Empirical formulae for oblique two-direction surveying

It is essentially the projection of each point of a graph in three-dimensional space onto a two-dimensional plane (in fact, the y-coordinate must be divided by 2, but this is not the point). After projecting the coordinates down to two dimensions, it is possible to draw each point on a two-dimensional paper, thus drawing out the entire graph.

(Oblique two-measurement, 200px side length of the cube)

When learning vectors, some transformation formulas were deduced (at that time the matrix multiplication was not known), and by the way, we studied how to draw a visual graph instead of feeling. Make a mess of a set of formulas, basically can be sure is right:

\ (\left\{\begin{matrix}
X ' = x+\frac{\sqrt{2}}{4}y \ \
Y ' = z+\frac{\sqrt{2}}{4}y
\end{matrix}\right.\)

(x, Y, z) represents the coordinates of the midpoint of the space, (x ', y ') that represents the coordinates of the midpoint of the coordinate system to the right and the y-axis (z-axis in the corresponding space) on the paper after the projection. In this coordinate system, you can draw the line y = x to represent the y-axis in space (the z-axis in the figure). Note that this y-axis is pointing to the upper right, which means that this is a left-handed system (in fact, the left-hand system is almost no different from the right-hand system in the coordinate operation, but the Vector Fork product formula is not the same):

(left-hand and right-hand, picture from network)

The above formula applies only to the left-hand system, but a minor modification should give another set of formulas that apply only to the right-hand system. But I am too lazy to push (try) again.

In the form of a transformation matrix:

\ (\begin{bmatrix}
X ' \ \
Y ' \ \
Z
\end{bmatrix}=\begin{bmatrix}
1 & \frac{\sqrt{2}}{4} & 0 \ \
0 & \frac{\sqrt{2}}{4} & 1 \ \
0 & 0 & 0
\end{bmatrix}\begin{bmatrix}
x\\
Y\\
Z
\end{bmatrix}\)

Empirical formulae for oblique two-direction surveying