Suppose I have a 3D scattered data
$$(A_1,B_1,Z_1)$$$$(A_2,B_3,Z_3)$$$$(A_3,B_3,Z_3)$$$$...$$$$(A_n,B_n,Z_n),$$ shown schematically in the picture below (left) and there is no order in the data.
Now, I build a map from 3D space to 2D space by dividing the first coordinates by the second coordinates, in the following way: $$(A_1,B_1,Z_1)\rightarrow(\frac{A_1}{B_1},Z_1)$$$$(A_2,B_3,Z_3)\rightarrow(\frac{A_2}{B_2},Z_2)$$$$(A_3,B_3,Z_3)\rightarrow(\frac{A_3}{B_3},Z_3)$$$$...$$$$(A_n,B_n,Z_n)\rightarrow(\frac{A_n}{B_n},Z_n)$$ So now I have a 2D scattered data with the corresponding plot schematically represented on the right of the figure.
What does this map represent mathematically? What operation does it correspond to? It looks very much like some kind of projection... In which case what kind of projection is that? Onto a plane, or onto a curved surface? Orthogonal or non-orthogonal? How can I insert 2D plane with coordinates ($\frac{A}{B},Z$) in the initial 3D space with coordinates $(A,B,Z)$?
From each data point $(A_i,B_i,Z_i,)$ find the nearest point on the $Z$ axis, which is $(0,0,Z_i).$ Construct the line between those two points. The point where that line intersects the plane $B = 1$ is $\left(\frac{A_i}{B_i}, 1, Z_i\right).$ You can eliminate the middle coordinate, which is always $1,$ and write the projected pont as $\left(\frac{A_i}{B_i}, Z_i\right).$ The axes for those coordinates are simply copies of the $A$ and $Z$ axes, translated from the plane $B=0$ to the plane $B=1;$ specifically, you can use the lines $B=1,Z=0$ and $A=0,B=1$ as the new axes.
This projection is a kind of hybrid of orthogonal and central projections onto the plane $B=1.$ In any plane containing the $Z$ axis, the projection takes points orthogonally onto a line parallel to the $Z$ axis. But in any plane parallel to the $A,B$ plane, the projection is a central projection through a point with coordinates $(0,0,Z).$