Vtyjkyuk

Edit to add: This clearly deals with projective geometry. If I figure it out well enough to post a satisfactory answer, I will do so. If someone else posts a decent answer before then, I will be happy to give you the brownie points.

My understanding of what has been said leading to the quoted statement is as follows:

A space of all allowable coordinate systems under the affine group of $mathbbR^n$ is called an affine space, or an $E_n$. The space of all allowable coordinate system under the centered affine group of $mathbbR^n$ is called a centered $E_n$.

The space of all allowable coordinate systems resulting from centered affine transformations of the solution set of the system of $n-p$ linearly independent equations

$$C_nu^ix^nu+C^i=0,$$

is called a flat sub-manifold of $E_n$, and is an $E_p$. Two $E_p$'s are said to be parallel if they can be transformed into each other by a translation. Two parallel $E_p$'s are said to have the same $p-textdirection.$

After introducing the concept of the intersection of $E_p$'s and $E_q$'s, Schouten states:

[W]e consider the points of $E_p$ 'at infinity' as an improper $E_p-1$ (or $E_p-1$ at infinity. Then two parallel $E_p-1$'s intersect in an improper $E_p-1,$ and from this we see that an improper $E_p-1$ may be considered as a $p-textdirection.$

That statement makes very little sense to me. Can this be stated in terms of a traditional Euclidean 2-plane in 3-space, in a way that clarifies Schouten's statement? Am I to envision translating the plane parallel to itself to "positive infinity"? Am I to consider the "boundary" of the plane at the infinite extremes of its coordinate axes?

The statement appears following equation 3.9 on page 5. It is available in the preview.