Understanding contraction in hyperplane arrangements.

Question

Here are two figures that shows a hyperplane arrangement's contraction (this is from McNulty book, "Matroids, a geometric introduction"):

I am not sure why a became a line in the right figure of 7.19, can someone explain this to me please?

Answer 1

In a hyperplane arrangement $\mathcal{A}$ in dimension $d$, the contraction of $\mathcal{A}$ with respect to $H\in \mathcal{A}$ gives a new hyperplane arrangement in dimension $d-1$, defined as $\mathcal{A}/H=\{H\cap H':H'\in \mathcal{A}\}$. This can be viewed as a hyperplane arrangement in the space $H$, in which case we do not consider $H$ an element of $\mathcal{A}/H$ (note that this agrees with matroid contraction, where contracting an element removes it from the ground set). In the case depicted, $a$ just represents the space the arrangement is in, it is no longer considered an element (a hyperplane) of the arrangement $\mathcal{A}/H$.