Angle of view calculations

[rocket_scientist]

Maybe I don't fully understand all of the specifications that are described when looking at a lens but I would like to better understand the angle of view specification a little more clearly.  When you look at say a 300mm prime lens on the Canon website, the diagonal angle of view is said to be 8° 15'.  I am fairly certain this is for a full frame sensor as the angle of view (I think) is dependent upon sensor size.  If this is so, how would I use this information to determine the distance needed to frame a 6ft tall person within a crop sensor (portrait or landscape)?  That is, if I wanted to shoot a football game and would like to shoot from the 50yd line, how could I calculate the focus distance that a 300mm lens would "reach" for a full body shot without much cropping?  I understand it should be fairly simple geometry but I am not 100% sure as to how to approach it.  Anyone with some knowledge? 

As the above paragraph is probably convoluted, I am trying to figure out what focal length lens I would need for specific sports of I know my distance form the athletes. So if I am at the 50yd line, would a player at the 20yd line take up the full frame?

[K-amps]

Is Neuro Online today? 

[rocket_scientist]

I was thinking he would probably give the best reply to this question.

[Random Orbits]

If you go to the-digital-picture.com and select a lens review and then select the link for "Lens Specification", it will break down the diagonal, horizontal and vertical AOVs.

The AOV is dependent on sensor size, so a APS-C camera will have a smaller AOV.  But the easier way will be to do all your math using FF sizes to determine which FL you need, and then dividing that number by 1.6 at the end.  So for your example, your distance would be 30 yd = 90 ft.  The VAOV is approx = 2*atan(3/90) = 3.8 deg (where 3 is half of the height).  A 300mm lens is close at approximately 4 deg of VAOV (8 deg diagonal AOV), so a 300mm lens would work on FF.  For your crop, a 300/1.6 = 188, so a 200mm lens should be sufficient.  If you intend to shoot portrait more than landscape, then you would need a longer FL because the HAOV is larger than the VOAV due to the sensor size.

Now, if the action were at the 50 yd line, then the 70mm end of a 70-200 would be more helpful, so a 70-200 or a 70-300mm lens would both work well for sports.  If you choose a 70-200, then you can always add a TC later if you wanted more reach.

[DoesNotFollow]

Cambridge in Colour has a neat focal length calculator to help determine the lens needed for a subject based on the size of the subject and distance. It also takes into account varying sensor sizes.

http://www.cambridgeincolour.com/tutorials/camera-lenses.htm#calculator

[shadowsatnight]

to get a guy who's 6' tall in to the frame at a distance of 30 yards (90'):
tan(theta) = opposite/adjacent
so, your lens will need to have a viewing angle of at least tan^-1(opposite/adjacent) for the guy to be in shot.
That's 2* tan^-1(3/90)  <- you're calculating the angle to see half of his height and then doubling the result
or ~3.8 degrees.

the 300mm prime lens has angles of view:
horizontal: 6deg50
vertical:4deg35
diagonal:8deg15
on a full frame sensor.

On a full frame, that means that you could either shoot portrait, with a viewing angle of 6deg50, and be fine or shoot landscape with a viewing angle of 4deg35 and be fine.

The sensor in the 7d has a crop factor of approximately 1.6, so:
portrait would give a viewing angle of 6deg50/1.6, or ~4.27 degrees, which would be fine.
landscape would give a viewing angle of 4deg35/1.6, or ~2.86 degrees, which would crop him at about the knees :-)

Or, in the more general case:

angle of view = 2 * tan^-1(sensor dimension/ 2 * focal length)
7D sensor height in landscape = 14.9mm
7D sensor height in portrait = 22.3mm

to get someone in shot,
angle of view >= angle subtended.

where
angle subtended = 2* tan^-1 (0.5 * their height/how far away they are)

so we solve:
2 * tan^-1(sensor dimension/ 2 * focal length) = 2 * tan^-1(3/distance to subject in feet)

simplifying both sides
sensor dimension/(2 * focal length) = 3/distance to subject in feet
sensor dimension / (3/distance to subject in feet) = 2 * focal length
(sensor dimension * distance got subject in feet) / 3 = 2 * focal length
(sensor dimension * distance to subject in feet) / 6 = focal length

plugging some numbers in:
at 30 yards (90 feet) in portrait orientation (22.3mm):
focal length = (22.3 * 90)/6
focal length = 334mm
so anything shorter than 334mm will be fine.

at 30 yards in landscape orientation:
focal length = (14.9*90)/6
focal length = 224mm
so anything shorter than 224mm will be fine.

Which, happily, agrees with what i said to start with.  though, no doubt, i'll have screwed up the math somewhere along the lines!

TLDR version:  Using zooms is easier :-)

-Evie

[neuroanatomist]

I use Bob Atkins' calculator.

http://www.bobatkins.com/photography/technical/field_of_view.html

[Drizzt321]

I use Bob Atkins' calculator.

http://www.bobatkins.com/photography/technical/field_of_view.html

Wow...that was a little disappointing from you Neuro. Although I don't blame you for pawning off this one on someone else who built in a calculator.

[serendipidy]

Or you could just have someone 6 foot tall stand still while you pace it off using your zoom lens until he fills the viewfinder. 

But the calculator is much faster and easier.

Using Bob Akins' calculator from Neuro's post, for a 6 foot person to fill the frame on a 1.6 crop camera with a 300mm focal length lens, you should be 80' in portrait and 120' in landscape orientation from the subject. Is that correct?

[FarQinell]

rocket_scientist:

The answer is quite simple and does not require advanced trigonometry!
To show you how to work it out I would need to know the distance to the target is it 50-20yds= 30yds?
Sorry - I am clueless on American football terminology!

[serendipidy]

So I guess the equation is:

Focal length(mm)/sensor height(mm)=distance to subject/height of subject

7D sensor height in landscape = 14.9mm
7D sensor height in portrait = 22.3mm

Example: d=300mm/14.9mm X 6'= 120.8' ; distance to 6' height subject to fill 7D frame using a 300mm focal length lens in landscape orientation