[LensBeginner]

Hi,
couldn't find anything with google (although there probably is something), so made a stab at it...
Suppose I crop a 16MP APS-C image taken with a 40mm into a 9.6MP image.
The FoV is equivalent to the one I would get with a 100mm on FF, true?

40mm * 1.5 (sensor crop factor) * ( 16 / 9.6 ) = 100mm

[BrianR]

It should be:

40mm * 1.5 (sensor crop factor) * SquareRoot ( 16 / 9.6 ) = 77.5mm

The 16/9.6 is ratio of the area of the original image to the area of the cropped image. You want to multiply the ratio of the side lengths, so you need to take the square root.

You could also just go to the pixel lengths of the sides directly (pick either length or width but be consistent):

16mp image=4928x3264
9.6mp image=3816x2527

40mm * 1.5 (sensor crop factor) * 4928/3816 = 77.5mm

[stevebrot]

Sorry, but that is sort of a silly question. FOV is measured in degrees of arc.

But, in the interest of being helpful, all you need to do is to calculate the "crop factor" for your final crop. In simplest terms that is the length of the diagonal of a 35mm FF frame to that of your cropped frame. The total pixel count is incidental. What you need to know are the pixel dimensions and frame dimensions for the sensor. This varies by sensor, even within Pentax.*

crop factor = FF diagonal length/final crop diagonal length

"Length" is some unit of linear measurement. This is not the same as the number of pixels. The original "crop factor" is not part of the calculation unless you want to complicate things. You end up running the same numbers twice.

To speed things along:

• The diagonal length of the 35mm frame is 29.4mm 43.3mm.
• The frame dimensions for the camera are in the camera's specifications. This varies between cameras. For the K-50, the diagonal is 19.3mm 28.4mm.
• The pixel dimensions of the original frame are whatever came out of the camera.
• The pixel dimensions of the final cropped frame are whatever they are. You use this to calculate the height and width (on the sensor) of your final crop. The number of megapickles is incidental.

Remember that it was assumed that the ratio of the final crop is still 3:2. If your aspect ratio changes, the FOV comparison is less valid.

As mentioned above, the number of pixels in the crop is incidental and a red herring. To do the actual calculation, Pythagoras is your friend. Remember that the linear measurement is the basis for your calculation, not the number of pixels.**


Steve

* The oft quoted 1.5x crop factor for Pentax is an estimate. The actual crop factor is 1.53 or 1.52 depending on model.

** FWIW, what is described above is essentially the same type of calculation used for figuring the actual lp/mm resolution for lens testing using a standardized target.

[Oldbayrunner]

A Mathematical Look at Focal Length and Crop Factor

[clicksworth]

It would be simpler to reckon it in angle of view.


Using one of the many available charts or calculators for Angle of view, find the horizontal angle of a 40mm lens on a 1.5 crop sensor. That would be 33.4 degrees.


Determine how much the image has been cropped by the pixel dimensions. If it was 5000 pixels to start with and now it is 2500 across, the crop was 0.5, so the angle of view of the image is 33.4 X 0.5 = 16.7 degrees.


Back to the chart, the closest standard lens with that angle on a full frame is 120mm giving 17.06 degrees of horizontal angle.

[BrianR]

crop factor = FF diagonal length/final crop diagonal length

Assuming the same aspect ratio (and maybe a big assumption on my part), you can just take the ratio of the widths. Or the ratio of the lengths.

The original "crop factor" is not part of the calculation unless you want to complicate things. You end up running the same numbers twice.

A crop of a crop is a crop... you can just multiply the crop factors of successive crops, this is simpler imo, but ymmv.

As mentioned above, the number of pixels in the crop is incidental and a red herring. To do the actual calculation, Pythagoras is your friend. Remember that the linear measurement is the basis for your calculation, not the number of pixels.**

# of pixels is fine to use when figuring out the added 'crop factor' of a bona fide crop of your image, like the OP was asking for. It's on the same sensor, so each pixel represents the same unit of length.

edit- check your Pythagoras again on your diagonal lengths

---------- Post added 10-26-14 at 06:34 PM ----------

Determine how much the image has been cropped by the pixel dimensions. If it was 5000 pixels to start with and now it is 2500 across, the crop was 0.5, so the angle of view of the image is 33.4 X 0.5 = 16.7 degrees.

Crops and angles of view don't work like that, halving the diagonal does not halve the angle of view. There's an arctan involved, see the formula in the post above yours. When the angle of view is small, the math you've done starts to be 'close to right', but it will fall apart for the wider angle stuff.

[stevebrot]

It's on the same sensor, so each pixel represents the same unit of length.

Absolutely, though the number you want is not the width. The pixel pitch on the diagonal is not the same as that for the height or width and it is the length of the diagonal on which the crop factor is based. FWIW, there is no firm guarantee that the pitch for height is the same as for width.

You use the pixel pitch (height and width) to determine the linear dimensions of the final crop and use that to calculate the length of the diagonal. The length of the diagonal is then used to calculate the crop factor.

edit- check your Pythagoras again on your diagonal lengths

My Pythagoras was up late and had a headache...I will edit...


Steve

---------- Post added 10-26-14 at 05:28 PM ----------

A Mathematical Look at Focal Length and Crop Factor

Good reference. Thanks for sharing. Things would have been so much easier if the OP had asked for FOV per sensor size instead of crop factor based on megapixel. I guess we can expect some new facet of "equivalence" based on this discussion


Steve

---

Last edited by stevebrot; 10-26-2014 at 05:31 PM.

[BrianR]

Absolutely, though the number you want is not the width. The pixel pitch on the diagonal is not the same as that for the height or width and it is the length of the diagonal on which the crop factor is based. FWIW, there is no firm guarantee that the pitch for height is the same as for width.

You use the pixel pitch (height and width) to determine the linear dimensions of the final crop and use that to calculate the length of the diagonal. The length of the diagonal is then used to calculate the crop factor.

Assuming they are the same aspect ratio (as I said) then the ratio of the diagonals will be the same as the ratio of the widths will be the same as the ratio of the heights. Use whichever dimension you prefer to find the crop factor (the easiest dimension to find is a good choice).

If the aspect ratio has changed, then sure find the diagonal. It won't tell the whole story as you've said before, and you'd perhaps want to also look at the horizontal and vertical FoV's in this case.

[clicksworth]

Crops and angles of view don't work like that, halving the diagonal does not halve the angle of view. There's an arctan involved.

That is true if the angles and crop factors are large enough it would make a practical difference. In the example, the difference between a rigorous calculation and my easy method is part of a degree, or 2mm in the end result. A 122mm lens on the FF rather than 120mm.

[BrianR]

That is true if the angles and crop factors are large enough it would make a practical difference. In the example, the difference between a rigorous calculation and my easy method is part of a degree, or 2mm in the end result. A 122mm lens on the FF rather than 120mm.

For sure, it's pretty accurate in the normal to telephoto ranges. The rigorous calculation is easy enough though, in your example it's just 40mm x 1.5 x 2 = 120mm (where 1.5 is the crop from aps-c to full frame and 2 is the crop from your final image to aps-c (again assuming aspect ratio remains the same)). No tables required.

[LensBeginner]

Wow... what a thorough answer...
I'll look into it some more soon (this evening), I had the doubt that I was missed something...
Thanks all, posts liked

Edit:
1. Yep same 3:2 (or 2:3) ratio.
2. Yep it would have been easier, but it's nicer to calculate by MP since it's the most readily available info when I crop pictures.
3. I know FoV is an angular measurement, but since the distance doesn't vary in a single shot...

[clicksworth]

No tables required.

Using angle tables might encourage people to think in terms of AOV, sparing them from their endless confusion with the "equivalent focal length" oxymoron.

[LensBeginner]

Using angle tables might encourage people to think in terms of AOV, sparing them from their endless confusion with the "equivalent focal length" oxymoron.

It's not much of an oxymoron, rather a convenience when you were born with film SLRs and are accustomed to think in those terms...
I know what's a 28mm, a 50mm, a 100mm and a 200mm and those numbers immediately conjure an image in my mind, while angles don't.
I know, it's backwards, but I'd say the majority of people has it this way...