I am reading some lecture slides about projective geometry in $\mathbb{R}^{3}$. In particular, given a source plane, $S$, an image plane, $I$, and a focal point, $f$, the issue at hand is the description "lonely lines" in $S$ and $I$ that can be found when $S$ and $I$ are not parallel.
Points $s$ in $S$ are projected onto $I$ by taking the intersection of $I$ with the line passing through $f$ and $s$.
By constructing a plane $S_{f}$ that is parallel to $S$ and contains $f$, the intersection of $S_{f}$ and $I$ defines a line in $I$ that is not the image of any line in $S$. The slides refer to this as the "lonely line in $I$."
Conversely, by constructing a plane $I_{f}$ that is parallel to $I$ and contains $f$, the intersection of $I_{f}$ and $S$ defines a line in $S$ that has no image in $I$. The slides refer to this as the "lonely line in $S$."
Sheafs are defined as any set of parallel lines. In particular, for any sheaf $H$ in $S$, the slides state that the images in $I$ of all lines in $H$ intersect at a point $h$ on the lonely line in $I$.
However, if the lonely line in $I$ is a set of points that are not the images of any points in $S$, then it would seem by definition that the images of lines in $S$ cannot intersect at any points on this line -- since they simply cannot map to any points on this line.
Conversely, the slides state that the pre-image of all the lines in some sheaf in $I$ intersect at a point in the lonely line in $S$; and this incorrect for the same reason -- namely that none of the points on the lonely line in $S$ have an image in $I$.
I am hoping someone can either correct my understanding or clarify the point the slides are trying to communicate.
The projectivity defined by $f$ maps lines in $S$ to sets in $I$ (with $f$ at certain points perhaps being undefined); for most lines of $S$, the resulting image under $f$ is contained in a unique line of $I$. (Some lines in $S$, if $S$ happens to contain $f$, might map to single points in $I$). So we can consider it as a kind of map from lines to lines.
The claim is that a pencil of lines (i.e., either all lines passing through a single point of $S$, or all lines parallel to a particular line of $S$) will map (in the sense above) to a pencil of lines in $I$; for "parallel pencils" (ones whose "intersection point" is at infinity) (which your author calls sheaves, for some reason), the focal point of the transformed pencil will like in the lonely line of $I$.