M (240) calibration

[120]

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If you calculate what happens between 10000 meter in object space and anything further, approaching an infinite distance, the focus error due to an error gain (rangefinder arm length error), compensated at 10000 meter with the rangefinder roller, expressed as a percentage of the distance to the camera in object space, climbs from 0% upward and approaches asymptotically 100% towards infinity. Now that looks like a real problem for taking sharp images you may say. But in image space the error climbs from 0 mm to 0.0002 mm and so you could never detect the error in the image. All other tolerances of the rangefinder mechanism and the optical tolerances, and resolution of the sensor are all much worse.

 

So if we step into the real world of a rangefinder photographer for a moment , the problem of infinity does not exist.

 

I will stick with 1000m because that is what you used. You are trying to model what the rangefinder actually does. It is just a matter of the geometry of the two adjustments, and the answer should be purely mathematical, without appeal to anything else. I'm sure you picked the wrong offset to compensate with...your graph looks odd at close-up distances, and it's way, way off when you get past 1000. As you just said, the error is 100 % at large distances and infinity. This behavior comes not from the rangefinder, but from the contrived solution. If you use the offset given earlier in the thread, you get a well-behaved graph and the % error is never more than

 

100 X (1 - g) / g,

 

where g is the gain. This is exactly what you were trying to show, that the compensation is REALLY GOOD. If the gain is 1.01 (one percent error in arm length), the error in the distance is never more than 1%. If the gain is 0.95, the error in the distance is never more than 6%, and so on.

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[Lindolfi]

Some remarks to see if we can sort this out

 

1) Chosing 10000 meter or 1000 meter does not change the principle, so no discussion needed

 

2) Choosing 1% or 7% does not change the principle, no discussion needed (the graphs are more clear with 7%, that's all)

 

3) I did not say 100%, I said it grows asymptotically to 100% as we approach infinity

 

4) Your expression for the percentual error in object space 100 X (1 - g) / g is not correct

 

 

To show the last point let's look at the change in v (object distance) for every change in b (image distance).

 

And we have to do that, since the rangefinder arm length influences b, not v.

 

If we call the focal length f, then we can derive the expression v=(f.b)/(b-f) from the lens formula 1/f=1/b+1/v

 

Now let's take the derivate of v with respect to b: dv/db = (-f^2)/((b-f)^2)

 

From this we see that the error in object space at a given small error in image space is not constant, but depends on b. In fact when b approaches f, dv/db climbs to infinity (division by 0 in the last formula), which explains the asymptotic growth of the error towards 100% when going to infinity.

[120]

Some remarks to see if we can sort this out

 

1) Chosing 10000 meter or 1000 meter does not change the principle, so no discussion needed

 

2) Choosing 1% or 7% does not change the principle, no discussion needed (the graphs are more clear with 7%, that's all)

 

3) I did not say 100%, I said it grows asymptotically to 100% as we approach infinity

 

4) Your expression for the percentual error in object space 100 X (1 - g) / g is not correct

 

 

To show the last point let's look at the change in v (object distance) for every change in b (image distance).

 

And we have to do that, since the rangefinder arm length influences b, not v.

 

If we call the focal length f, then we can derive the expression v=(f.b)/(b-f) from the lens formula 1/f=1/b+1/v

 

Now let's take the derivate of v with respect to b: dv/db = (-f^2)/((b-f)^2)

 

From this we see that the error in object space at a given small error in image space is not constant, but depends on b. In fact when b approaches f, dv/db climbs to infinity (division by 0 in the last formula), which explains the asymptotic growth of the error towards 100% when going to infinity.

 

if it helps

1. I don't think basing an offset on either is natural; I have explained why.

2. I don't know what this is in response to...

3. Yes, your error quickly grows to 100% once you are past 1000.

4. That's not an expression for the percent error; that's a bound, as I mentioned three times in the post ("the error is never more than...") and also in a previous post. I don't think differentiating the lens formula will help get either my bound or yours; I get that from the limit of the expression for the % error.

[Lindolfi]

Dear 120, thanks very much for your contributions to this subject.

[120]

The two adjustments are represented by a linear function; the two coefficients are the gain and the offset. The adjustment at the roller contributes to the offset. When you rack the lens to infinity and the image distance is f = 50mm, and then adjust the roller to compensate a gain error, there is only one correct offset coefficient representing the situation. There is only one answer to the problem. You are compensating at the image distance f, not at the image distance corresponding to 1000m.

 

The linear function is defined everywhere. The fact that the function to go back and forth between object and image distances is not defined at distance = f is just what you want; you want the corresponding distance to be infinite.

 

The graph below shows a 7% gain error compensated at infinity (i.e. image distance = f), and also compensated at 1000m. They are obviously not the same. The second graph is a close-up of the first.

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[120]

close-up of graph

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[Lindolfi]

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The graphs you posted are very similar to the graphs I posted on the 5th of november 2010:

 

http://www.l-camera-forum.com/leica-forum/leica-m9-forum/147259-focussing-issues-3.html#post1507158

 

And this was the figure:

 

 

There is only one difference: you assume implicitly that you can turn the lens to infinity on the barrel and that the distance of the lens to the sensor is equal to the focal length. Often that is the case, but you can not rely on it. What I did is select an image distance from the lens to the sensor that focusses on a point at 1000 m (to be verified using the EVF, or by trial and error) and then adjusting the rangefinder to get perfect overlap at that object at 1000 meter by adjusting the roller in the presence of a rangefinder arm length error.

 

Since that small difference drowns in DOF in those conditions anyway, this is a purely academic discussion point.

[120]

The graphs you posted are very similar to the graphs I posted on the 5th of november 2010...

 

 

Yes, the graph corresponding to 1000m is exactly your graph, and I borrowed your constant "bcenter = 0.0513" meters. Credit also to Julian T. for the primordial graph.

 

...

 

There is only one difference: you assume implicitly that you can turn the lens to infinity on the barrel and that the distance of the lens to the sensor is equal to the focal length. Often that is the case, but you can not rely on it...

 

Yes, I assume that the lens and camera are set up correctly, excluding the rangefinder.

 

I am satisfied; thank you for your patience and all the comments.