Consider the following diagram:
The blue segment is projected on the orange projection screen from a specific point of view. The projection is shown at the bottom of the image. About the blue segment is known the following: the orthogonal distance between the point of view and each of the segment's edges, d1 and d2.
Now we select a point on the projected segment, such that it splits the projected segment into 2 segments of length p1 and p2.
Is there a way, with that information, to calculate the orthogonal distance d from the point of view to the actual point represented by the point on the projected segment?
If so, how? If not, what info would be required?
Notes:
The complexity comes from the fact that the proportions of the projected segment do no respect the proportions of the original segment (notice how the original segment is roughly split in half, while the projected segment is not split in equal halves, i.e. p1 is not equal to p2).
Since the segment is straight, I wonder if there is a formula that would give the ratio on the segment based on the ratio on the projected segment. I've done some empirical tests to find the p1/(p1+p2) ratio that represents the exact middle of the original segment. x1 and x2 are the other coordinate of the left and right segment, respectively.
| x1 | d1 | x2 | d2 | ratio | p1/(p1+p2) |
|---|---|---|---|---|---|
| -0.5 | 1 | 0.5 | 2 | 0.5 | 2/3 |
| -0.5 | 1 | 0.5 | 3 | 0.5 | 6/8 |
| -0.5 | 1 | 0.5 | 4 | 0.5 | 8/10 |
Here is the recipe, which would help you to find an expression for $d$ in terms of $d_1$, $d_2$, $p_1$, and $p_2$. Throughout this exercise, you will be using properties of a pair of similar triangles stated in Theorem Euclid VI. 4.
First, as shown in the diagram, let the orthogonal distance between the orange screen and the point of view be $d_0$. You do not need to worry about its length, because it will be used only to facilitate the derivation and will not appear in the final expression.
Two right-angled triangles $CMD$ and $CLA$ are similar. Apply the mentioned theorem to them to obtain an expression for $a$ in terms of $p_1$, $d_0$, and $d_1$.
Similarly, the pair of right-angled triangles $JMC$ and $BLC$ are also similar. Apply the mentioned theorem to them to obtain an expression for $b$ in terms of $p_2$, $d_0$, and $d_1$.
Two right-angled triangles $JKE$ and $JMC$ are similar. Apply the mentioned theorem to them to obtain an expression for $c$ in terms of $p_2$, $d_0$, $d_1$ and $d_2$.
Use the three formulae derived above to express $e$ in terms of $p_1$, $p_2$, $d_0$, $d_1$ and $d_2$.
Two right-angled triangles $DKE$ and $DMN$ are similar. Apply the mentioned theorem to them to obtain an expression for $f$ in terms of $p_1$, $p_2$, $d_1$ and $d_2$.
Finally, use the formula derived last, to obtain the sought expression for $d$ in terms of $p_1$, $p_2$, $d_1$ and $d_2$.