Rifle Scopes Does scope FOV scale linearly throughout the mag range?

[carbonbased]

(Edit to subject line to be more precise)

First, imagine a graph with feet on the y-axis and magnification on the x-axis.

If one was to plot a scope’s field of view on this graph, would it be sufficient to take the low mag:feet@100 yds point and draw a straight line to the high mag:feet@100 yds point? In other words, is the relationship linear?

Or does a typical scope FOV vary non-linearly throughout its mag range? In this case, the line on its FOV chart would be curved.

My guess is that most scopes have a linear FOV relationship due to me spot checking a couple of my scopes at different mag ranges. Straight lines would be very easy for me to plot by hand.

Why I’m asking this question is that I suck at math and want to quickly know what any scope’s FOV is at any point in its mag range. Keep in mind I probably don’t own the scope, and even if I do I’d really not rather actually go out and physically measure its FOV or compute angles etc. For me, the accuracy just needs to be close-ish. Like to a half a foot?

Importantly, it would be very easy to compare FOV visually vs just numbers on a page.

 

[beetroot]

Yes the FOV scales linearly with magnification, unless the scope tunnels in which case after the tunneling it will scale correctly.

To work out the FOV for any magnification:
multiply the maximum mag FOV spec by the magnification then divide by the magnification you want.

For example, to find the FOV on the Razor Gen 3 at 27x to compare agaisnt the Gen 2 Razor.
3.5ft X 36x / 27x = 4.667ft @ 100 yards.

Two caveats ;
-Always use the maximum magnifcation to work this out on scopes that tunnel, if you aren't sure if a scope tunnels work it out using both max and min specs, if the max spec show a much higher number then there is a good chance it's due to tunnelling (try this with a 5-25 PM2 to see what I mean, as it is a scope well known to tunnel)
-Using the max magnification for the calculation has more error than the minimum due to more significant rounding errors.
If you uses the Razor Gen 3 info you'll see you get slightly different numbers, more than likely it's due to rounding errors in the higher mag FOV spec as even a small rounding error multipled by 36x can make a significant difference.

 

[Neurotic]

Assuming no shenanigans:

FOV2 = FOV1*(mag1/mag2)

 

[carbonbased]

Thanks to you guys, here's what I've come up with for myself to quickly compare FOV across scopes. I have different layers I can turn on/off and a bunch of scopes plotted out (by hand, this is all by hand). Anyway, for someone like me this is WAY more intuitive and quick to understand the FOV relationships across mag ranges and scopes.

The "18x magik line" is my little joke to myself. Varmint shooting I tend to wind up at 18x, so it's an impt magnification for me.

And oh yeah, this isn't taking into consideration of any tunneling. I'm not real concerned about low magnifications, and the plotting here is just "close enough". It works for my comparative purposes. Now that I know scope FOV scales linearly, I just find published min/max and draw a line. If I can't find such numbers (like I can't convert meters at 100m to feet at 100yds) then I will delve into formulae.

 

[carbonbased]

 

[beetroot]

I've done something to work out the compartive FOV of scopes but at 10x using excel.

Basically enter the FOV spec and magnification and it spits out the FOV at 10x (could be 18x if I wanted).

Using excel means you can list a bunch of scopes and order them from greatest FOV to lowest, if you are into that kinda thing (which it seems I am).

 

[Neurotic]

Thanks to you guys, here's what I've come up with for myself to quickly compare FOV across scopes. I have different layers I can turn on/off and a bunch of scopes plotted out (by hand, this is all by hand). Anyway, for someone like me this is WAY more intuitive and quick to understand the FOV relationships across mag ranges and scopes.

The "18x magik line" is my little joke to myself. Varmint shooting I tend to wind up at 18x, so it's an impt magnification for me.

And oh yeah, this isn't taking into consideration of any tunneling. I'm not real concerned about low magnifications, and the plotting here is just "close enough". It works for my comparative purposes. Now that I know scope FOV scales linearly, I just find published min/max and draw a line. If I can't find such numbers (like I can't convert meters at 100m to feet at 100yds) then I will delve into formulae.

View attachment 7785554

FOV is not linear with magnification, it's linear with the change in magnification; following the equation I posted. A doubling in magnification results in halving the FOV. The charts should look something like this:

 

[carbonbased]

FOV is not linear with magnification, it's linear with the change in magnification; following the equation I posted. A doubling in magnification results in halving the FOV. The charts should look something like this:

View attachment 7785659

I see. I did not attempt using your formula because I am a fool when it comes to maths and I attempt to avoid stuff like that.

Now, since I haven’t seen your posts before around here, please do not be offended if I call in some others to doublecheck what you are saying.

@koshkin @Denys could either of you help out here?

 

[beetroot]

I see. I did not attempt using your formula because I am a fool when it comes to maths and I attempt to avoid stuff like that.

Now, since I haven’t seen your posts before around here, please do not be offended if I call in some others to doublecheck what you are saying.

@koshkin @Denys could either of you help out here?

He's right, FOV scales with magnification but has an inverse relationship, so that as magnification goes up FOV goes down.

But FOV scales linearly (i think I'm using correctly language) In that you can just draw a graph of have a simple inverse formula and it be correctly.

Maybe is actually some sort of log scale or something, either way the formula he gave is the same as what I said and holds true.

 

[carbonbased]

He's right, FOV scales with magnification but has an inverse relationship, so that as magnification goes up FOV goes down.

But FOV scales linearly (i think I'm using correctly language) In that you can just draw a graph of have a simple inverse formula and it be correctly.

Maybe is actually some sort of log scale or something, either way the formula he gave is the same as what I said and holds true.

Ok, I’ll assume you dudes are right unless I hear otherwise.

What sort of graphing tool could a layman use to plot this and get a result like @Neurotic posted? I’m very visual and dislike just seeing how a batch of different scope’s FOV compares at just one or two magnifications. I want to see the whole thing and see how the entire mag range looks compared to other scopes.

Drawing curved lines by hand, especially curved lines in which I would have to work out many individual point via math, simply isn’t a good use of my time. Not much in optics is very intuitive! (to me, anyway).

I tried to find a vid that koshkin might have made on this issue but failed to find one.

 

[carbonbased]

FOV is not linear with magnification, it's linear with the change in magnification; following the equation I posted. A doubling in magnification results in halving the FOV. The charts should look something like this:

View attachment 7785659

Something occurred to me. Go easy, people, I’m a word and art guy, not a math guy. I have not done what a math fellow would, which is plug a bunch of number into your formula and see what happens (allergic), so bear with me.

The relationship scales linearly, just not in the way I thought. So instead of straight lines, they will all be curved, and curved exactly in the same arc.

I use a program called Adobe Illustrator. It would seem if I could make one long curve that has the right “ratio” (or whatever) then I could reuse that curved line again and again and just cut the ends off of it at the appropriate low & high magnifications.

Tell me your thoughts. This solution would allow me to avoid finding some new app and dealing with the inevitable complications that entails.

Thanks a lot, btw!

Edit: looking some more at your graph, I bet I’m totally wrong.

 

[beetroot]

I'd use Excel, not because it's the best just that I know how to use it.
If you set up a formula in excel you can work out the actual FOV numbers per magnification, you can then turn that into a graph in excel or it'll give you the info you need to make a graph in other programs.

I guess where your graph went wrong is by only having two data points, so adding a few more will make the graph appear how it should.

 

[beetroot]

Rather than take out word for it you can always test it for yourself, just set up foot markers at 100yards, or closer so long as you scale it correctly (25yards is 1/4 the distance so would be quarter the FOV foot value.

Or convert the FOV specs into a MIL/MOA value and measure it in your scope reticle if it has the appropriate hash marks.

 

[carbonbased]

Rather than take out word for it you can always test it for yourself, just set up foot markers at 100yards, or closer so long as you scale it correctly (25yards is 1/4 the distance so would be quarter the FOV foot value.

Or convert the FOV specs into a MIL/MOA value and measure it in your scope reticle if it has the appropriate hash marks.

Oh I totally believe you guys. I'm just trying to figure out a way to plot this thing quickly and in a flexible way using some tools I am an expert in (edit: I'm no excel master).

Other paths I'm exploring:
Some people use LaTeX with Illustrator to produce graphs, but I have never done so and don't exactly know what LaTeX is.

Also, I found this graphing calculator but have no idea how to use it.

 

[AIKNOWU]

Scope​	FOV in Feet​	@ Power​	Magik Power​	FOV in Feet​	FOV in Inches​	FOV in Mils​
NF 4-16​	6.9​	16​	18​	6.1​	73.6​	20.4​
TT 5-25​	4.77​	25​	18​	6.6​	79.5​	22.1​
NF 7-35​	3.4​	35​	18​	6.6​	79.3​	22.0​
March 5-42​	4.68​	28​	18​	7.3​	87.4​	24.3​
Razor 6-36​	3.5​	36​	18​	7.0​	84.0​	23.3​
LHT 4.5-22​	4.7​	22​	18​	5.7​	68.9​	19.1​
Minox 5-25​	4.8​	25​	18​	6.7​	80.0​	22.2​
Razor 4.5-27​	4.4​	27​	18​	6.6​	79.2​	22.0​
SB 5-45​	2.7​	45​	18​	6.8​	81.0​	22.5​
​	Column B​	Column C​	Column D​	Column E​	Column F​	​
Formulas:​	input​	input​	input​	=b12*(c12/d12)​	=e12*12​	=f12/3.6​

 

[JB.IC]

Algebra teachers across the nation looking wide eyed right now.

If it were linear, it would look something like this for the Gen3 6-36x:
y = mx + b
where m = -17/30
where b = 23.9
(6,20.5) & (36,3.5)

Mag is on the x axis and FOV is on the y axis.
16x FOV would = 14.8ft at 100yds
Or at 20x = 12.56ft at 100yds

When you look at that and compare it to say the FOV of a NF 4-16x and see that its FOV at 16x is 6.9ft at 100yds. Or a ZCO 4-20x and the FOV at 20x is 6ft at 100yds. This huge (basically doubling) disparity between the assumed linear relationship tells me that it is nonlinear.

I would need more ordered pairs to create a nonlinear equation.

 

[carbonbased]

Algebra teachers across the nation looking wide eyed right now.

I know, right?! Ugh! Talk about feeling dumb as a brick.

Last math class I had was calculus in high school about 34 yrs ago…got an A-. Never, ever, EVER used a lick of any math since except veeerrry basic algebra. I’m in fields that simply don’t require much math. I just didn’t want to use Neurotic’s formula unless I had to; it’s not hard at all to plot one point.

It’s the plotting of a smooth curve that’s stumping me (and then onto adding 30 or so scopes to one chart and being able to turn on/off individual scopes).

I’ve gotten a spreadsheet program to spit out the FOV @ X Mag numbers and I’m enlisting my teenage sons who seem to know how to do the graphing part of it via some free online graphing calculators.

I’m sort of surprised there isn’t some program that just takes min/max magnification and the FOV at each end and just bangs out the curved line. Should’ve known, I guess, that there’s not much demand for such a thing.

@Neurotic if you could pass along what app you used to make that chart with the nice smoooooth curves, do let me know. (And thx for the help!)

 

[AIKNOWU]

Bro. Excel.

 

[AIKNOWU]

Use the info from my first spreadsheet. Make it look like the spreadsheet in this post. Make chart from results. Nix a few magnifications or something if you want your plotted graph less cluttered looking.
The formula is called a function. It's called that because the *formula* is performing an action (or a "function") on the inputs. You put an input (some magnification) into the function, and it spits out a result. You're wanting to graph the result of that function. That should be plenty to give you a hand with your search engine.

​	Razor 6-36​	SB 5-45​	LHT 4.5-22​	March 5-42​	NF 7-35​	TT 5-25​	NF 4-6​	Razor 4.5-27​
4​	31.5​	30.4​	25.9​	32.8​	29.8​	29.8​	27.6​	30.4​
6​	21.0​	20.3​	17.2​	21.8​	19.8​	19.9​	18.4​	20.3​
8​	15.8​	15.2​	12.9​	16.4​	14.9​	14.9​	13.8​	15.2​
10​	12.6​	12.2​	10.3​	13.1​	11.9​	11.9​	11.0​	12.2​
12​	10.5​	10.1​	8.6​	10.9​	9.9​	9.9​	9.2​	10.1​
14​	9.0​	8.7​	7.4​	9.4​	8.5​	8.5​	7.9​	8.7​
16​	7.9​	7.6​	6.5​	8.2​	7.4​	7.5​	6.9​	7.6​
18​	7.0​	6.8​	5.7​	7.3​	6.6​	6.6​	6.1​	6.8​
20​	6.3​	6.1​	5.2​	6.6​	6.0​	6.0​	5.5​	6.1​
22​	5.7​	5.5​	4.7​	6.0​	5.4​	5.4​	5.0​	5.5​
24​	5.3​	5.1​	4.3​	5.5​	5.0​	5.0​	4.6​	5.1​
26​	4.8​	4.7​	4.0​	5.0​	4.6​	4.6​	4.2​	4.7​
28​	4.5​	4.3​	3.7​	4.7​	4.3​	4.3​	3.9​	4.3​
30​	4.2​	4.1​	3.4​	4.4​	4.0​	4.0​	3.7​	4.1​
32​	3.9​	3.8​	3.2​	4.1​	3.7​	3.7​	3.5​	3.8​
34​	3.7​	3.6​	3.0​	3.9​	3.5​	3.5​	3.2​	3.6​
36​	3.5​	3.4​	2.9​	3.6​	3.3​	3.3​	3.1​	3.4​

 

[carbonbased]

By Jove, I do believe I've got it!

But easy on the ol' man. Remember…I don't even use Excel (or in this case Numbers for the mac), like, ever.

Thx for the help.

 

[AIKNOWU]

Bingo!
Now that you've got that figured out, buy the most expensive one.

 

[JB.IC]

I know, right?! Ugh! Talk about feeling dumb as a brick.

Last math class I had was calculus in high school about 34 yrs ago…got an A-. Never, ever, EVER used a lick of any math since except veeerrry basic algebra. I’m in fields that simply don’t require much math. I just didn’t want to use Neurotic’s formula unless I had to; it’s not hard at all to plot one point.

It’s the plotting of a smooth curve that’s stumping me (and then onto adding 30 or so scopes to one chart and being able to turn on/off individual scopes).

I’ve gotten a spreadsheet program to spit out the FOV @ X Mag numbers and I’m enlisting my teenage sons who seem to know how to do the graphing part of it via some free online graphing calculators.

I’m sort of surprised there isn’t some program that just takes min/max magnification and the FOV at each end and just bangs out the curved line. Should’ve known, I guess, that there’s not much demand for such a thing.

@Neurotic if you could pass along what app you used to make that chart with the nice smoooooth curves, do let me know. (And thx for the help!)

I wouldn’t say I’m a math nerd but I use math quite often for analytics and analysis. Mainly statistics. But even then it’s not that much to have any sort of claim to being an expert.

I’d have to research it for myself but I think for a nonlinear relationship, we would need more than two ordered pairs to actually graph the curve. Or at least know the coefficients and the value in which the variables are raised to and then just substitute the x value in to get the y value.

Someone else might know but I don’t know how to model nonlinear equations with just two ordered pairs.

 

[koshkin]

I see. I did not attempt using your formula because I am a fool when it comes to maths and I attempt to avoid stuff like that.

Now, since I haven’t seen your posts before around here, please do not be offended if I call in some others to doublecheck what you are saying.

@koshkin @Denys could either of you help out here?

He is correct. Aside from tunnelling, his calculation is reasonably accurate within the small angle approximation, i.e. damn close for precision scope magnifications.

 

[carbonbased]

He is correct. Aside from tunnelling, his calculation is reasonably accurate within the small angle approximation, i.e. damn close for precision scope magnifications.

Btw just joined your DLO site. Figured I should help support you since I've watched a number of your YT vids and read your postings for a number of years now.

 

[koshkin]

Btw just joined your DLO site. Figured I should help support you since I've watched a number of your YT vids and read your postings for a number of years now.

Thank you.
I appreciate that.
I'll be doing SHOT updates there in the evening.
ILya

 

[carbonbased]

He is correct. Aside from tunnelling, his calculation is reasonably accurate within the small angle approximation, i.e. damn close for precision scope magnifications.

Hey, could you (or anyone) take a look at the NF NX8 4-32 (and 2.5-20) which don't seem to work great with the math in this thread. For example, I tried calculating the 4-32 and the high-mag FOV is off by way more than other scopes (should be 4.6ft/100yds, but the math calcs it to 3.2ft/100yds).

So, because I am an idiot, I plotted the same scope twice on the same chart (see attached). For the blue line, I started the calc with the low mag 4x. With the red line, I did the opposite.

The "correct" line should start where the blue line begins and finish where the red line ends.

The official NF numbers for the FFP NX8 4-32 are:
4x: 26.1 ft
32x: 4.6 ft

NX8 - 4-32x50mm F1 - Nightforce Optics

A vast magnification range, with superb clarity and resolution at every step, means that one riflescope will give you a world of versatility and confidence in any situation.

www.nightforceoptics.com

 

[Neurotic]

The math doesn't quite work out for these 8 times erectors.

The 2.5-20s I've looked at are all off at one end or the other. Where using maximum mag for the input gives a lower value than the published minimum mag FOV. I assume these must be fisheye-ing or something at the low end which increases minimum magnification FOV.

This looks like opposite, tunnelling. Where using maximum mag for the input gives a higher value than the published minimum mag FOV.

 

[Im2bent]

Scope​	FOV in Feet​	@ Power​	Magik Power​	FOV in Feet​	FOV in Inches​	FOV in Mils​
NF 4-16​	6.9​	16​	18​	6.1​	73.6​	20.4​
TT 5-25​	4.77​	25​	18​	6.6​	79.5​	22.1​
NF 7-35​	3.4​	35​	18​	6.6​	79.3​	22.0​
March 5-42​	4.68​	28​	18​	7.3​	87.4​	24.3​
Razor 6-36​	3.5​	36​	18​	7.0​	84.0​	23.3​
LHT 4.5-22​	4.7​	22​	18​	5.7​	68.9​	19.1​
Minox 5-25​	4.8​	25​	18​	6.7​	80.0​	22.2​
Razor 4.5-27​	4.4​	27​	18​	6.6​	79.2​	22.0​
SB 5-45​	2.7​	45​	18​	6.8​	81.0​	22.5​
​	Column B​	Column C​	Column D​	Column E​	Column F​	​
Formulas:​	input​	input​	input​	=b12*(c12/d12)​	=e12*12​	=f12/3.6​

Is there a poors version?

 

[carbonbased]

Assuming no shenanigans:

FOV2 = FOV1*(mag1/mag2)

The math doesn't quite work out for these 8 times erectors.

shenanigans found!!! lol

 

[Neurotic]

shenanigans found!!! lol

Totally...

 

[koshkin]

Hey, could you (or anyone) take a look at the NF NX8 4-32 (and 2.5-20) which don't seem to work great with the math in this thread. For example, I tried calculating the 4-32 and the high-mag FOV is off by way more than other scopes (should be 4.6ft/100yds, but the math calcs it to 3.2ft/100yds).

So, because I am an idiot, I plotted the same scope twice on the same chart (see attached). For the blue line, I started the calc with the low mag 4x. With the red line, I did the opposite.

The "correct" line should start where the blue line begins and finish where the red line ends.

The official NF numbers for the FFP NX8 4-32 are:
4x: 26.1 ft
32x: 4.6 ft

NX8 - 4-32x50mm F1 - Nightforce Optics

A vast magnification range, with superb clarity and resolution at every step, means that one riflescope will give you a world of versatility and confidence in any situation.

www.nightforceoptics.com

NX8 tunnels on low power, so it reaches its widest magnification right under 5x. I just had a friend of mine who has one glance at it.

The ratio of FOVs as listed is 5.67. The FOV on max power is usually listed accurately (not always and manufacturers get upset when I call them out on it, but that's a story for another day). If 32x FOV is really 4.6ft at 100 yards, then it should achieve the widest magnification around 5.67x. That is a bit more than he is observing, so we can approach this from another angle.

Now, one thing that is nice about mil reticles is that you have a ruler in your scope to measure FOV. My friend's NX8 has FOV of just over 12.9mrad on 32x, which translates to ~3.89 ft at 100yards. That's a bit lower than Nightforce claims, so it would be good for a couple more people with NX8s to check that.

If we assume the widest FOV is indeed 26.1ft, then that widest FOV should be on 4.77x which is very close to what he is observing.

If any of you have this scope on hand, set it on 32x and see what the FOV in mrad is.

ILya

 

[TheSunPope]

NX8 tunnels on low power, so it reaches its widest magnification right under 5x. I just had a friend of mine who has one glance at it.

The ratio of FOVs as listed is 5.67. The FOV on max power is usually listed accurately (not always and manufacturers get upset when I call them out on it, but that's a story for another day). If 32x FOV is really 4.6ft at 100 yards, then it should achieve the widest magnification around 5.67x. That is a bit more than he is observing, so we can approach this from another angle.

Now, one thing that is nice about mil reticles is that you have a ruler in your scope to measure FOV. My friend's NX8 has FOV of just over 12.9mrad on 32x, which translates to ~3.89 ft at 100yards. That's a bit lower than Nightforce claims, so it would be good for a couple more people with NX8s to check that.

If we assume the widest FOV is indeed 26.1ft, then that widest FOV should be on 4.77x which is very close to what he is observing.

If any of you have this scope on hand, set it on 32x and see what the FOV in mrad is.

ILya

Just checked mine, looks like a hair over 6.4 MRAD in each direction left to right. So 12.9 sounds correct.

 

[pineoak]

Now, one thing that is nice about mil reticles is that you have a ruler in your scope to measure FOV.

That's so obvious! I never considered that.

Like wondering where something is made... and your friend flips it over and it says "Made in China" under it.

 

[koshkin]

Just checked mine, looks like a hair over 6.4 MRAD in each direction left to right. So 12.9 sounds correct.

Thank You. It sounds like Nightforce has a typo in the specs. What is the lowest marked magnification that you can measure the FOV on?

Also, would you happen to have a 2.5-20x50 NX8 as well? If memory serves me right, that one is off as well, but it would be nice to get it measured.

ILya

 

[TheSunPope]

Thank You. It sounds like Nightforce has a typo in the specs. What is the lowest marked magnification that you can measure the FOV on?

Also, would you happen to have a 2.5-20x50 NX8 as well? If memory serves me right, that one is off as well, but it would be nice to get it measured.

ILya

The lowest marked magnification I can still measure is at 6X. I can see just under 34 Milradian down on the vertical line crosshair with the MilXT reticle. Mine starts to tunnel about 1/3rd of the way between marked 4x and 6x. I wish I could help on the 2.5-20, but I only have the one NX8.

 

[koshkin]

The lowest marked magnification I can still measure is at 6X. I can see just under 34 Milradian down on the vertical line crosshair with the MilXT reticle. Mine starts to tunnel about 1/3rd of the way between marked 4x and 6x. I wish I could help on the 2.5-20, but I only have the one NX8.

Thank You.
That matches the ~3.89 ft @ 100yds on 32x almost perfectly.
We can conclude with reasonable certainty, then that the 4-32x50 NX8 tunnels below ~4.8x. Above that magnification, the Apparent FOV is ~23.3 degrees at both 6x and 32x. That's a pretty wide angle eyepiece.

ILya