Namespaces
Variants
Actions

Difference between revisions of "Osculating plane"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
(Category:Differential geometry)
 
Line 24: Line 24:
  
 
====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>
+
<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>
 +
 
 +
[[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=34532
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article