# Height of the focal point

intro

### A-series paper size

First we look at the dimensions of an A series paper size. The crucial property is that if you fold it in half, the ratio of the long and short sides stays the same. If $a$ is the length of the short side and $b$ is the length of the long side:

$\frac{a}{b} = \frac{\frac{1}{2}b}{a}$

Therefore

$b^2 = 2 a^2$

We do not have to take square roots here to get the lengths since we are using rational trigonometry. Separations between points are represented as quadrances. So,

$A = a^2, B = b^2$

Therefore, for an A paper we have

$B = 2 A$

For the diagonal $C$ we use Pythagoras’ theorem, so

$C = A + B = A+ 2 A = 3 A$

We focus now on the short segment with quadrance $A$, but the resulting formula is equally applicable to the long segment $B$ and the diagonal $C$. Simply multiply with the appropriate factor.

For the exact dimensions you can use the fact that A0 is 1.0 $m^2$, A1 is A0 folded in half, etcetera. I usually use A4 paper.

### Central projection

The segment with quadrance $A$ lies on the ground plane. The focal point of the camera is a point above the plane. The lines connecting the endpoints of $A$ and the focal point have quadrances $D_1$ and $D_2$. This forms a triangle and the spread between $D_1$ and $D_2$ is $p_{12}$.

We can now apply the cross law

$(D_1 + D_2 - A)^2 = 4 D_1 D_2 (1-p_{12})$

If we drop a line from the focal point to the ground plane and define our height quadrance $H$. This line forms two new triangles with the endpoints of the segment with quadrance $A$. The spreads between the line with quadrance $H$ and the two other lines with quadrances $D_1$ and $D_2$ are $q_1$ and $q_2$ respectively. We can apply the spread law to these right triangles

$\frac{1-q_1}{H} = \frac{1}{D_1}$

$\frac{1-q_2}{H} = \frac{1}{D_2}$

We can solve these for $D_1$ and $D_2$ and insert them into the equation we derived from the cross law

$(\frac{H}{1-q_1} + \frac{H}{1-q_2} - A)^2 = 4 \frac{H^2}{(1-q_1)(1-q_2)} (1-p_{12})$

We can rearrange this equation by taking a square root and multiplying by $(1-q_1)(1-q_2)$

$[(1-q_1)+(1-q_2) \pm 2 \sqrt{(1-p_{12})(1-q_1)(1-q_2)}]H = (1-q_1)(1-q_2) A$

with all terms of $H$ on one side.

### Solution

If we solve for $H$ we get

$H = \frac{(1-q_1)(1-q_2)}{(1-q_1)+(1-q_2) \pm 2 \sqrt{(1-p_{12})(1-q_1)(1-q_2)}} A$

Note that there are 2 solutions. It turns out that only relevant solution in our context is for the spread $p_{12}$ being acute, and therefore the one with a minus sign.

So, the height quadrance $H$ for a given segment with quadrance $A$ on the ground plane, visual spread $p_{12}$, and spreads with the gravity direction $q_1$ and $q_2$ is

$H = \frac{(1-q_1)(1-q_2)}{(1-q_1)+(1-q_2) - 2 \sqrt{(1-p_{12})(1-q_1)(1-q_2)}} A$

I can not imagine how to derive this in classical trigonometry.