# The distance function from osculating plane to the origin under the Frenet-Serret frame

I'm curious about how to compute the distance function from an osculating plane to the origin under the Frenet-Serret frame. Any idea will be appreciated, thanks a lot!

Suppose ${\bf r}(t)$ is the trajectory. If we have ${\bf r}(t_0)$ and the binormal vector ${\bf B}(t_0)$ associated with the trajectory at that point, it is a straightforward and standard exercise to find the distance from the osculating plane to any point in $\mathbb{R}^3$. (See here, for example, and note that ${\bf B}$ is normal to the osculating plane. Thus, the osculating plane is given by ${\bf B}\cdot (\langle x,y,z\rangle-{\bf r}(t_0))=0$.)