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Difference between revisions of "Osculating sphere"

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''at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705801.png" /> of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705802.png" />''
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''at a point $M$ of a curve $l$''
  
The sphere having contact of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705803.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705804.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705805.png" /> (see [[Osculation|Osculation]]). The osculating sphere can also be defined as the limit of a variable sphere passing through four points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705806.png" /> as these points approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705807.png" />. If the radius of curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705808.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o0705809.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o07058010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o07058011.png" /> is the torsion, then the formula for calculating the radius of the osculating sphere has the form
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o07058012.png" /></td> </tr></table>
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$$R=\sqrt{\rho^2+\frac{1}{\sigma^2}\left(\frac{d\rho}{ds}\right)^2},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o07058013.png" /> denotes the differential along an arc of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070580/o07058014.png" />.
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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=19214
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article