# Differential Geometry/Osculating Plane

The term *osculating plane*, which was first used by Tinseau in 1780, of a curve C parametrized by a function **f**(t) at a point **f**(a) is the plane that is approached when it is spanned by two vectors **f**(x)-**f**(a) and **f**(y)-**f**(a) when x and y both approach a.

First, assume that the curve C is at least of class 2.

Then consider the points , , and and consider the points , , and . The segments and are the vectors . If these vectors are linearly independent, then they span a plane.

We can divide each of those vectors by , meaning that the plane is also spanned by the vectors .

We can also replace the second vector by and it is easy to see that and w span the same plane as the original vectors.

Using Taylor's formula, we get

.

This indicates that and that

Thus, as both and approach 0, approaches f'(x) and approaches f*(x). The osculating plane is consequently spanned by f'(x) and f*(x), and consequently contains the tangent line.

Consider the position vector of any point on the osculating plane.

Then it is obvious that the following scalar triple product is equal to 0:

.

The intersection of the osculating plane and the normal plane is called the principal normal line.

If it happens to be the case that f'(x) and f*(x) are linearly dependent, then we can consider every plane containing the tangent line to be the osculating plane.*