Last night, I posted a method for converting between linear FOV and angular FOV. I have since simplified the procedure greatly. Here is the bottom line:

Let L represent linear FOV, that is, number of feet per 1000 yards (L=305 for my 9×63 binoculars).

Let A represent angular FOV in degrees (A=5.8 for my 9×63 binoculars).

Then:

A = 0.0191 × L

and

L = 52.4 × A

(I hope the “×” symbol works on all browsers–it is supposed to be the “x”, the multiplication symbol).

Ok–for those interested, here is how to “derive” those formulas (which are approximations, but good ones).

Recall that, assuming you have an arctan() function that takes a ratio and returns an angle in radians, the following formula gives you the angular FOV exactly:

A = 2*arctan((L/2)/3000)*180/pi

where pi is 3.1415926535….

The Taylor Series for arctan is:

arctan(x) = x – x^3/3 + x^5/5 – x^7/7 + x^9/9 – ….

Note that because x is very small (typically less than 0.1), x^3/3 (and the higher-order terms) are nearly zero. Thus, arctan(x) is approximately equal to x radians in this situation.

So, the formula becomes:

A = 2*((L/2)/3000)*180/3.1415926535

which simplifies to approximately 0.0191*L.

Now, to get the formula for L given A, just take the reciprocal of 0.0191 to get approximately 52.4.

So, we now see that Freshman Calculus was useful after all!