# Osculation

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## Osculation

The osculation of a curve *q* with a curve *l* at a given point *M* is a geometric concept meaning that *q* at *M* has contact with *l* of maximum order by comparison with the other curves in some preassigned family of curves *{q}* containing *q*.

The order of contact of *q* and *l* is *n* if the length of the segment *QL* is an infinitesimal of order *n* + 1 relative to the length of the segment *MK* (see Figure 1). Here, *QL* is perpendicular, when extended, to the common tangent of *q* and *l* at *M*. Thus, with respect to the curves in *{q}*, the curve that is closest to *l*—that is, the curve for which the length of *QL* is an infinitesimal of maximum order—is the osculating curve of *l* at *M*. For example, the osculating circle of *l* at *M* is the circle whose order of contact with *l* is greater than that of any other circle.

The osculation of a surface *q* belonging to a given family of surfaces *{q}* with some curve / or with some surface at the point *M* of the curve or surface can be defined in much the same way. The order of contact here is defined in a manner similar to the above. We need only replace the tangent line *MK* shown in Figure 1 by the tangent plane of *q* at *M*.

### REFERENCES

La Vallée Poussin, C.-J. de.*Kurs analiza beskonechno malykh*, vol. 2. Leningrad-Moscow, 1933. (Translated from French.)

Il’in, V. A., and E. G. Pozniak.

*Osnovy matematicheskogo analiza*, 3rd ed., part 1. Moscow, 1971.