Difference between revisions of "Osculating sphere"
From Encyclopedia of Mathematics
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| − | ''at a point | + | {{TEX|done}} |
| + | ''at a point $M$ of a curve $l$'' | ||
| − | The sphere having contact of order | + | The sphere having contact of order $n\geq3$ with $l$ at $M$ (see [[Osculation|Osculation]]). The osculating sphere can also be defined as the limit of a variable sphere passing through four points of $l$ as these points approach $M$. If the radius of curvature of $l$ at $M$ is equal to $\rho$ and $\sigma$ is the torsion, then the formula for calculating the radius of the osculating sphere has the form |
| − | + | $$R=\sqrt{\rho^2+\frac{1}{\sigma^2}\left(\frac{d\rho}{ds}\right)^2},$$ | |
| − | where | + | where $ds$ denotes the differential along an arc of $l$. |
Latest revision as of 13:20, 29 April 2014
at a point $M$ of a curve $l$
The sphere having contact of order $n\geq3$ with $l$ at $M$ (see Osculation). The osculating sphere can also be defined as the limit of a variable sphere passing through four points of $l$ as these points approach $M$. If the radius of curvature of $l$ at $M$ is equal to $\rho$ and $\sigma$ is the torsion, then the formula for calculating the radius of the osculating sphere has the form
$$R=\sqrt{\rho^2+\frac{1}{\sigma^2}\left(\frac{d\rho}{ds}\right)^2},$$
where $ds$ denotes the differential along an arc of $l$.
Comments
References
| [a1] | R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1979) pp. 39 |
| [a2] | D.J. Struik, "Lectures on classical differential geometry" , Dover, reprint (1988) pp. 25 |
How to Cite This Entry:
Osculating sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osculating_sphere&oldid=31967
Osculating sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osculating_sphere&oldid=31967
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article