Normal, Rectifying, and Osculating Planes

Normal Planes

Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function that represents the smooth curve $C$ for $t \in [a, b]$, and let $P(x_0, y_0, z_0)$ be a point on $C$ corresponding to $\vec{r}(t_0)$. Then the Normal Plane of $C$ at point $P$ is the plane spanned by $\hat{N}(t_0)$ and $\hat{B}(t_0)$ with normal vector $\hat{T}(t_0)$.

Rectifying Planes

Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function that represents the smooth curve $C$ for $t \in [a, b]$, and let $P(x_0, y_0, z_0)$ be a point on $C$ corresponding to $\vec{r}(t_0)$. Then the Rectifying Plane of $C$ at point $P$ is the plane spanned by $\hat{B}(t_0)$ and $\hat{T}(t_0)$ with normal vector $\hat{N}(t_0)$.

Osculating Planes

Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function that represents the smooth curve $C$ for $t \in [a, b]$, and let $P(x_0, y_0, z_0)$ be a point on $C$ corresponding to $\vec{r}(t_0)$. Then the Osculating Plane of $C$ at point $P$ is the plane spanned by $\hat{T}(t_0)$ and $\hat{N}(t_0)$ with normal vector $\hat{B}(t_0)$.