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 ${\displaystyle x}$, ${\displaystyle x+h_{1}}$, and ${\displaystyle x+h_{2}}$ and consider the points ${\displaystyle P=f(x)}$, ${\displaystyle P_{1}=f(x+h_{1})}$, and ${\displaystyle P_{2}=f(x+h_{2})}$. The segments ${\displaystyle PP_{1}}$ and ${\displaystyle PP_{2}}$ are the vectors ${\displaystyle s_{i}=f(x+h_{i})-f(x)}$. If these vectors are linearly independent, then they span a plane.

We can divide each of those vectors by ${\displaystyle h_{i}}$, meaning that the plane is also spanned by the vectors ${\displaystyle a_{i}={\frac {s_{i}}{h_{i}}}}$.

We can also replace the second vector by ${\displaystyle b={\frac {2(a_{2}-a_{1})}{h_{2}-h_{1}}}}$ and it is easy to see that ${\displaystyle a_{1}}$ and w span the same plane as the original vectors.

Using Taylor's formula, we get

${\displaystyle f(x+h_{i})-f(x)=h_{i}f'(x)+{\frac {h_{i}}{2!}}f''(x)+o(h_{i}^{2})}$.

This indicates that ${\displaystyle a_{1}=f'(x)+{\frac {h_{1}}{2}}f''(x)+o(h_{1})}$ and that ${\displaystyle b=f''(x)+o(1)}$

Thus, as both ${\displaystyle h_{1}}$ and ${\displaystyle h_{2}}$ approach 0, ${\displaystyle a_{1}}$ approaches f'(x) and ${\displaystyle b}$ 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 ${\displaystyle x_{1}}$ of any point on the osculating plane.

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

${\displaystyle |(x_{1}-f(x))f'(x)f''(x)|=0}$.

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.