osculating plane[′äs·kyə‚lād·iŋ ′plān]
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 alsoDIFFERENTIAL GEOMETRY.)