Namespaces
Variants
Actions

Difference between revisions of "Osculation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
''of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705901.png" /> with a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705902.png" /> at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705903.png" />''
+
{{TEX|done}}
 +
''of a curve $q$ with a curve $l$ at a given point $M$''
  
A geometrical concept, meaning that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705904.png" /> has contact of maximal order with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705905.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705906.png" /> in comparison with any curve in some given family of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705907.png" /> including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705908.png" />. The order of contact of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o0705909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059010.png" /> is said to be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059011.png" /> if the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059012.png" /> is a variable of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059013.png" />-st order of smallness with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059014.png" /> (see Fig., where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059015.png" /> is perpendicular to the common tangent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059017.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059018.png" />).
+
A geometrical concept, meaning that $q$ has contact of maximal order with $l$ at $M$ in comparison with any curve in some given family of curves $\{q\}$ including $q$. The order of contact of $q$ and $l$ is said to be equal to $n$ if the segment $QL$ is a variable of $(n+1)$-st order of smallness with respect to $MK$ (see Fig., where $QL$ is perpendicular to the common tangent of $q$ and $l$ at $M$).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o070590a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o070590a.gif" />
Line 7: Line 8:
 
Figure: o070590a
 
Figure: o070590a
  
Thus, of all the curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059019.png" />, the curve having osculation with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059020.png" /> is the one which is most closely adjacent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059021.png" /> (that is, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059022.png" /> has maximal order of smallness). The curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059023.png" /> having osculation with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059024.png" /> at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059025.png" /> is called the osculating curve of the given family at this point. E.g., the [[Osculating circle|osculating circle]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059026.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059027.png" /> is the circle having maximal order of contact with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059028.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059029.png" /> in comparison with any other circle.
+
Thus, of all the curves in $\{q\}$, the curve having osculation with $l$ is the one which is most closely adjacent to $l$ (that is, for which $QL$ has maximal order of smallness). The curve in $\{q\}$ having osculation with $l$ at a given point $M$ is called the osculating curve of the given family at this point. E.g., the [[Osculating circle|osculating circle]] of $l$ at $M$ is the circle having maximal order of contact with $l$ at $M$ in comparison with any other circle.
  
Similarly one can define the concept of osculation of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059030.png" /> in a given family of surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059031.png" /> with a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059032.png" /> (or with a surface) at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059033.png" /> of it. Here the order of contact is defined similarly, except that one must examine the tangent plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059034.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059035.png" /> instead of the tangent line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059036.png" /> in the figure.
+
Similarly one can define the concept of osculation of a surface $S$ in a given family of surfaces $\{S\}$ with a curve $l$ (or with a surface) at some point $M$ of it. Here the order of contact is defined similarly, except that one must examine the tangent plane of $S$ at $M$ instead of the tangent line $MK$ in the figure.
  
 
====References====
 
====References====
Line 17: Line 18:
  
 
====Comments====
 
====Comments====
The phrase  "QL is a variable of the variable of n+1-st order of smallness with respect to another variablen+1-st order of smallness with respect to MK"  means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059037.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059038.png" /> approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070590/o07059039.png" />.
+
The phrase  "QL is a variable of the variable of n+1-st order of smallness with respect to another variablen+1-st order of smallness with respect to MK"  means that $|QL|=O(|MK|^{n+1})$ as $K$ approaches $M$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Hsiung,  "A first course in differential geometry" , Wiley  (1988)  pp. Chapt. 2, Sect. 1.4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Hsiung,  "A first course in differential geometry" , Wiley  (1988)  pp. Chapt. 2, Sect. 1.4</TD></TR></table>

Latest revision as of 13:16, 29 April 2014

of a curve $q$ with a curve $l$ at a given point $M$

A geometrical concept, meaning that $q$ has contact of maximal order with $l$ at $M$ in comparison with any curve in some given family of curves $\{q\}$ including $q$. The order of contact of $q$ and $l$ is said to be equal to $n$ if the segment $QL$ is a variable of $(n+1)$-st order of smallness with respect to $MK$ (see Fig., where $QL$ is perpendicular to the common tangent of $q$ and $l$ at $M$).

Figure: o070590a

Thus, of all the curves in $\{q\}$, the curve having osculation with $l$ is the one which is most closely adjacent to $l$ (that is, for which $QL$ has maximal order of smallness). The curve in $\{q\}$ having osculation with $l$ at a given point $M$ is called the osculating curve of the given family at this point. E.g., the osculating circle of $l$ at $M$ is the circle having maximal order of contact with $l$ at $M$ in comparison with any other circle.

Similarly one can define the concept of osculation of a surface $S$ in a given family of surfaces $\{S\}$ with a curve $l$ (or with a surface) at some point $M$ of it. Here the order of contact is defined similarly, except that one must examine the tangent plane of $S$ at $M$ instead of the tangent line $MK$ in the figure.

References

[1] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1 , MIR (1982) (Translated from Russian)
[2] P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)
[3] J. Favard, "Cours de géométrie différentielle locale" , Gauthier-Villars (1957)
[4] V.A. Zalgaller, "The theory of envelopes" , Moscow (1975) (In Russian)


Comments

The phrase "QL is a variable of the variable of n+1-st order of smallness with respect to another variablen+1-st order of smallness with respect to MK" means that $|QL|=O(|MK|^{n+1})$ as $K$ approaches $M$.

References

[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1988) pp. Chapt. 2, Sect. 1.4
How to Cite This Entry:
Osculation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osculation&oldid=15619
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article