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''of a surface at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705501.png" />''
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''of a surface at a point $P$''
  
The paraboloid that reproduces the shape of the surface near this point up to variables of the second order of smallness with respect to the distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705502.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705503.png" /> be a paraboloid (see Fig.) with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705504.png" /> and tangent to the surface at this point, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705506.png" /> be the distances of an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705507.png" /> on the paraboloid to the surface and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705508.png" />, respectively.
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The paraboloid that reproduces the shape of the surface near this point up to variables of the second order of smallness with respect to the distance from $P$. Let $\Phi$ be a paraboloid (see Fig.) with vertex $P$ and tangent to the surface at this point, and let $h$ and $d$ be the distances of an arbitrary point $Q$ on the paraboloid to the surface and to $P$, respectively.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o070550a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o070550a.gif" />
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Figure: o070550a
 
Figure: o070550a
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o0705509.png" /> is said to osculate if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055010.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055011.png" />. This does not exclude the degeneration of the paraboloid into a parabolic cylinder or plane. At every point of a regular surface there is a unique osculating paraboloid. Osculating paraboloids can be used to classify the points on a surface (see [[Elliptic point|Elliptic point]]; [[Hyperbolic point|Hyperbolic point]]; [[Parabolic point|Parabolic point]]; [[Flat point|Flat point]]).
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Then $\Phi$ is said to osculate if $h/d^2\to0$ as $Q\to P$. This does not exclude the degeneration of the paraboloid into a parabolic cylinder or plane. At every point of a regular surface there is a unique osculating paraboloid. Osculating paraboloids can be used to classify the points on a surface (see [[Elliptic point|Elliptic point]]; [[Hyperbolic point|Hyperbolic point]]; [[Parabolic point|Parabolic point]]; [[Flat point|Flat point]]).
  
  
  
 
====Comments====
 
====Comments====
The osculating paraboloid at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055012.png" /> to the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055013.png" /> has contact of order three with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055014.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055015.png" />, i.e. the derivatives up to and including order 2 of the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055016.png" /> of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055018.png" /> describing the paraboloid and the surface are all zero at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070550/o07055020.png" />.
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The osculating paraboloid at $P$ to the surface $S$ has contact of order three with $S$ at $P$, i.e. the derivatives up to and including order 2 of the difference $p(x,y)-s(x,y)$ of the functions $p(x,y)$ and $s(x,y)$ describing the paraboloid and the surface are all zero at $(x_0,y_0)$, where $P=p(x_0,y_0)=s(x_0,y_0)$.
  
 
====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. 138</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.S. Millman,  G.D. Parker,  "Elements of differential geometry" , Prentice-Hall  (1977)  pp. 138</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR></table>

Latest revision as of 13:35, 29 April 2014

of a surface at a point $P$

The paraboloid that reproduces the shape of the surface near this point up to variables of the second order of smallness with respect to the distance from $P$. Let $\Phi$ be a paraboloid (see Fig.) with vertex $P$ and tangent to the surface at this point, and let $h$ and $d$ be the distances of an arbitrary point $Q$ on the paraboloid to the surface and to $P$, respectively.

Figure: o070550a

Then $\Phi$ is said to osculate if $h/d^2\to0$ as $Q\to P$. This does not exclude the degeneration of the paraboloid into a parabolic cylinder or plane. At every point of a regular surface there is a unique osculating paraboloid. Osculating paraboloids can be used to classify the points on a surface (see Elliptic point; Hyperbolic point; Parabolic point; Flat point).


Comments

The osculating paraboloid at $P$ to the surface $S$ has contact of order three with $S$ at $P$, i.e. the derivatives up to and including order 2 of the difference $p(x,y)-s(x,y)$ of the functions $p(x,y)$ and $s(x,y)$ describing the paraboloid and the surface are all zero at $(x_0,y_0)$, where $P=p(x_0,y_0)=s(x_0,y_0)$.

References

[a1] R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 138
[a2] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
How to Cite This Entry:
Osculating paraboloid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osculating_paraboloid&oldid=31971
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article