I have a plane. This plane consists of a grid of 25 points arranged equally spaced in a 5x5-grid (alternatively: I have 16 squares arranged to a 4x4-square). I'm only interested in the coordinates of these 25 points

I tilt the grid away from me, so the square changes to a trapez.

During this process, the upper rows get closer together. Basically the distance between the lowest row and the row above is greater than the distance between the second row from down and the third.

My question: Is there an easy way to compute the distances/the coordinates between the points, given only the coordinates of the 4 outer points?
Especially, is the following method correct?

Hard numbers:

$$p_0 = (0,0), p_1=(10,0), p_2=(7.5,5), p_3=(2.5,5)$$ Between $p_0$ and $p_1$ the points are equally spaced and easy to compute. Likewise between $p_2$ and $p_3$.

I understand that the center of the square lies in the intersection of the diagonals. As far as I understood this holds true for the trapez as well.
Let's call this point $p_m$, I found this point to lie in $$p_m = (5, 3.\overline{3}) = (5, 5*\frac{2}{3})$$

That means that 4 of those points must be at $y=3.\overline{3}$. Now I construct the diagonals from the trapez below this $y$ and above this $y$ and note these values as further lines.

This means our $y$-values are (from bottom to top): $$y_0=0, y_1\approx1.90476,y_2=3.\overline{3},y_3\approx4.28571,y_4=5$$ (Note that I do not need the numbers to this degree of accuracy, my calculator happens to limit to 5 digits after the comma)

I can now compute the $x$-values by interpolating linear between the lower and the upper x-coordinates with respect to the computed $y$-value. $$x_{bottom} + (x_{top}-x_{bottom})/height * y$$ Example for leftmost line: $$0 + (2.5 - 0)/5 * y$$

The result I get from this seems visually correct. Is it mathematically correct?

Latex-Rendering with computed points (Please ignore the ticks to the left and right. They are equidistant along the outer edges, which I needed for another picture and forgot to remove.)

  • $\begingroup$ Feel free to add tags as appropriate. If a section is awkwardly formulated, please excuse, English is not my native language and I usually don't talk Math, so there are two language-barriers in place. $\endgroup$ – J_F_B_M Jan 13 '16 at 12:48

It is probably correct but you may be making this harder than necessary.

First note that there are many possible perspectives and you have chosen a particular one that leaves lines in one particular direction (left-right) parallel.

In that perspective foreshortening of left-right distances only depends on the distance between the line that carries the segment and the observer; thus both the front and the back horizontal line of your grid can be divided into 4 equal segments each, and you can connect the corresponding points.

The intersection of one of the diagonals with the above connection lines give you the position of the other horizontal lines (height, and therefore horizontal coordinates of their end points). The points on those horizontal lines can be most easily obtained by dividing each horizontal line in 4 equal segments.

  • $\begingroup$ So you mean the diagonal above is not just accidentally going through the corners of the squares? In other words: It does not only go through the (former) center of the square, but all points of the former diagonal? $\endgroup$ – J_F_B_M Jan 13 '16 at 13:03
  • $\begingroup$ No it isn't and yes it does. In my construction it is even defining the corners of some of the squares. In general, any meaningful perspective would have to preserve at least incidence, i.e., if three lines intersect in the real world then they should intersect in perspective. In your particular perspective we have the additional benefit that straight lines in the real world are also straight lines in perspective. $\endgroup$ – Justpassingby Jan 13 '16 at 13:04
  • $\begingroup$ After reading my (edited) comment, your comment and answer I wonder how I didn't saw that before. Thank you for opening my eyes. $\endgroup$ – J_F_B_M Jan 13 '16 at 13:06

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