Osculation
of a curve with a curve
at a given point
A geometrical concept, meaning that has contact of maximal order with
at
in comparison with any curve in some given family of curves
including
. The order of contact of
and
is said to be equal to
if the segment
is a variable of
-st order of smallness with respect to
(see Fig., where
is perpendicular to the common tangent of
and
at
).
Figure: o070590a
Thus, of all the curves in , the curve having osculation with
is the one which is most closely adjacent to
(that is, for which
has maximal order of smallness). The curve in
having osculation with
at a given point
is called the osculating curve of the given family at this point. E.g., the osculating circle of
at
is the circle having maximal order of contact with
at
in comparison with any other circle.
Similarly one can define the concept of osculation of a surface in a given family of surfaces
with a curve
(or with a surface) at some point
of it. Here the order of contact is defined similarly, except that one must examine the tangent plane of
at
instead of the tangent line
in the figure.
References
[1] | V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1 , MIR (1982) (Translated from Russian) |
[2] | P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian) |
[3] | J. Favard, "Cours de géométrie différentielle locale" , Gauthier-Villars (1957) |
[4] | V.A. Zalgaller, "The theory of envelopes" , Moscow (1975) (In Russian) |
Comments
The phrase "QL is a variable of the variable of n+1-st order of smallness with respect to another variablen+1-st order of smallness with respect to MK" means that as
approaches
.
References
[a1] | C.C. Hsiung, "A first course in differential geometry" , Wiley (1988) pp. Chapt. 2, Sect. 1.4 |
Osculation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osculation&oldid=15619