Field of View

field of viewField of view (FoV), often expressed as an angle, is the extent of the observable world that is seen at a given distance from the observer. Technically, when stated as an angle we are describing the Angle of View (AoV) or AFoV, which may be confusing. To separate them, we define the FoV as a horizontal distance covered by the width of what can be seen by the sensor/person. The AoV is the angle that the sensor sees out from the center. One can derive the FoV from the AFoV by using a little bit of trigonometry and understanding the relationship between focal length, magnification, and aperature.

The trigonometry used to describe FoV is broken down into right triangles in the following diagram with angle of view A degrees, Distance D from the observer, and width (FoV) 2W where W is the small opposite side of the right triangle. With this breakdown if we have two of the three values we can use them to solve for the third.

angle of view

The relation is derived in the following way. We split the angle down the middle and work with A/2 because this give us a right triangle. By definition, tangent of A/2 is equivalent to the ratio of Opposite over Adjacent and we have:

tan(A/2)= W/D

So, if provided the distance to target and field of view then we can calculate the angle of view easily.

arctan(tan(A/2)) = A/2 = arctan(W/D)

We double both sides because we’re solving for A

A = 2 * A/2 = 2*arctan(W/D) = AFoV

Using the original formula, we will always be able to solve for one of the three values if provided the other two. Their respective solutions are:

W = D * tan(A/2)

and

D = W/{tan(A/2)}

It is important to remember that these relations are easily solved because they are right triangles and while D the distance has the correct scalar, both the angle and FoV are half of what the real value is. So, make sure to double the final values and keep in mind that AFoV = A and FoV =  2W.

Using this formula we’ve created a small table of Fields of View for reference. It’s a linear relationship independent of units, we used yards for our example but it can be feet, meters, miles, or any other unit of distance.

AfoVFoV @ 50 yardsFoV @ 100 yardsFoV @ 150 yards
3 degrees2.62 yards5.24 yards7.86 yards
5 degrees4.37 yards8.74 yards13.10 yards
12 degrees10.51 yards21.02 yards31.53 yards
40 degrees36.40 yards72.80 yards109.2 yards

Not all optical systems use circular lenses, they can be rectangular or even oddly not-uniform at all. Note the square sensor below. The calculations for these systems get much more complicated quickly.

rectangular lensFor sport optics, field of view depends on the ratios of the magnification and the focal length of the objective/eye piece. This is why we can change the FoV and thus magnification of optical systems by moving the lenses in particular ratios from each other.

 

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**Note that the diameter of the objective lens does not affect FoV.