What kind of projection does a specific map (3D -> 2D) correspond to?

Question

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 and there is no order in the data. Now, I build a map from 3D space to 2D space by subtracting the second coordinates from the first coordinates, in the following way: $$(A_1,B_1,Z_1)\rightarrow(A_1-B_1,Z_1)$$$$(A_2,B_3,Z_3)\rightarrow(A_2-B_2,Z_2)$$$$(A_3,B_3,Z_3)\rightarrow(A_3-B_3,Z_3)$$$$...$$$$(A_n,B_n,Z_n)\rightarrow(A_n-B_n,Z_n)$$ So now I have a 2D scattered data and the plot looks like this: 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 ($A-B,Z$) in the initial 3D space with coordinates $(A,B,Z)$? If these are the family of surfaces onto which a projection takes place, then one surface would be enough to show in the 3D space.

Answer 1

Vectors that lie in the A-Z plane are unmoved by this transformation. A vector in the direction of B-axis is sent to a vector in the A-axis of the same magnitude but opposite direction (ie change of sign).

Contrary to what gimusi says, it can also be regarded as projection, if you embed the image plane in 3d-space differently as below: That is interpreting your map as $(A,B,Z)\mapsto (A-B, 0, Z)$, it would be a non-orthogonal projection, of course.

Answer 2

The transformation $\mathbb{R^3}\to \mathbb{R^2}$ corresponds to the multiplication by the matrix

$$M=\begin{bmatrix}1&-1&0\\0&0&1\end{bmatrix}$$

We can also view it as a transformation $\mathbb{R^3}\to \mathbb{R^3}$ by the matrix

$$N=\begin{bmatrix}1&-1&0\\0&0&1\\0&0&0\end{bmatrix}$$

which is not a projection, indeed $N^2\neq N$.