geometry - Understanding cross ratio and harmonic conjugates

I'm studying projective geometry and I'm really having trouble with ''grokking'' what's it all about.

Is there an easy/intuitive/visual way to understand cross ratio? I understand that it's important because it's an invariant of projective transformations, I just don't understand how someone could discover that: oh, if I just take the ratio of these two points, and ratio of these two points...

Not understanding cross ratio, of course I don't understand harmonic conjugates (I'm aware of complete quadrangle definition too, but it doesn't makes things easier for me) and why would someone want to define such a thing.

I'm aware that this is maybe one of those situations where some concepts and definitions start to make sense after several new definitions and theorems, and you get that Ooooh, that's what we are trying to do here moment.
On the other hand, roots of projective geometry are in the drawings and art, so I have a feeling that these concepts maybe could be ''explainable'' in some easier way.

Answer

One of the most basic aspects of projective geometry is that the projective transformations act transitively on triples of distinct points in the projective line. That is, if $P,Q,R$ are three distinct points on a projective line, and $P',Q',R'$ are three other distinct points, then there exists a projective transformation of the line mapping $P$ to $P'$, $Q$ to $Q'$, and $R$ to $R'$.

Therefore, when considering the relationship between four points $P,Q,R,S$, it makes sense to use a projective transformation to map the first three points $P,Q,R$ to some "standard position". Thus usual choice is to map $P$ to $0$, $Q$ to $1$, and $R$ to $\infty$, in which case the image of $S$ is the cross ratio
$$
(P,Q;R,S) \;=\; \frac{(S-P)(Q-R)}{(Q-P)(S-R)}.
$$
Another way of saying this is that the map
$$
x \;\mapsto\; (P,Q;R,x)
$$
is the unique projective transformation of $\mathbb{R}P^1$ that maps $P$ to $0$, $Q$ to $1$, and $R$ to $\infty$.

In terms of perspective drawing, if $P$, $Q$, and $R$ are points on a line $L$ in a painting, where $R$ lies on the "horizon", then the cross ratio defines a linear scale on $L$, using the distance from $P$ to $Q$ as a unit of length.