Osculating Plane

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.

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.
References in periodicals archive ?
In this paper we have interested in Tzitzeica elliptic cylindrical curves in Minkowski 3-Space, more precisely we ask in what conditions a cylindrical curve is a Tzitzeica one, namely the function t [right arrow] [tau](t)/[d.sup.2(t) is constant, where d(t) is the distance from origin to the osculating plane of curve.
Further, we define the so called osculating plane of r spanned by the vectors r'(x) and r"(x) in the same point.
At each of the curve, the planes spanned by {T, N}, {T, B} and {N, B} are known respectively as the osculating plane, the rectifying plane and the normal plane[5].
Namely, rectifying, normal, and osculating planes of such curves always contain a particular point.