"Maths is at only one remove from magic."
― Neel Burton
A while back I was invited to participate in a podcast for work on the topic of Field of View (FOV). In a nutshell, there are three ways sport optics companies list it:
- Linear FOV (most common)
- Angular FOV (next most common)
- Apparent FOV (rarely ever)
Each means something slightly different, but essentially all work with the same angle. With binoculars and spotting scopes, the Linear FOV is expressed as a number of feet at 1,000 yards. For example, you've probably seen something like "360 feet at 1,000 yards." Linear FOV can be converted to Angular FOV simply by dividing the 360 by 52.5. The reason we use that number is a single degree at 1,000 yards gives a linear distance of 52.5 feet. It's actually 52.365, but 52.5 is accepted.
In my reduced hypothetical FOV diagram, the optic would have the following specifications:
Linear FOV = 210 feet at 1,000 yards
Angular FOV = 4 degrees
And the formula to calculate Apparent FOV:
Apparent FOV = Angular FOV x Magnification
Thus, if the above were an 8x binocular, the Apparent FOV would be 32 degrees from 8x times 4 degrees, so all FOV specifications would be:
Linear FOV = 210 feet at 1,000 yards
Angular FOV = 4 degrees
Apparent FOV = 32 degrees
Here are the formulas to convert one FOV type to another:
Linear FOV = Angular FOV x 52.5 *
Angular FOV = Linear FOV / 52.5 *
Apparent FOV = Angular FOV x Magnification
There's no magic or voodoo here — it's just basic math and trigonometry. Apparent FOV is merely the "naked eye" FOV multiplied by the magnification. In other words, if you were looking at the two ends of the optic's FOV angle at 1,000 yards without the binocular, that distance is the FOV as a pie slice taken away from 360 degrees.
Anyway, the following comment was made to the podcast:
1 degree at 1,000 yards "is" 1 degree. It "subtends" to 52.5 feet.
I replied:
Another way to state it would be "1 degree at 1,000 yards in linear feet is 52.5". Technically the conversion is linear FOV to angular FOV and vice versa, which was the point of offering the 52.5 figure.
Which rendered this response from the same individual:
I respectfully disagree. Degrees are units of angle, not units of length or distance. 1 degree at 2769 yards "is" 1 degree. It isn't some length of inches or any other unit of length. Not being picky or snooty. It's just better when technical information transfer is done literally, not colloquially. When I first got into shooting and optics in a serious way and was trying to learn from all the different "experts" on YouTube and from manufacturers' websites, I found it very frustrating trying to understand what they were trying to communicate, simply because they were speaking colloquially rather than literally. Take for example all of the YouTube videos you can find going into all this detail and talking in circles describing "What is MOA or minute of angle." One even wrongly related it to minutes on a clock, and another related it to a compass. What? Gosh, it's just simply an angle that is one minute in size, one angular minute, 1/60th of one degree. Even so, I greatly appreciate everyone's time and effort being great guys and giving away their knowledge.
There's nothing to disagree with here. He's kind of correct, but on the wrong track. I responded by stating the 3 FOV types, conversion formulas, and provided the following examples:
When an optic has a linear field of view of 390 feet at 1,000 yards, what we're saying is that the diameter of the circle is that many feet across when observing something at that particular distance. Think of a drawing of a subtended angle and at some distance draw a straight line across it — it's the linear length of that line. This same linear number can be converted to the subtended angle as you say with the number 52.5. With 390, it's 7.42. If you look at any Sport Optic manufacturer's website, you will see you can convert the linear, angular, and apparent using these formulas — I didn't invent them; they're an industry accepted standard.
Randomly picking Maven Optics and their B.2 9x45 binocular, they list the following FOV specifications on their product page:
Angular FOV = 7.2 degrees
Linear FOV = 377 feet at 1,000 yards
Showing the work:
7.2 x 52.365 = 377
It works!
Let's say you have a binocular that you don't have the FOV specification for. You can set a yardstick 10 feet away from it, note how many inches you can see from one side of the field to the other, and then use a proportion to get the linear FOV at 1,000 yards. The angle is the same all the way out to infinity, but the linear changes depending on how far out one is observing. The 1,000 yards for binoculars and spotting scopes is just a standard distance that was chosen decades ago.
Doubling down, the individual replied:
Obviously, that's not what I'm talking about. I'm talking about the unfortunate common misuse of units in descriptions like "2 minutes of angle equals about 2 inches at 100 yards." They do not "equal" any number of inches or other length of measure. Units of degrees and portions of degrees of angle "equal" an equivalent number of radians or other unit of angle, but they do not "equal" any unit of length. In your using colloquial language to address technical topics, you make it difficult for new users who are thinking people and who are trying to figure out the meaning of your colloquial descriptions. Colloquial language is great for chilled out casual discussions. But when someone is new and trying to figure out technical things by listening to your technical descriptions, they are depending on your words to carry their correct meanings. I think you would sell more products, especially to new customers who will become lifelong customers, if you, along with the other merchants, wouldn't make things so much harder to understand than they need to be. As is, it's like being dumped into a strange land with a different language and learning the new language without the benefit of a teacher who knows both languages. Okay, I'm finished with this. Listen and improve you program and your sales, or don't. Your choice. I don't care either way.
Wow.
I replied one final time:
I think it's quite easy to understand even for a novice when picturing FOV as an isosceles triangle — any line drawn across the triangle from one leg to another has a measurable length. Linear FOV is analogous to one of those lines, one that just happens to be 1,000 yards from the optics user. Naturally, the angle is constant from the user to infinity. What changes is the factor you would use to run the various FOV formulas. For 1,000 yards, that number just happens to be 52.365 (rounded to 52.5). Because, at 1,000 yards, 1 of those degrees across that line is equal to 52.365 feet. If you wanted to measure linear FOV at some other distance rather than 1,000 yards, it wouldn't be 52.365. But you could get all the pertinent numbers using the formulas above. The angle is exactly the same and the math will still work out — this is basic trigonometry and nothing more and it's how FOV is measured in the Sport Optics Industry. It's exceedingly easy to picture, and easy run the conversion formulas. They work, and that's all optics users need to know.
And that was the end of the thread.
Alright, so take another look at my diagram above. Each "dx" linear value is 52.5 feet only at 1,000 yards. Add them up and you get the Linear FOV of 210' at 1,000 yards. Of course, each angle is 1 degree (4 degrees divided by 4 angles) all the way out. The point of expressing FOV as a linear number in sport optics is to offer a way for end users to visualize and compare brands and models. What I think the commenter is missing is the red line I've drawn across the triangle — those really are linear distances. We're not redefining what a degree is with any of the FOV specifications; we're measuring linear distances of a line intersecting the FOV triangle.
Make sense? :)
In a future blog post I'm going to trash the junk phrase: Let's agree to disagree.
FOV image © 2023 Mike McDowell