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(Category:Differential geometry)
 
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''at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o0705601.png" /> of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o0705602.png" />''
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''at a point $M$ of a curve $l$''
  
The plane having contact of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o0705603.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o0705604.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o0705605.png" /> (see [[Osculation|Osculation]]). The osculating plane can also be defined as the limit of a variable plane passing through three points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o0705606.png" /> as these points approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o0705607.png" />. Usually, a curve intersects the osculating plane at the point of contact (see Fig.).
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The plane having contact of order $n\geq2$ with $l$ at $M$ (see [[Osculation|Osculation]]). The osculating plane can also be defined as the limit of a variable plane passing through three points of $l$ as these points approach $M$. Usually, a curve intersects the osculating plane at the point of contact (see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o070560a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o070560a.gif" />
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Figure: o070560a
 
Figure: o070560a
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o0705608.png" /> is given by equations
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If $l$ is given by equations
  
<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/o070560/o0705609.png" /></td> </tr></table>
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$$x=x(u),\quad y=y(u),\quad z=z(u),$$
  
 
then the equation of the osculating plane has the form
 
then the equation of the osculating plane 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/o070560/o07056010.png" /></td> </tr></table>
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$$\begin{vmatrix}X-x&Y-y&Z-z\\x'&y'&z'\\x''&y''&z''\end{vmatrix}=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o07056011.png" /> are moving coordinates and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o07056012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o07056013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o07056014.png" /> are calculated at the point of contact. If all three coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070560/o07056015.png" /> in the equation of the osculating plane vanish, then the osculating plane becomes indefinite (and can coincide with any plane through the tangent line).
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where $X,Y,Z$ are moving coordinates and $x,y,z$, $x',y',z'$, $x'',y'',z''$ are calculated at the point of contact. If all three coefficients of $X,Y,Z$ in the equation of the osculating plane vanish, then the osculating plane becomes indefinite (and can coincide with any plane through the tangent line).
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.S. Millman,  G.D. Parker,  "Elements of differential geometry" , Prentice-Hall  (1977)  pp. 31–35</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.J. Struik,  "Lectures on classical differential geometry" , Dover, reprint  (1988)  pp. 10ff</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.S. Millman,  G.D. Parker,  "Elements of differential geometry" , Prentice-Hall  (1977)  pp. 31–35</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  D.J. Struik,  "Lectures on classical differential geometry" , Dover, reprint  (1988)  pp. 10ff</TD></TR>
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</table>
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[[Category:Differential geometry]]

Latest revision as of 19:51, 15 November 2014

at a point $M$ of a curve $l$

The plane having contact of order $n\geq2$ with $l$ at $M$ (see Osculation). The osculating plane can also be defined as the limit of a variable plane passing through three points of $l$ as these points approach $M$. Usually, a curve intersects the osculating plane at the point of contact (see Fig.).

Figure: o070560a

If $l$ is given by equations

$$x=x(u),\quad y=y(u),\quad z=z(u),$$

then the equation of the osculating plane has the form

$$\begin{vmatrix}X-x&Y-y&Z-z\\x'&y'&z'\\x''&y''&z''\end{vmatrix}=0,$$

where $X,Y,Z$ are moving coordinates and $x,y,z$, $x',y',z'$, $x'',y'',z''$ are calculated at the point of contact. If all three coefficients of $X,Y,Z$ in the equation of the osculating plane vanish, then the osculating plane becomes indefinite (and can coincide with any plane through the tangent line).


Comments

References

[a1] R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 31–35
[a2] D.J. Struik, "Lectures on classical differential geometry" , Dover, reprint (1988) pp. 10ff
How to Cite This Entry:
Osculating plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osculating_plane&oldid=12865
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article