Differential Geometry/Osculating Plane

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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 x, x+h_1, and x+h_2 and consider the points P=f(x), P_1=f(x+h_1), and P_2=f(x+h_2). The segments PP_1 and PP_2 are the vectors 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 h_i, meaning that the plane is also spanned by the vectors a_i=\frac{s_i}{h_i}.

We can also replace the second vector by b=\frac{2(a_2-a_1)}{h_2-h_1} and it is easy to see that a_1 and w span the same plane as the original vectors.

Using Taylor's formula, we get

f(x+h_i)-f(x)=h_if'(x)+\frac{h_i}{2!}f''(x)+o(h_i^2).

This indicates that a_1=f'(x)+\frac{h_1}{2}f''(x)+o(h_1) and that b=f''(x)+o(1)

Thus, as both h_1 and h_2 approach 0, a_1 approaches f'(x) and 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 x_1 of any point on the osculating plane.

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

|(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.