# Math behind the scale

It's simple.

```Define
delta = delta on rail
delta' = delta on ring
D = delta'/delta
N = min. f-number required
c = circle of confusion
M = magnification
Av = Apeture value in APEX notation

Givens					Goal

N = delta/2/c/(1+M)		Conversion from delta to delta'
D = delta'/delta
-------------------------------------------------------
delta' = D*N*2*c*(1+M)

Basically that's it.
Because I want to calibrate scales in 1/3 steps, using APEX notation,

Av = log(N^2)/log(2)
--------------------------------------------------------
delta' = D*N*2*c*(1+M)
N = 10^(1/2*log(2))*Av
--------------------------------------------------------
delta' = D*10^(1/2*log(2))*Av*2*c*(1+M)

For non-macro work, M is much less than 1, so making M ~ 0, giving:

delta' = D*10^(1/2*log(2))*Av*2*c where Av = { 1/3 stop increments }

This equation is used in making the scale.
It is strictly correct at infinity focus (M = 0).

For arbitrary N and non-zero M,

N(M) = delta/2/c/(1+M)

while

N(M=0) = delta/2/c

so

N(M) = 1/(1+M)*N(M=0)	---- (1)

This equation accounts for the facts explained in Good things to know section:

The min. f-numbers calibrated are correct at magnification M = 0, infinity focus. However, at any magnification, the min. f-number
required for a given delta will be less than that at infinity focus. Therefore you're always guaranteed to get min. dof for a given
delta at any magnification.
At magnification M = 1 (life-size, 1:1), the min. f-numbers N on scale are half. At any magnification in between 0 and 1 (infinity
focus), the min. f-number required will fall in between the f-number scales and their corresponding halves. For an arbitrary N, use
N(M) = 1/(1+M)*N(M=0) to figure out exact min. f-number required at M. For example, at M = 0.5, N(M=0.5) = 2/3*N(M=0). So multiply the
f-numbers on scale by 2/3.

To prove backing off 2 stops on the scale is equivalent to delta/2,

Givens				Goal

(1)				backing off 2 stops on the scale is equivalent to delta/2
-------------------------------------------------------
N(M)/2 is equivalent to delta/2
-------------------------------------------------------
Using (1) and dividing both sides by two,

N(M)/2 = 1/(1+M)*N(M=0)/2 = 1/(1+M)*delta/2/c/2 = const*delta/2
```
So backing off 2 stops on the scale is equivalent to delta/2 or delta'/2; delta'= D*delta, D can be factored into const in the above equation.
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