What does transversal to the flow mean?

Question

The Poinare section is a hyper plane which is transversal to a trajectory.

What does "transversal" mean here? Does it mean that the trajectory goes through this plane?

Answer 1

It means that the tangent vector to the trajecory is not tangent to the hyperplane (where they have points in common). This corresponds to the intuition of passing through (instead of lying in) the hyperplane, yes.

Answer 2

Typically, by transversal we mean a curve that is not tangent to any trajectory. Thus a hyperplane "which is transversal to a trajectory" simply means nowhere is the trajectory tangent to the plane (or more specifically, for all time $t$, $\dot{x}(t)$ is not in the tangent space of the hyperplane).