osculating_circle&oldid=14753
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article