Scheimpflug crisis

[Charles Twist]

I have now checked my equations and I am confident there are no more mistakes. I have to say that it is very enjoyable doing the maths and understanding what matters. They are a good complement to the movies that Jo mentioned on the Merklinger site. What the graph I have just added shows is:
(i) that the angle of the plane of critical focus can bend downward as well as upward when you tilt the lens forward. This is news to me - worth checking out in the field.
(ii) This effect is not shown on the Merklinger movies. In the second movie, the bellows have considerable extension, as much as 1.5 times the focal length, which is actually macro regime (half life size). Looking at the above graph it is clear, he would not reach a negative angle in these conditions. These movies now have obvious flaws: for example, the tripod is ridiculously short - about 10% of normal length.
(iii) This effect depends on the bellows extension: the shorter the bellows (further focus), the greater the effect of lens plane tilt on the angle of the plane of critical focus and the sooner it goes negative. Because of the strong dependence on the angle of tilt when focusing far, I can see the point of Merklinger's assertion that focusing far helps to bring the foreground into focus (see Shutterbug article in Jo's message). The angle of the plane of critical focus is more immediately flat.
(iv) The graph also shows that as you extend the bellows, there comes a point where the angle of the plane of critical focus goes negative. This effect is shown in the first Merklinger videos.
(v) Richard Kelham's rule of thumb is still out by some distance and the rule of thumb I posited yesterday is tosh. I don't see a rule of thumb appearing any time soon.
(vi) I don't understand the behaviour when the bellows are extended for focus at infinity, f=i (relative extension of 1). The movies stay clear of showing what happens when you focus on infinity (f=i). Anybody any offers?
I am still struggling to understand why the ratio of the distance between the centre of the film board and the Scheimpflug point divided by the distance between the centre of the lens board and the hinge point tends to 1 at low tilt angles. Is it because I am dividing two very large numbers? The difference between the two values is in full agreement with the angle graph given above, however.
Finally Marc, if you look at the Merklinger movies, you'll see the point of focus in line with the camera varies with the angle of tilt. Sorry about that.
Well, enough fun for one evening. Over to you,
Charles

[masch]

Finally Marc, if you look at the Merklinger movies, you'll see the point of focus in line with the camera varies with the angle of tilt.

Yes, it does... This is because the focus distance is along the optical axis of the lens, and not that of the camera.
If you look at movie 2 (tilting lens), you can quite easily see the rotation around an approximately stable point along the main camera axis. this point slowly drops with increase in tilt angle. I didn't want to overcomplicate the matter, but give a quick overview to show the limiting factors.

Marc

[Charles Twist]

Do I understand that the angle of view tilts downward somewhat so that the point of focus is on the line of critical focus (distance o conserved)?
Charles

[Ole Tjugen]

It is possible with some cameras to adjust the tilt axis relative to the lens so that a point stays in focus throughout the tilting operations. This is quite an interesting exercise, but sadly impossible on all but a very few cameras (anyone coming to Norway? Give me a call and I'll set up a demonstration).

In the meantime (when I think about the rest of the questions), here's a picture shot using lot of swing and a little tilt. The right side of the picture is focussed beyond infinity, and you clearly see the "wedge of sharpness" travelling through the picture:

[Charles Twist]

Hello Ole,
You're jumping the gun a bit with dual movements! I have spent some time going through the mighty Merklinger's pdf's. When it comes to dual movements, he acknowledges the case but says there is no simple treatment. And leaves it at that apparently - he might have worked on it privately or elsewhere but I have not found it yet. Any suggestions on how to treat the dual movement case?
The case of f=i amused him too, but he states that the OP angle of 0 at tilt = 0 is nothing to worry about as the hinge line is infinitally far. How convenient! How amusing indeed!
Marc: He mentions the line of focus being different to the line of sight. I'll read up some more and get back. He also mentions that the effective focal length varies with tilt, which probably plays a role. The effective focal length is the nominal focal length divided by the cosine of the tilt. Strangely I had noticed this effect but never dwelt on it. Interestingly his movies do not take this effect into account. If he had, movie2 (lens tilt) would have shown the line of focus going negative at large tilt values.
I look forward to hearing your ideas.
Charles

[masch]

When it comes to dual movements, he acknowledges the case but says there is no simple treatment.

There is, I think....
A tilt in two planes is identical to a tilt in one plane and a rotation around the direct viewing line of the camera ("boresight" or whatever Merklinger calls it.). It's a simple coordinate transformation.
The problem is that it is difficult to visualise, since the Scheimpflug line then sits at an oblique angle to your visual projection...
I might have this wrong, though...

Marc: He mentions the line of focus being different to the line of sight. I'll read up some more and get back. He also mentions that the effective focal length varies with tilt, which probably plays a role.

Yep. Variopus reasons, but mainly non planarity of field at larger angles, and longer optical pathlengths...

Marc

[Ole Tjugen]

I think you're correct Marc, that's how I've always envisioned the "multiple movements". I don't find it particulary difficult though, just don't ask me to work out the maths!

As to part II of your post, it is exactly the planarity of the field that causes the longer focal length at larger angles. If the lens had a consistent focal length over the whole field of view, you would have a curved image plane with a radius equal to the focal length.

Some older lenses actually have this, which is how some early photographers managed to shoot street scenes with both sides of the street sharp all the way to infinity, and blurry lampposts up the middle of the picture: Simple "landscape lenses", a meniscus mounted behind the aperture.

[Charles Twist]

I like the sound of your simple lens, Ole. Not sure how much mileage it would get nowadays but very instructive in an optics class. As to the problem of dual movements, I basically had the same vision as both of you and had been thinking in that direction. Comforted by your words, I set about calculating the transformation of one plane by the other. I struggled a bit with the centres of reference, complementary angles and general mathematical rustiness, but I think I got the transformation right. In the graphs below, I have set the centre of the image as (0,0,0). Imagine that the axis from -75 to 75 is left to right and 0 to 500 is from the camera to the yonder. Units are the same on the 3 axes: they're all in meters (defined by the focal length which I calculated in meters). You start off with movements on just one axis, let us say a forward tilt of 5 degrees (the lens is a 135mm focal and the extension is 159mm), to obtain:

Then you apply a swing of 10 degrees to the left to obtain:

Does that look about right? Is that what you had in mind? It's what I visualised anyway. Contact me if you need more points.
Assuming that is correct, what is interesting to note is that the swing effectively raised the right-hand side. Has anyone used this kind of effect to get a tall object on one side into acceptable focus? I am sure we all have consciously or not.
And now for some serious head-ache: I thought it would be interesting to revisit my previous graph where I give the angle of the plane of critical focus relative to the tilt angle for different bellows extensions. That is all well and good, but what we really need is the distance from the camera of that plane. Merklinger refers to the depth of the hinge point. I have a lot of trouble visualising the number of metres beneath me, but the number of metres along the ground I can do. So, I calculated where the plane of focus emerges from the depths of the earth: the number of metres from the camera is on the y axis; the lens tilt is on the x axis. For this calculation, I assumed that the film plane is vertical to the ground and that the centre of the image is 1 metre above the ground (waist height). The lens focal length is 100mm. The bellows extension i are expressed as f/i (as above); the graph compensates for the non-linearity of the focal length, but f/i is relative to the nominal focal length. This is what I obtain:

Good, eh? It surprised me. I am amazed how non-linear it is. I am pretty confident I made no mistakes. What I notice is that within the first 10 to 20 degrees of tilt things are pretty predictable (depending on the focal length: the greater f, the greater the predictable range). Which is handy for all those people with field cameras. I also notice that the shorter the extension, the less predictable things are, which correlates with the strong effect of tilt observed above. I was curious to know whether anybody with experience of monorail cameras and much wider movements, has noticed anything that this graph might relate to.
I don't know why there is a nodal point - any ideas?
I look forward to hearing from you,
Charles

[Ole Tjugen]

...
I don't know why there is a nodal point - any ideas?...

I assume you mean the nodal point at 5.71 degrees?

I wondered about that, had a suspicion and calculated arctan(0.1) and got 5.71 degrees (rounded).

It's because your 0-point is the location of the film, while you're tilting the lens. And the height of the camera was at 1m, extension 100mm. So with a 5.71 degrees tilt the lens plane intersects the film plane at ground level.