Osculating plane
From Encyclopedia of Mathematics
at a point 
of a curve 
The plane having contact of order 
with 
at 
(see Osculation). The osculating plane can also be defined as the limit of a variable plane passing through three points of 
as these points approach 
. Usually, a curve intersects the osculating plane at the point of contact (see Fig.).
Figure: o070560a
If 
is given by equations
 |
then the equation of the osculating plane has the form
 |
where 
are moving coordinates and 
, 
, 
are calculated at the point of contact. If all three coefficients of 
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=32637
Osculating plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osculating_plane&oldid=32637
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article