Let $\pi: P \rightarrow M$ be a principal $G$-bundle. For any point $x \in M$, let $p \in P_x$ be a point in the fiber $P_x = \pi^{-1}(x)$.
In Hamilton's Mathematical Gauge Theory he defines:
The vertical tangent space $V_p$ of the total space $P$ in the point $p$ is the tangent space $T_p(P_x)$ to the fiber.
and
A horizontal tangent space in $p \in P$ is a subspace $H_p$ of $T_pP$ complementary to the vertical tangent space $V_p$, so that $$T_pP = V_p \oplus H_p.$$
I am trying to develop some geometric intuition for these spaces but I am having trouble visualizing them. Is there any reason these tangent spaces are called vertical and horizontal?
On the Wikipedia page (https://en.wikipedia.org/wiki/Vertical_and_horizontal_bundles) they state:
The name is motivated by low-dimensional examples like the trivial line bundle over a circle, which is sometimes depicted as a vertical cylinder projecting to a horizontal circle.
In this case the fibers are vertical, but what about the tangent spaces to the fibers are vertical or the complement horizontal?