let $\alpha : I \to \mathbb{R}^3$ be a curve parametrized by arc length with curvature $k(s) \neq 0$, $s \in I$.

Question

Let $P$ be a plane satisfying both of the following conditions:

1. $P$ contains the tangent line at s.
2. Given any neighborhood $J \subset I$ of s, there exist points of $\alpha (J)$ in both sides of $P$.

Prove that $P$ is the osculating plane of $\alpha$ at s.

Well, I took this question from Do Carmo, pg 29.

I know that the local canonical form of $\alpha$, in a neighborhood of $s_0$ is

$$\alpha(s) = \alpha(s_0) + x(s)t(s_0) + y(s) n(s_0) + z(s) b(s_0)$$ where,

$x(s) = (s-s_0) + k(s_0)^2 \dfrac{(s-s_0)^3}{6} + R_x(s)$

$y(s) = k(s_0)(s-s_0)^2/2+k'(s_0)(s-s_0)^3/6 + R_y(s) $

$z(s) = -k(s_0)\tau(s_0)(s-s_0)^3/6+ R_z(s) $

$R(s) = (R_x(s),R_y(s),R_z(s))$.

Any help would be nice.

Answer 1

Let us write Taylor formula for $M(s)=(x(s),y(s),z(s)$.

$M(s_0+h)=M(s_0)+h t(s_0)+k(s_0){h^2\over 2} n(s_0)+o(h^2)$

Let $l(x)=0$ be an affine equation of the plane affine osculating plane at $M(s_0)$, say $P$. We choose $M(s_0)$ as the origin of coordinates.

We have $l(M(s_0+h)=k(s_0){h^2\over 2} l(n(s))+ o(h^2)={h^2\over 2} (k(s_0) l(n(s))+ o(1))$.

The sign of $l$ gives the position relative to $P$.

If $l(n(s_0) \not =0$, this sign remains constant as $k(s_0)l(n(s_0)\not = 0$.

Hence if the sign changes, $l(n(s_0))$ must vanish, and the plane is the osculating plane, as it contains the tangent and normal vector.