Osculation

of a curve $q$ with a curve $l$ at a given point $M$

A geometrical concept, meaning that $q$ has contact of maximal order with $l$ at $M$ in comparison with any curve in some given family of curves $\{q\}$ including $q$. The order of contact of $q$ and $l$ is said to be equal to $n$ if the segment $QL$ is a variable of $(n+1)$-st order of smallness with respect to $MK$ (see Fig., where $QL$ is perpendicular to the common tangent of $q$ and $l$ at $M$).

Figure: o070590a

Thus, of all the curves in $\{q\}$, the curve having osculation with $l$ is the one which is most closely adjacent to $l$ (that is, for which $QL$ has maximal order of smallness). The curve in $\{q\}$ having osculation with $l$ at a given point $M$ is called the osculating curve of the given family at this point. E.g., the osculating circle of $l$ at $M$ is the circle having maximal order of contact with $l$ at $M$ in comparison with any other circle.

Similarly one can define the concept of osculation of a surface $S$ in a given family of surfaces $\{S\}$ with a curve $l$ (or with a surface) at some point $M$ of it. Here the order of contact is defined similarly, except that one must examine the tangent plane of $S$ at $M$ instead of the tangent line $MK$ 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 $|QL|=O(|MK|^{n+1})$ as $K$ approaches $M$.

References

[a1]	C.C. Hsiung, "A first course in differential geometry" , Wiley (1988) pp. Chapt. 2, Sect. 1.4