How can I figure out the length of a rectangle when I know the width of it

Question

I have a picture of a rectangle which looks like a trapezium due to my photo angle. I know the width of this rectangle which is 55 cm. Is there any way I can find the length of it considering where the point the picture is taken?

Answer 1

Interesting problem.

It has a negative answer, which seems unfortunate, but the answer is mathematically interesting.

Consider all rectangles of width 55 (I'll suppress the units of centimeters). They can have arbitrary length. Choosing any two lengths $s,t$, your question can be reworded: Is it possible for an $s \times 55$ rectangle and a $t \times 55$ rectangle to appear the same in a photograph?

To translate this into mathematics, we use projective geometry. Consider two rectangles $S,T$ in Euclidean 3-space $\mathbb{R}^3$. Let $S_1,T_1$ be congruent rectanges in the Euclidean plane $\mathbb{R}^2$. Then take the usual injection of $\mathbb{R}^2$ into the real projective plane $\mathbb{R}P^2$. We can ask: are $S_1,T_1$ projectively equivalent? What this means is that that there is a projective transformation $f : \mathbb{R}P^2 \to \mathbb{R}P^2$ such that $f(S_1)=T_1$.

The key fact needed (discovered by Renaissance artists, and translated into modern language) is that $S,T$ "can appear the same in a photograph" if and only if $S_1,T_1$ are projectively equivalent.

So, if $S_1 = [0,s] \times [0,55]$ and $T_1 = [0,t] \times [0,55]$, does there exist a projective transformation taking $S_1$ to $T_1$? And the answer is "yes", because the transformation that multiplies the horizontal coordinate by $t/s$ and leaves the vertical coordinate unchanged is a projective transformation.