Spacing of perspective points

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?

(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.)

• 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. – J_F_B_M Jan 13 '16 at 12:48