Perspectivities seem quite useful, but are they really the perspective-shifting maps we've been looking for?
Let's look at some examples of one dimensional projectivities -- that is, projectivities between (extended) lines in P³.
That example was rather simple. Can imagine a slightly more complicated scenario?
Now let's make things more interesting, by imagining projectivites from the real line to itself. We can then think of these projectivities as functions!
And now for something special...a little theorem that many important results will rest upon!