What are projection transformations?

Question

As much as I know a projection transformation is the mathematical conversion of a map from one projected coordinate system to another, generally used to integrate maps from two or more projected coordinate systems into a GIS.

But I lack the mathematical rigor, especially in terms of proving/disproving if some transformation is a projective transformation. In particular, I have been racking my brains with the following figure, where I have to decide upon if there is a projective transformation $r(A_1,B_1,C_1)\bar{\wedge}s(A_2,B_2,C_2)$.

How does one check if there is such a projective transformation?

Answer 1

There's a projective transformation $T$ from line $s$ to line $r$ defined by projection from the point $P$. This transformation takes $B_2$ to $B_1$, and $C_2$ to $C_1$, because the general rule is "for any point $X$ on $s$, draw the line from $P$ through $X$ and see where it hits $r$; that intersection point will be $T(X)$."

What happens if we apply this rule to the point $X = A_2$? We draw a line from $P$ through $A_2$ and see where it hits the line $r$. Well, since $A_2$ happens to be on $r$, it hits that line at ... $A_2$! So $T(A_2) = A_2$ (which happens to also be called $A_1$ in your diagram).

If you wanted a transformation from $r$ to $s$ instead, just replace all the "2"s in my explanation with $1$s, and vice-versa.

One challenge in answering this question is knowing what approach to projective geometry you're taking. ("real projective geometry", "synthetic projective geometry", some mix of these, or maybe "complex projective geometry"). For further questions, it might help to say "I've been reading Coxeter's book on projective geometry, and in chapter 2 it says ..."; that'll help establish context for folks trying to answer.