Manifolds Unfolded

Kishalay Sarkar

May 22, 2025

Curves in space – The Osculating Plane and the Frenet-Serret Frame

1. Curves in space – The Osculating Plane and the Frenet-Serret Frame: For a unit speed curve γ in space, if T is the unit velocity vector/tangent vector to the curve, and if the acceleration vector is,

Then the plane spanned by the velocity / tangent vector and the acceleration vector, i.e.

is called the osculating plane of the motion of the unit speed curve γ. The unit vector B defined by,

is called the Binormal vector of the motion of the curve.

The orthonormal frame

is called the Frenet-Serret Frame of the curve γ, and it is often denoted by,

Moving Frenet Frame and the osculating Plane for a parametrized curve in 3-space

The *Frenet-Serret Frame* and the *osculating plane* of the curve γ in space.

Remark: For curves drawn on a surface,

(a) if the osculating plane (of a curve γ on a surface S) is perpendicular to the tangent plane of the surface, then the total curvature or the acceleration vector points normally to the tangent space, and hence the total curvature is contributed by the normal curvature vector, whereas the geodesic curvature component of the acceleration vector vanishes. Thus, they curve is a geodesic in this case.

In this picture, C is a great circle (geodesic) of the sphere S. At any point p on C, the osculating plane and the tangent plane are perpendicular to each other, thus the total curvature is equal to the normal curvature as the acceleration vector is along the surface normal n, whereas the geodesic curvature of C is zero.

On the other hand,

(b) if the osculating plane is almost coincident to the tangent plane of the surface, then the total curvature or acceleration vector lies almost in the tangent plane, perpendicular to the tangent vector T of γ. Thus, the geodesic curvature vector contributes to the majority of the total curvature vector, and the normal component almost vanishes in this case (see the picture below). In particular, for a planar curve, the normal curvature is zero.

If we draw a very small circular curve C’ on the surface of the sphere, then the *osculating plane is almost coincident with the tangent plane*, hence the *normal curvature almost vanishes* and the *geodesic curvature is almost equal to the total curvature*.

(c) For a generic curve on a surface, the scenario is somewhere in the middle of the two situations described above, with both geodesic and normal curvatures contributing to the total curvature.