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.).
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).
|[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=12865