How to project 3D plots on a 2D coordinate system without losing the metric scale?

Question

I've been collection data of a river bank last week and I need to plot the cross sections of the data.

The issue is, that the data taken consists of 3 coordinates: easting, northing and elevation. Here is an example data set:

Easting     Northing    Elevation
408797.964  5620093.075 738.943
408797.892  5620092.864 738.718
408797.564  5620092.021 738.567
408797.118  5620090.892 738.497
408796.672  5620089.418 738.570
408796.612  5620089.275 738.776
408796.400  5620088.506 739.217

Which looks plotted like that (plotted with R):

What I did so far is plotting the 3D points to a 2D plane by dividing easting and northing.

$$x = \frac{easting}{northing} \cdot n + c$$

with $x$ being the new x-value (width of the river), $n$ being an zoom factor and $c$ the center (of the channel). So far so good.

But the zoom factor now is an issue for me as I have to maintain the scale of both axes in meter. I achieved to plot this image above using a zoom factor of $20,000,000$. But this is only an approximation of the real zoom factor.

Now my question is, how to determin a correct zoom factor to maintain the scale of both the x- and the y-axis to be $1$ meter?

Answer 1

Ok, this is quite simple, I finally figured out.

You just have to calculate the ratio of the distance of two known points both with the real coordinates and the projected coordinates along the scaled zoom-axis (here: $x$).

Let's say there is point $P(408796.400;5620088.506)$ at the left side of the river and point $Q(408797.964;5620093.075)$ on the right side. Projecting the easting and northing into a new $x$-axis like described in the question above results new $x$-values for the points. The elevation can be ignored.

$$q = \frac{Q_{easting}}{Q_{northing}} = \frac{408796.400}{5620088.506} = 0.07273842743998950112$$ $$p = \frac{P_{easting}}{P_{northing}} = \frac{408797.964}{5620093.075} = 0.07273864659261003431$$

Now we just calculate the distance of $|\overline{QP}|$ and $|\overline{qp}|$:

$$ |\overline{QP}| = \sqrt{(P_x - Q_x)^2 + (P_y - Q_y)^2} = 4.82927085593674275535 $$ $$ |\overline{qp}| = | p - q | = 2.1915262053319 \cdot 10^{-7}$$

The wanted zoom factor is the ratio of the distances:

$$ n = \frac{|\overline{QP}|}{|\overline{qp}|} = 22036108.1888380354757 $$

The guessed $20kk$ wasn't that bad at all.