Does FL effect DOF for the same Magnification? Going nuts here!

[Arjen van Andel]

The second link is less usefull, as the auther has made some math
errors. according to him, everything between the focus point and
the camera should be in focus, and we should always focus on
infinity!

you can't have to much information, as long as it is all correct!

If you think about it you'll find that Merklinger is (also) correct. His technique is useful if you need optimal sharpness at great distance. Since objects closeby can deal better with being out of focus than similar objects at the horizon Merklingers technique can improve landscape shots. I use it all the time, even for things much closer, like buildings and find that the results are good, better than HPD with less work. The hyperfocal technique is just as good but is harder to use and is often less reliable.

There's really no need to do calculations with his trick. Focus at something you want critically sharp (tree at visible horizon, areal, doorpost of the house in the distance, etc.) and make sure the aperture is between the best value for the lens and the largest f-number where diffraction becomes visible. No need to think, calculate or to lookup in a DOF table.

Arjen.

 

[Dredman]

Again, the flowers are of different magnification. Do some
controlled test shots and look at the OOF area with same
magnification. I'm sure you should see they are about the same.

http://www.luminous-landscape.com/tutorials/dof2.shtml

Hi Ray

well, Bob Atkins (who is well respected for his knowlege of physical optics) prefers the deffinition: "The depth of field is the range of distances reproduced in a print over which the image is not unacceptably less sharp than the sharpest part of the image". DOF is defined "in a print" . My OP stated that the main subject would have the same magnification, and if you make two prints with the main subject at the same magnification, then there should be no need to discuss different magnification of the background, if we are limiting ourselves to the subject of DOF. the question is, is the point in the background reproduced smaller then the COC for that print, if yes, it is in the DOF, if no, it is not. Regardless of the size of the object reproduced.

In the mean time, I found my answer. See Doug Kerr's reply below:
http://forums.dpreview.com/forums/read.asp?forum=1029&message=17512417

or the link that arjen pointed me to:
http://www.vanwalree.com/optics/dof.html

Where they state that principle with the qualifier "when the focus distance is not close to the HFD"

Yes, for all practical purposes, at relativly close focus distances, the DOF is the same if the magnification and the f/stop is the same. but not when using focus distances close to or at the HFD.

Yes, I did go shoot some test images for myself, yes, I confirm the rule at short focus distances, and disprove it at focus distances close to the HFD. I just wish such sources of knowlege as the Luminous landscape (link above, yes I had read that already) and the popular photo mag's included this qualifier when they stated their "rule of DOF" so they would be accurate

thanks for the info!

D

 

[Dredman]

ahhh... right, I wasn't reading his page quite right... I thought that he was saying that everything would be equaly in focus, but he was saying that the spot size would be all equal, and the objects more in the forground would have larger reproduced spot size on the print (more OOF) but he didn't care, as he wanted the town to be the sharpest part of the image.

I use HFD mostly as an idea of about what range I can get "sharp" when shooting deep landscapes, where I really want both the flowers in the field and the mountains at infinity to appear sharp. I konw that if I shoot at 18mm, at f/8, most everything will be in focus if I set the focus point at 5-6 meters.

anyway, thanks for the info! Great!

D

 

[Dredman]

The upshot of all this is that, for given values of N, C and M,
longer lenses have more near DOF, shorter lenses have more far DOF,
and shorter lenses have more total DOF. However, the last effect
is weaker than the first two, since F appears in a quadratic term
in the equation for total DOF.

Although this is based on simple lens theory, it is valid for all
sensor sizes, magnfications and compositions (macro & landscapes).

thanks! quite true! I guess my primary need for DOF is behind the subject, to include people/trees/mountains in the background. I guess this is partly why it seems that I get much more DOF with shorter lenses.

Thanks!

D

 

[Dredman]

Greetings all,

Thanks for the informative responces! I found through your posts and links (as well as some personal photographic experimentation) that the rule "If the magnification at the plane of best focus is the same, and the f/stop is the same, then the DOF will be the same for lenses with different focal lenths." is a simplification that is true at relativly short focus distances and wide f/stops, but not true when the focus distance gets close to the HFD. Close to the HFD, the shorter focal lenths have more depth of field behind the subject, less DOF infront, and more total DOF .

Thanks everyone for the info!

I just wish that Outdoor photographer, Popular photography, The Luminous Landscape, and other such founts of knowlege would include this qualifier when they repeated this "rule".

D

 

[Lee Jay]

Actually, I think all lenses, regardless of focal length, have the same near and far DOF fractions for a focus distance that's a given fraction of the hyperfocal distance. Since the hyperfocal distance goes up with focal length, this is sort of what you said as well. As for Doug's comment about short focus distances, this comparison should always be made relative to the hyperfocal distance.

--
Lee Jay
(see profile for equipment)

 

[Lee Jay]

The usual DOF blanket-statements that many of us make always have the implicit assumption that the focus distance

--
Lee Jay
(see profile for equipment)

 

[Dredman]

The usual DOF blanket-statements that many of us make always have
the implicit assumption that the focus distance
distance. If this is not true, then those statements are not
necessarily true.

Yeah, it is just that I love shooting landscapes at wide angles, usualy with small apratures, so I run into HFD range often and was just trying to make sense of what I read!

BTW what other "blanket-statements" are you thinking of?

D

 

[Lee Jay]

BTW what other "blanket-statements" are you thinking of?

Oh boy...there's a ton of them you can generate if you want to.

--
Lee Jay
(see profile for equipment)

 

[Doug Kerr]

Hi, Helmuth,

I have not seen the DoF equations in the convenient form you describe. I will try to derive them myself. (Or do you have a reference to a derivation?)

1. I have a couple of questions on your notation:

Starting from the basic equations found at dofmaster.com, one can
show that the "near DOF", defined as the distance between the focus
distance and the nearest point in acceptable focus (not the same as
"near distance of acceptable sharpness" as used by dofmaster)
varies with the focal distance as 1/(1 + NC/FM), where

1.1 Do you by any chance mean here "varies with focal length" (rather than "varies with focal distance")?

Or is F in fact the focus distance? In that case, the conclusions you draw regarding change of DoF with change in focal length certainly do not follow in any visible way from those expressions.

1.2 You say the near (or far) DoF "varies as" the expression you give. I assume that means that the actual values, for a particular starting situation, is this expression times some constant (which must have the dimensionality of distance). Would that constant be the distance of perfect focus? Or what?

2. I cannot get anything like your results from numerical examples calculated under the "usual" DoF equations.

For example, with these paramaters:

F: 100 mm
C: 0.02 mm
N: 8.0 (f/8.0)
M: 0.11 (which would result from a focus distance of 1000 mm)

Then I calculate the near DoF as 0.0142 m and the far DoF as 0.0179 m. The ratio of the far to near DoF is 1.262.

If I assume that the actual far and near DoF values from your two expressions result from them both multiplying the same constant, the for evaluating your two expressions gives:

Near DoF: 0.986 K (where K is that constant)

Far DoF: 1.0147 K

with a ratio of near to far of 1.029.

That does not comport with the calclulations using the conventional form of the DoF equations.

Perhaps I am misunderstanding your notation.

3. I am a little startled by some implications of your result.

For example, as we increase the circle of confusion diameter limit (C), other factors remaining constant, we usually expect the depth of field to increase (near, far, or total) . This is because, by adopting a greater value of C, we have in effect made ourselves accept greater blurring, which will then admit greater departures of the object distance from the distance of perfect focus (the definition of depth of field).

However, your expression for the near DoF decreases with increasing values of C.

We also expect that, all other factors remining constant, with increase in the f/number, the depth of field (near, far, and total) would increase.

However, your expression for near depth of field decreases with increasing values of N.

What am I missing here?

4. It is intersting to note that with your two expressions, if NC=FM, then the near DOF would be K/2, while the far one would be infinite.

This is tantalizingly evocative of the situation for hyperfocal distance, if K were in fact the hyperfocal distance.

Best regards,

Doug

 

[Doug Kerr]

Him Helmuth,

Well, I see what is probably part of my problem. Evidently, from your language (now that I look at it a bit more closely), these expressions are not the general expressions for the near and far depth of field as functions of N, C, F, and M.

Rather, they are expressions for the relative variations of the near and far depth of field with F for certain values of N, C, and M, which are parameters of the expression.

Thus. my comments about the apparently-anomalous relationship between the near DoF and N and C are not meaningful.

I am, however, still dismayed by the fact that, for any given (constant) values of the parameters N, C, and M, and the variable F, the relative magnitudes of your expressions for near and far DoF do not seem consistent with the relative magnitudes of those two quantities as calculated by the actual expressions for DoF as a function of consistent values of N, C, F, and M.

Is it possible that the "scale" of the two expressions is not the same; that is, the value of K that goes with them is not the same? (It's a little hard to imagine that, given the symmetry of the whole thing, but not having derived those expressions I certainly couldn't say for certain!)

Or maybe I just made an error in evaluating your expressions for my test case!

In any event, I have to assume that the condition NC=FM, where the far DoF becomes infinite, corresponds to the situation in which the hyperfocal distance is equal to the assumed focus distance, P (which is of course determined by F and M, both present in the expression). And in fact, that indeed seems to be the case, based on a numerical test.

Interesting.

Best regards,

Doug

Visit The Pumpkin, a library of my technical articles on photography, optics, and other topics:

http://doug.kerr.home.att.net/pumpkin

'Make everything as simple as possible, but no simpler.'

 

[Helmuth Schumann]

Hi Doug:

I have not seen the DoF equations in the convenient form you
describe. I will try to derive them myself. (Or do you have a
reference to a derivation?)

I derived them myself, starting from the dofmaster forms. I substituted dofmaster's expression for the HF distance, expressed the focus distance (S, I think dofmaster calls it) as F(1+1/M), changed his "nearest/furthest point in acceptable focus" to my "near/far DOF" as I defined earlier, and did some algebraic manipulation to end up with:

Near DOF = (Nc/M)(1+1/M) (1+Nc/FM)
Far DOF = (Nc/M)(1+1/M) (1-Nc/FM)
where N = f-stop
c = circle of confusion
M = magnification
F = focal length.

1. I have a couple of questions on your notation:
1.1 Do you by any chance mean here "varies with focal length"
(rather than "varies with focal distance")?

Doh! Yes, of course, I should have said focal length. Sorry for wasting your time trying to figure out why I said distance.

1.2 You say the near (or far) DoF "varies as" the expression you
give. I assume that means that the actual values, for a particular
starting situation, is this expression times some constant (which
must have the dimensionality of distance). Would that constant be
the distance of perfect focus? Or what?

The constant is, as shown by the full equations above, (Nc/M)(1+1/M), which, as you correctly point out, has the dimensionality of distance. (But it isn't the focus distance, of course.)

2. I cannot get anything like your results from numerical examples
calculated under the "usual" DoF equations.

Could you give me these equations, and I'll look into it.

3. I am a little startled by some implications of your result.

For example, as we increase the circle of confusion diameter limit
(C), other factors remaining constant, we usually expect the depth
of field to increase (near, far, or total) . This is because, by
adopting a greater value of C, we have in effect made ourselves
accept greater blurring, which will then admit greater departures
of the object distance from the distance of perfect focus (the
definition of depth of field).

However, your expression for the near DoF decreases with increasing
values of C.

We also expect that, all other factors remining constant, with
increase in the f/number, the depth of field (near, far, and
total) would increase.

However, your expression for near depth of field decreases with
increasing values of N.

What am I missing here?

What you're missing is that, in my first post, I said "varies with F, for given values of c, N and M" or words to that effect. This means that the expression given in that post doesn't necessarily tell you how DOF varies with c, N or M. In fact, as you can see above, c, N and M appear in the "constant" (i.e. the part of the expression NOT containing F) in such a fashion as to make the whole expression agree with photographic intuition.

4. It is intersting to note that with your two expressions, if
NC=FM, then the near DOF would be K/2, while the far one would be
infinite.

This is tantalizingly evocative of the situation for hyperfocal
distance, if K were in fact the hyperfocal distance.

Yes. The hyperfocal situation is characterised by Nc/FM = 1, in which case

K = (Nc/M)(1+1/M) = F(1+1/M) = S, the focus distance. And dofmaster's HF distance expression becomes:
H = F^2/Nc + F = F^2/FM + F = F(1/M+1) = S
So K = H in this special case, but not generally.

Cheers,
Helmuth.

 

[Doug Kerr]

Hi, Helmuth,

I have not seen the DoF equations in the convenient form you
describe. I will try to derive them myself. (Or do you have a
reference to a derivation?)

I derived them myself, starting from the dofmaster forms. I
substituted dofmaster's expression for the HF distance, expressed
the focus distance (S, I think dofmaster calls it) as F(1+1/M),
changed his "nearest/furthest point in acceptable focus" to my
"near/far DOF" as I defined earlier, and did some algebraic
manipulation to end up with:

Near DOF = (Nc/M)(1+1/M) (1+Nc/FM)
Far DOF = (Nc/M)(1+1/M) (1-Nc/FM)
where N = f-stop
c = circle of confusion
M = magnification
F = focal length.

Indeed. Then, to get your (simpler) expressions of the dependnce of the DoF values on F, you dropped out all the appearances of the other variables that did not interact with F.

1. I have a couple of questions on your notation:
1.1 Do you by any chance mean here "varies with focal length"
(rather than "varies with focal distance")?

Doh! Yes, of course, I should have said focal length. Sorry for
wasting your time trying to figure out why I said distance.

It's easy to do!

1.2 You say the near (or far) DoF "varies as" the expression you
give. I assume that means that the actual values, for a particular
starting situation, is this expression times some constant (which
must have the dimensionality of distance). Would that constant be
the distance of perfect focus? Or what?

The constant is, as shown by the full equations above,
(Nc/M)(1+1/M), which, as you correctly point out, has the
dimensionality of distance. (But it isn't the focus distance, of
course.)

Yes, of course. There is no reason it should have been!

2. I cannot get anything like your results from numerical examples
calculated under the "usual" DoF equations.

Could you give me these equations, and I'll look into it.

They are just the "precise" equations given on the Dofmaster site.

However, for confirmation, I'll send those to you a little later, and as well I will reconstruct my work with them, which of course may have some errors. (It was not done under the most scrupulous of circumstances!)

I'll also work through the reckoning using your "full" equations, which based on their provenance should be quite equivalent to the ones I use.

3. I am a little startled by some implications of your result.

We also expect that, all other factors remining constant, with
increase in the f/number, the depth of field (near, far, and
total) would increase.

However, your expression for near depth of field decreases with
increasing values of N.

What am I missing here?

What you're missing is that, in my first post, I said "varies with
F, for given values of c, N and M" or words to that effect. This
means that the expression given in that post doesn't necessarily
tell you how DOF varies with c, N or M. In fact, as you can see
above, c, N and M appear in the "constant" (i.e. the part of the
expression NOT containing F) in such a fashion as to make the whole
expression agree with photographic intuition.

Indeed. As you see from my second message, I realized that eventually. (You were quite clear in your presentation of that, in fact - I just decided to ignore it!.)

4. It is intersting to note that with your two expressions, if
NC=FM, then the near DOF would be K/2, while the far one would be
infinite.

This is tantalizingly evocative of the situation for hyperfocal
distance, if K were in fact the hyperfocal distance.

Yes. The hyperfocal situation is characterised by Nc/FM = 1, in
which case
K = (Nc/M)(1+1/M) = F(1+1/M) = S, the focus distance. And
dofmaster's HF distance expression becomes:
H = F^2/Nc + F = F^2/FM + F = F(1/M+1) = S
So K = H in this special case, but not generally.

Indeed.

Nice work, and thanks for helping me in such a clear way to understand what is happening. I enjoy your clear outlook on all of this, and your style of presentation. Thanks for coming out to play.

Best regards,

Doug

Visit The Pumpkin, a library of my technical articles on photography, optics, and other topics:

http://doug.kerr.home.att.net/pumpkin

'Make everything as simple as possible, but no simpler.'

 

[Helmuth Schumann]

Hello Doug:

2. I cannot get anything like your results from numerical examples
calculated under the "usual" DoF equations.

Could you give me these equations, and I'll look into it.

They are just the "precise" equations given on the Dofmaster site.

However, for confirmation, I'll send those to you a little later,
and as well I will reconstruct my work with them, which of course
may have some errors. (It was not done under the most scrupulous of
circumstances!)

I'll also work through the reckoning using your "full" equations,
which based on their provenance should be quite equivalent to the
ones I use.

I've done the calculations with the dofmaster expressions for near and far distances of acceptable focus and the results agree perfectly with those coming out of my equations for near and far DOF. The reason you (and I, initially!) got discrepancies is that we used an approximate value for M = 0.11 (a calculated, rather than given, number) which, at 2 significant figures, is accurate enough to give only about 2 significant figures of accuracy in the results when used in my formulae. Which it does, but then there is a discrepancy (beyond 2 digits) with the results of dofmaster's formulae. The reason these latter calculations provide a result accurate to more than 2 significant digits is that the intermediate H one calculates is a large number (in mm) so one retains a large number of its significant digits when continuing with the rest of the calculation. So there isn't any 2-digit inaccuracy in the final result.

Thanks Doug, for the feedback and interaction. It's good to know someone out there checks up on my posts!

Helmuth.

 

[Doug Kerr]

Hi, Helmuth.

I evaluated your two complete expressions for this configuration:

F: 100 mm
c: 0.02 mm
N: 8 (f/8)
S: 1000 mm (does not appear directly)

M: 1/9 (from F and S; expressed as an integer fraction to avoid rounding complications)

My results from your complete expressions were:

Near DoF: 14.6183 mm
Far DoF: 14.6104 mm

Calculating that same configuration using my spreadsheet calculator (which should be based on the "precise" equations as given by Dofmaster - but I need to confirm that!) gives:

Near DoF: 14.1960 mm [vs 14.6183 mm from you expression]

Not bad agreement, perhaps you adopted some simplification.

Far DoF: 17.9150 mm [vs 14.6104 mm from you expression]

Not good agreement at all.

I indeed need to confirm that there is no error in implanting the "precise" Dofmaster equations into the spreadsheet. I'll do that as soon as I can.

The equations I hoped I had used (!) were (using your symbols):

Dh = (f^2/Nc) + F

where Dh represents the hyperfocal distance.

Then:

Dn = (S(Dh-f)) (Hd + 2F + S )

where Dn is the distance to the near lmit of the depth of field (not the near depth of field)

and

Df = (S(Dh-f)) (Hd - 2F + S)

where Dn is the distance to thefar limit of the depth of field (not the far depth of field)

I need to confirm that these are correct. I will probably rederive them, or check for consistency with VanWalree's equations (which are in a different form).

Bizarrely enought, In that regard, I just looked at the presentation of the equations on the Dofmaster site, and I see that in the expression for the far limit, the term "-2F" in the denominator is missing compared to what I have, which I had thought was a transcription of what was on that site some while ago! Perhaps that current equation is correct, and I mistranscribed it.

However, testing with the Dofmaster online calculator produces the same result as my calculator for a couple of cases (with the equation as I have it, and substantial disagreement when I modified teh equation to what is now shown on their site), so perhaps their presentation of the equation for the far limit has suffered an editorial inadvertence!

Again, I need to reconstruct a lot of this!

Oh, great!

Best regards,

Doug

Visit The Pumpkin, a library of my technical articles on photography, optics, and other topics:

http://doug.kerr.home.att.net/pumpkin

'Make everything as simple as possible, but no simpler.'

 

[Doug Kerr]

Hi, Helmuth,

I've done the calculations with the dofmaster expressions for near
and far distances of acceptable focus and the results agree
perfectly with those coming out of my equations for near and far
DOF. The reason you (and I, initially!) got discrepancies is that
we used an approximate value for M = 0.11 (a calculated, rather
than given, number) which, at 2 significant figures, is accurate
enough to give only about 2 significant figures of accuracy in the
results when used in my formulae. Which it does, but then there is
a discrepancy (beyond 2 digits) with the results of dofmaster's
formulae. The reason these latter calculations provide a result
accurate to more than 2 significant digits is that the intermediate
H one calculates is a large number (in mm) so one retains a large
number of its significant digits when continuing with the rest of
the calculation. So there isn't any 2-digit inaccuracy in the
final result.

Well, as you will see from my recent message, I got the discrepancy using M=1/9 (exactly) in your expressions for my case (F=100, S = 1000, etc.).

But (as you will also see there) there is the possibility that the equations I used (embodied in my spreadsheet ca,culator) are incorrect.

On the other hand, it also looks as if the current equation for the far limit of the depth of field on teh Dofmaster site is errored! (I don;lt know which onbe you atarted with).

And in any case, I got a discrepancy for the near limit/distance as well (but not very great).

Thanks Doug, for the feedback and interaction. It's good to know
someone out there checks up on my posts!

Well, and vice-versa!

Best regards,

Doug

 

[Helmuth Schumann]

Hi, Helmuth.

I evaluated your two complete expressions for this configuration:

F: 100 mm
c: 0.02 mm
N: 8 (f/8)
S: 1000 mm (does not appear directly)
M: 1/9 (from F and S; expressed as an integer fraction to avoid
rounding complications)

My results from your complete expressions were:

Near DoF: 14.6183 mm

No. I get 14.1956. Pls check your calculation.

Far DoF: 14.6104 mm

Yes.

Calculating that same configuration using my spreadsheet calculator
(which should be based on the "precise" equations as given by
Dofmaster - but I need to confirm that!) gives:

Near DoF: 14.1960 mm [vs 14.6183 mm from you expression]

Not bad agreement, perhaps you adopted some simplification.

No simplification. I got agreement, as above.

Far DoF: 17.9150 mm [vs 14.6104 mm from you expression]

I suspect your spreadsheet is wrong here. Again, I got agreement between my formulae and dofmaster's.

Not good agreement at all.

I indeed need to confirm that there is no error in implanting the
"precise" Dofmaster equations into the spreadsheet. I'll do that as
soon as I can.

 

[Doug Kerr]

Hi, Helmuth,

I had made a few small errors in my calculations. Now that I have that straightened out, I am able to get precise agreement between your near DOF formula value and the same quantity calculated from the Dofmaster near equation. It is also in quite good agreement with the result from my spreadsheet.

With regard to your far DoF formula, I am able to get precise agreement between its result and the same quantity calculated from the Dofmaster far equation as it currently appears.

I am unable to get agreement between the value from your formula and the value given by my spreadsheet.

I next used a slightly different case to examine the result from the Dofmaster online calculator (one that produced results for which fair precision was available in the Dofmaster 2-decimal place output): the same case as before but with S=10 m rather than 1 m.

For the distance to the near limit of DoF, there was excellent agreement between the value I calculated using the Dofmaster equation and the result from the Dofmaster online calculator, There was also excellent agreeement between both of those results and the result from my spreadsheet for that case.

For the distance to the far limit, the result calculated under the Dofmaster equation (as currently presented) did not agree well at all with the result from the Dofmaster online calculator.

The value from the Dofmaster online calculator agreed well with the result from my spreadsheet for that case.

Accordingly, as a review of my original paper on this topic had suggested, the equation currently stated on the Dofmaster site for the far distance appears to be in error. I believe that it should be (using their notation):

Df = (s(H-f)) (H-s-2f)

not

Df = (s(H-f)) (H-s)

as currently shown.

(This of course assumes that the operation of the Dofmaster online calculator remains correct.)

I assume this was one of those improvident resuts of an accidental glancing blow with the mouse to an initially-correct formula. It happens to me all the time!

It is fascinating the lovely symmetry evident between your two equations for the near and far distances, given that one was apparently derived from an errrneous original equation! Perhaps this is good art trying to make up for bad math!

I have separately been working to reconstruct your derivation, and it sure is no fun!

Best regards,

Doug

Visit The Pumpkin, a library of my technical articles on photography, optics, and other topics:

http://doug.kerr.home.att.net/pumpkin

'Make everything as simple as possible, but no simpler.'

 

[Doug Kerr]

Hi, Helmuth,

Well, I have just confirmed by calculations made from fundamental principles that both formulas on the Dofmaster site are in fact apparently correct.

I'm not sure how I got off the track.

A corollary is that my spreadsheet has had an error in it for some while. I'll be reissuing it shortly under a separate transmittal explaining the situation. Very embarrassing.

I will need to reconstruct the tests I did earlier today comparing the Dofmaster online calculator and calculations made with the Dofmaster formulas to see where I went wrong in that interpretation.

Thanks again for giving me the context in which to get straightened out in this regard.

Best regards,

Doug

Visit The Pumpkin, a library of my technical articles on photography, optics, and other topics:

http://doug.kerr.home.att.net/pumpkin

'Make everything as simple as possible, but no simpler.'

 

[Helmuth Schumann]

Hello Doug:

If I'd been quick enough to respond to your second-last posting, where you said:

Accordingly, as a review of my original paper on this topic had suggested, > the equation currently stated on the Dofmaster site for the far distance > appears to be in error. I believe that it should be (using their notation):
Df = (s(H-f)) (H-s-2f)
not
Df = (s(H-f)) (H-s) ,

I would've observed that the first form cannot be correct since, when s=H, by definition, Df has to be infinite and this is not the case with that form.

Anyhow, it's gratifying that we've got all this sorted out in the end!

Cheers,
Helmuth.