How would you plot the axes of a 3D orthonormal cartesian coordinate system?
Assuming the axes are labeled $x, y, z$ (see figure below) and that if you look at the $yz$-plane you see a uniform grid (squares), what should be the length of the unit vector of axis $x$ you plot. Is there any specifics or standards for the angle between axes $x$ and $y$ (in the plot of course). Should it be 135 degrees for example?
On
This is an answer to a self-asked question, so very comments are welcome to enrich and discuss the content.
It turns out that this view of the Cartesian coordinate system corresponds to the so-called "Oblique projection" of the 3D objects we want to represent. This projection has the advantage of representing one side (here the projection into the $yz$-plane) of the considered 3D object with its true, non distorted, form (scaled of course). Even though this type of projection is not the most realistic (see perspective projection) it seems it is the easiest for representing 3D objects in 2D.
Oblique projections are characterized by two parameters: the angle $\theta$ at which we draw the side of the 3D object and the scaling factor (aka foreshortening factor) $s$, which is the ratio between the plotted length and the real length of the 3D object along the viewing direction. The choices of $\theta$ and $s$ may be arbitrary and the projection still would be oblique. Nonetheless, some specific values have been used until their use spread. I cite cavalier and cabinet (with its variants) projections. Bellow, I give two examples of such projections. What it represented is a cube with side $a$.
Characterized specifically by a scaling factor $s=1$ and an angle $\theta= 45^\circ$.
Characterized by a scaling factor $s=\frac{1}{2}$ and angle $\theta= 45^\circ$. Other Cabinet projections exist also (see here).
If you use graphing paper, or any similar method with a regular square base grid, the three I prefer are
because of the ease in both drawing and perception; each one of the above shows a typical unit cube drawn in $(0,0,0)-(1,1,1)$.
The center one has been particularly common in computer games, and is close to military projection. The vertical Z axis units in military projection match the X and Y axis units (as measured in the projection), but in the above image, the Z axis units projected length are only $1:\sqrt{2}$ or about 70% of the X and Y axis units projected length.
The left and right projections are trimetric, with Z axis units $\sqrt{4/5}$ or about 89% of the X and Y axis units. It turns out that this difference is not as perceptible. The angles in the left and right trimetric projections, above, are 90° (top), 116.565°, and 153.435°.
If $x$ increases down left, $y$ down right, $z$ up, with the dotted-line vertex as the origin, and projected $x^{,}$ right and $y^{,}$ down, then $$\begin{cases} x^{,} = y - 2 x \\ y^{,} = 2 z - x - 2 y \end{cases}$$ for the leftmost projection; $$\begin{cases} x^{,} = y - x \\ y^{,} = z - x - y \end{cases}$$ for the center projection; and $$\begin{cases} x^{,} = 2 y - x \\ y^{,} = 2 z - 2 x - y \end{cases}$$ for the rightmost projection.