# Osculating Plane

## osculating plane

[′äs·kyə‚lād·iŋ ′plān]*C*at some point

*p*, this is the limiting plane obtained from taking planes through the tangent to

*C*at

*p*and containing some variable point

*p*′ and then letting

*p*′ approach

*p*along

*C*.

## Osculating Plane

The osculating plane of a curve *l* at a point *M* is the plane having contact of order *n* ≥ 2 with *l* at *M*. The oscillating plane can also be defined as the plane in the limiting

position of the plane through three points of *l* as the points approach *M*.

From the standpoint of mechanics, the osculating plane can be characterized as the acceleration plane. As a mass point moves arbitrarily along *l*, the acceleration vector lies in the osculating plane.

Except for special cases, *l* usually penetrates the osculating plane at *M* (see Figure 1). If *l* is defined by the equations *x* = *x(u*), *y* = *y(u*), and *z* = *z(u*), the equation of the osculating plane is of the form

Here, *X, Y*, and Z are the moving coordinates, and *x*, *y*, *z*, *x*’, *y*’, *z*’, *x*”, *y*”, and *z*” are computed at *M*. If all three coefficients of *X*, *Y*, and Z in the equation vanish, the osculating plane is undefined—it can coincide with any plane through the tangent. (*See also*DIFFERENTIAL GEOMETRY.)

### REFERENCE

Rashevskii, P. K.*Kurs differentsial’noi geomelrii*, 4th ed. Moscow, 1956.