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Trying to come across the idea of the cross ratio naturally by thinking about the projective plane $\mathbb{R} \mathbb{P}^2$, using ideas from Brannan's Geometry book: given 4 collinear points $A,B,C,D$ we note (for the vectors respresenting these points) that $$C = aA + bB$$ $$D = cA + dB$$

so that the cross ratio is defined as $(b/a)/(d/c)$. Why is this obvious, and why is it obvious it should be a projective invariant without big calculations?

My guess is that you take

$$C = aA + bB = a[A + (b/a)B] \sim A + (b/a)B$$ and $$D \sim A + (d/c)B$$

so that, for some reason, we want to define the discrepancy between these terms, i.e. we want to find the $\lambda$ that would turn $(b/a)$ into $(d/c)$:

$$(b/a) = \lambda (d/c)$$

so that

$$\lambda = (b/a)/(d/c)$$

tells us that, given two generators of a line ($A$ and $B$) we have a third point specified by the ratio $b/a$ and a fourth point specified by the ratio $d/c$ but because we just consider the factor that turns, for our given starting generators, a third point into a fourth point we expect it to be preserved when we project between lines:

enter image description here

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This is the best I can do to make the statement that the ratio of ratio's (Stillwell's 4 Pillar's book) is preserved under projections, because it seems to be saying you use this term to turn a third point into a fourth point given two staring points. Still not 100% clear on why it should also make sense when you start taking projections from points in perspective :\

Any thoughts? Any pictures to make it nicer? Any ideas on making the ratio of ratio's, along with the cross-ratio, more obvious?

There are some great answers on here, such as https://math.stackexchange.com/a/1023055/82615 or https://math.stackexchange.com/a/627396/82615 but I have not come away with a child-like grasp of this yet, lets hope it can be achieved!

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I'm sure I've seen this phenomenon before: one starts with some quantity defined in terms of coordinates and asks how anyone would think of that and how it could be natural. This is too "up close."

For example, is it better to understand the determinant of a transformation on a real vector space as a change in volume, or is it more valuable to dig into a hideous mess of addition and multiplication of entries in matrices? (and why is this computation invariant under similarity transformations?!) Better to forget about the matrix and focus on the transformation and the single quantity (the determinant).

The change of perspective that fixes this is to go a little more abstract and realize that the 'mysterious' definition in terms of coordinates is just a computational corollary of the abstract picture.

Try this

Here's another version of the definition of the cross-ratio:

Let $A,B,C,D$ be coplanar in $\Bbb R^3$ such that $\langle A\rangle, \langle B\rangle, \langle C\rangle, \langle D\rangle$ are distinct subspaces, and $A+B=C$. Then there is a unique $\lambda\in R$ such that $\langle D\rangle =\langle A+\lambda B\rangle $. This scalar $\lambda$ is called the cross ratio of the $4$-tuple $(A, B; C, D)$. By identifying these lines as points of the projective plane, we have defined the cross-ratio for collinear projective points.

(I may have misremembered the position of $\lambda$ and accidentally adopted a nonstandard version of the cross-ratio: I'll have to check later.)

We also know that since $PGL(2,\Bbb R)\cong GL(3, \Bbb R)/\Bbb R^\times$, by deciding upon a line at infinity, you can identify every projective transformation as a linear transformation on $\Bbb R^3$ working on homogeneous coordinates.

If $T$ is this transformation, then by linearity $T(A)+T(B)=T(A+B)=T(C)$ and if $\alpha D= A+\lambda B$, then $\alpha T(D)=T(A+\lambda B)=T(A)+\lambda T(B)$, so that the cross ratio of $(T(A),T(B);T(C),T(D))$ is also $\lambda$. Thus $\lambda$ is invariant under the action of the projective group.

Conclusion

From here, you get to the familiar definition with ratios of four numbers by fixing a homogeneous coordinate system and doing computations within the framework of homogeneous coordinates.

As I mentioned at the outset, I'm not sure there is any real intuitive value in trying to make rationalizations of the quotient of quotients directly (beyond what you've already said.) It's probably best to grok the single, coordinate-free quantity rather than the method used to find the quantity, just as one can understand the determinant of a transformation in terms of volume change rather than some weird computation with matrix entries.

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