Finding counter example to the statement $\forall t \in \mathbb{R}, N(t) = \frac{\gamma^{\prime\prime}(t)}{||\gamma^{\prime\prime\prime\prime}(t)||}$

Question

Let $\gamma : \mathbb{R} \rightarrow \mathbb{R^3}$ be a parametric curve with components $f, g, h$ so that $\gamma(t) = (f(t), g(t), h(t)).$ I am looking for a counter example to the statement $\forall t \in \mathbb{R}$, the unit normal vector, if it exists, $N(t) = \frac{\gamma^{\prime\prime}(t)}{||\gamma^{\prime\prime\prime\prime}(t)||}$ I have the idea that if $||\gamma^{\prime\prime}(t)||$ is not a scalar value but a function of $t$ then the above statement is not necessarily true, but I am having a hard time coming up with a counter example that is not too complicated to show.

Answer 1

Counter example construction

Let's take $\gamma''(t)=(1,0,0)$, then $\gamma'(t)=(t+b_x,b_y,b_z)$, $\gamma(t)=(t^2/2+b_xt+c_x,b_yt+c_y,b_zt+c_z)$, then $\gamma''(t)\cdot\gamma'(t)=t+b_x$. This expression is not zero at almost any $t\in\mathbb{R}$, so $\gamma''(t)$ is not normal to curve, therefore $N(t)$ is not normal to curve at almost any $t$.