Osculating Plane

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osculating plane

[′äs·kyə‚lād·iŋ ′plān]
For a curve 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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

Figure 1

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 alsoDIFFERENTIAL GEOMETRY.)


Rashevskii, P. K. Kurs differentsial’noi geomelrii, 4th ed. Moscow, 1956.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.