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Keystone angle formula and graph


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A few days ago I started a thread about what I thought was distortion but turned out to be keystoning. I caused some confusion for certain members when I asked how much keystoning gets worse for wider lenses. Alan G. gave a great graphic illustration of how keystoning gets more severe at the edge of the frame for wider lenses.

 

I decided to calculate the equation for keystoning as a function of camera tilt using basic optics, which of cause, assumes perfect optical design with no distortions or corrections.

 

Anyway I found the equation to be

 

Tangent(keystone angle)=Tangent(image angle)xTangent(tilt angle)

 

where the keystone angle is measured from a vertical line, the image angle is the angle of view the object makes with the lens axis, and the tilt angle is measured from horizontal.

 

The attached two graphes show the keystone angle as a function of tilt for various angles of view. I have chosen the angles of view corresponding to the angles of the 24, 28, 35, 50, and 90mm Leica lenses. Therefore the curves represent the keystoning at the edge of the frame for these lenses.

 

The first graph is for full frame, and the second graph takes into account of the M8 sensor crop factor of 1.33 which reduces the lens angle. Hopefully I did not make any mistakes in Excel making the graphs.

 

Nothing new here, but thought you techies may be interested.

 

Alan

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I am intrigued by your graphs but not sure if they are correct. How did you come up with this calculation?

 

The reason I question it goes back to my gut instinct that the only time there'd be a 1:1 ratio of camera tilt to keystoning would be with a 90 degree horizontal angle of view. Your graph shows it occurs with a 47 degree angle of view (diagonally?) Now I haven't done any calculations and I'm not sure I can. (I may try to read through one of my old texts on optics and see if I can still understand it.)

 

But I just looked through my full frame slr camera and when I use a 20mm lens (about 90 degrees) and tilt it 45 degrees, I seem to get the keystoning to go about 45 degrees too. Plus once I tilt up past 45 degrees I think the keystoning diminishes or changes in a manner we can no longer call "keystoning." There doesn't seem to be additional keystoning past a 45 degree tilt. I'll have to think this over.

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Alan S--

Pretty work. But unless I misunderstand completely what you're doing, there's something wrong in your figures for field of view for M8:

 

with 24mm, M8 diagonal fov is 68°

with 28mm, M8 diagonal fov is 60°

with 35mm, M8 diagonal fov is 50°

with 50mm, M8 diagonal fov is 36°

with 90mm, M8 diagonal fov is 20°

 

(M8 sensor is 18x27 mm; diagonal = 32.45 mm)

 

But interestingly, the amount of error diminishes as focal length increases:

You show 82° for a 24, but that angle fits an 18.7mm lens.

You show 71.6° for a 28, but that angle fits a 22.5mm lens.

You show 57° for a 35, but that angle fits a 30mm lens.

You show 38.8° for a 50, but that angle fits a 46mm lens.

You show 20.9° for a 90, but that angle fits an 88mm lens.

 

I think we both accepted AlanG's diagram the other day. But if the curves for a 90mm lens curve the other way from those of the shorter lenses in your chart, doesn't that falsify his diagram? (Speculation on my part.)

 

Two more things to consider:

You say correctly that the 24mm has an 84° field of view on the 35 mm format. That's the diagonal field, so your figures would not be accurate for the edges of the frame, but for the corners. (That part I'm sure of. The following is just a general feeling::)) Wouldn't you have to use the half-angle for your calculation, since lines on opposite sides of center converge?

 

--HC

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Alan S--

 

with 24mm, M8 diagonal fov is 68°

with 28mm, M8 diagonal fov is 60°

with 35mm, M8 diagonal fov is 50°

with 50mm, M8 diagonal fov is 36°

with 90mm, M8 diagonal fov is 20°

 

(M8 sensor is 18x27 mm; diagonal = 32.45 mm)

 

 

You show 82° for a 24, but that angle fits an 18.7mm lens.

You show 71.6° for a 28, but that angle fits a 22.5mm lens.

You show 57° for a 35, but that angle fits a 30mm lens.

You show 38.8° for a 50, but that angle fits a 46mm lens.

You show 20.9° for a 90, but that angle fits an 88mm lens.

 

 

--HC

 

Thanks, you must be right. I stand corrected. Did not get them from a reliable source.

 

Alan

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Alan--

"Reliable source"? :rolleyes:

 

I thought this was all out of your noggin and was impressed! :)

 

I meant I did not get the angles of the M8 lenses from an official source, just some approximations. Did the calculations myself, but obviously dropped a factor of 2 as I said earlier.

 

Alan

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I found a calculator for this. Here's what you need for the M8:

 

I think you want to base your calculations on the horizontal field of view and not the diagonal field of view. (We don't shoot too many pictures on a diagonal and then look for keystoning.) You could also do it for the vertical field of view if you like.

 

Width = 18 mm, Length = 27 mm, Diagonal = 32.45 mm

 

f Hor Vert Diag H/V

15.0 83.9744 61.9275 94.4932 1.3560

21.0 65.4705 46.3972 75.3806 1.4111

24.0 58.7155 41.1121 68.1206 1.4282

28.0 51.4814 35.6378 60.1815 1.4446

35.0 42.1847 28.8415 49.7421 1.4626

50.0 30.2192 20.4079 35.9565 1.4808

75.0 20.4079 13.6855 24.4137 1.4912

90.0 17.0615 11.4212 20.4388 1.4938

 

Here it is for full frame:

 

Width = 24 mm, Length = 36 mm, Diagonal = 43.2666 mm

 

f Hor Vert Diag H/V

21.0 81.2026 59.4898 91.7021 1.3650

24.0 73.7398 53.1301 84.0622 1.3879

28.0 65.4705 46.3972 75.3806 1.4111

35.0 54.4322 37.8493 63.4400 1.4381

50.0 39.5978 26.9915 46.7930 1.4670

75.0 26.9915 18.1806 32.1798 1.4846

90.0 22.6199 15.1893 27.0316 1.4892

 

I'm not sure what any of us will do with the graphs and formula once we get them but I'm kind of curious at this point.

 

This forum removed the formatting so the titles are squashed together. The first number is focal lenght, the next is the horizontal angle, then the vertical angle, then the diagonal angle and final the horizontal to vertical ratio.

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Well, I did miss a term in my math. Blame it on late night and wine. Anyway the equation and graphs in my first post were wrong. Here are the correct equation (hope I didn't miss any other term) and graphs. On second thought I'll just post the graphs since the math eqaution is quite complicated and hard to type here. I'll pm anyone really interested.

 

I used the horizontal angle Alan G posted. Thanks Alan and HoCo for pointing out my mistakes. Don't think anyone else is interested. Guess I'm the only nerd here, and maybe you two as well :)

 

Yes, this is absolutely useless in actual photography. Just curious. I promise this will be my absolute last post on this subject:D

 

Alan

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Looks great, Alan! That set of curves looks very reasonable and makes good sense.

 

And yes, I guess it's just you and AlanG and I left. We seem to have scared the others off. :)

 

Diagramming always helps me understand things, and I'm glad you did the work.

 

One annoying thing I found out is that we've all (myself included) been using the wrong term: This is not keystoning, but simple perspective.

 

I was unhappy with the Merriam Webster definition of 'keystone' (didn't even recognize it as a verb); went to Wikipedia, whose definition is woefully short (doesn't include the 6x6 Hasselblad slide projector with anti-keystoning optics, seems to think anti-keystoning can only be done digitally)--but with the Wikipedia article I began to realize I had been using the term incorrectly.

 

Finally went to the OED. Upshot: "Keystoning" is what happens when you tilt a projector upward (for example), i.e. the part of the image furthest away is the largest--just the opposite of what we've been looking at. :(

 

FWIW, the concept of a keystone that interests us shows these dates, all very recent:

 

'keystoned' adjective first used in English 1887, meaning "having a keystone"

 

'keystone' noun used attributively first appears in English 1914 in cinematography (as in "keystone effect")

 

'keystone' transitive verb first used in English 1940 in TV, meaning "to subject to a keystone effect" (as in "the shape will be keystoned")

 

--hence the verbal noun "keystoning"

 

The OED of course cites the sources of these uses. The 1914 cinema usage is the first time the term was used as we use it. The quotation says something to the effect that 'if you angle the projector upward toward the screen, the top of the image will be enlarged, and if you angle the projector sideways toward the screen, the keystone effect will be horizontal rather than vertical.'

 

So it occurs to me: When we angle the camera upward, we get a kind of anti-keystoning because those higher points are further away than the lower points of the image. Now all we need to do is angle the projector upward by the same amount, thus introducing actual keystoning that will counteract the contraction of the top of the image as we shot it.

 

As you said, Alan: Maybe not useful, but certainly interesting!

 

Thanks again for all the effort!

 

--HC

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