Difference between revisions of "Osculating plane"
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− | <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]] |
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 |
Osculating plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osculating_plane&oldid=32637