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Scheimpflug and Hinges - the parts Dave Chew wisely left out.

MGrayson

Subscriber and Workshop Member
In Dave Chew's excellent article on using camera movements in the field, he concentrates on what actually happens to the focus plane (focus wedge, actually) as lens tilt and focus are adjusted. This is what a photographer needs to know.

Unfortunately, I'm a mathematician. Mathematicians care about three things: what is true, why it is true, and the proof that it's true. To publish papers and be famous (famous mathematician, now THERE'S an oxymoron), one only needs the "what" and the "proof". But some of us are absurdly attached to the "why".

I gave my "why" for tilting elsewhere, but it makes no reference to the two things that all other treatments treat as fundamental: Scheimpflug and the Hinge (bad rock group name?). The great thing about these entities is that they don't exist (or are hiding at infinity) for normal untilted lens/sensors. But if they DO exist, which is whenever the lens and sensor are not parallel, then they have a truly wonderful property. THEY ARE BOTH IN FOCUS. Good optics having no field curvature *cough*, we then know that the entire line between these two points is in focus.

All treatments tell you WHAT these points are: Scheimpflug is the intersection of the sensor plane and the lens plane (Lines in all the pictures. We're just doing tilt, no swings!). Did you ever think how a point ON THE SENSOR could be in focus? I mean, I guess they all are if you ignore the lens, but that's not a really useful fact. But how can a point ON THE LENS PLANE be in focus. Its light can't go through the lens. We'll get to that.

The harder point to describe, even if it's more useful (see Dave above), is the Hinge. The Hinge is the intersection of a vertical line through the lens center (we're keeping the sensor vertical and tilting the lens) and a weird line H which is parallel to the lens but one focal length in FRONT of it. Why is it called the Hinge? Because it doesn't depend on the sensor distance, and so as you rack the sensor back and forth, the Hinge doesn't move, and the plane of focus rotates around it as if it were ... a hinge.

Now the real world is a mess, so physicists to some extent, and mathematicians to a much greater extent, simplify it. (ASIDE: Physicists are very very bad about distinguishing their simplified models from reality, and so we get books on superstrings and the multiverse as if these are "real". They are no more real than the music of the spheres, which was a good theory 600 years ago.) So we simplify what a lens is and where focus is. These are close enough - and are necessary for the above mentioned rules to hold, so we'll use them.

A "Lens of focal length f" is a magical object. In our simplified world, it lies in a plane. Since we're looking at all this from the side, the lens will lie on a line. What is magical about it? Take a point P (more than distance f in front of the lens - we'll deal with closer points later). Then there is a point Q on the other side of the lens such that EVERY ray of light emanating from P that hits the lens will bend and then pass through Q. Wait, what was that about the focal length? If parallel light rays hit the lens (P is off at infinity), then Q is distance f from the lens plane.

Two facts (axioms) about the lens will let us find Q given P, and vice-versa (this is called "focusing"). One - any light ray passing through the center of the lens does not bend at all. Two - light rays entering the lens perpendicular to its plane will pass through the center line a distance f behind the lens. This very important point we call F. Aim the lens at the sun and this point F gets very very hot.. These two rules are enough to explain both Scheimpflug and the Hinge with surprisingly little effort. Suspension of disbelief? You betcha. Effort, not so much. Quick corollary - if a bunch of P's lie on a plane parallel to the lens, then all their Q's will too.

Scheimpflug is really easy, because if P is on the lens line, then Q and P are the same point. Wait, WHAT? How can they possibly be the same point? Because our definitions make sense under normal circumstances, but are also meaningful in these weird circumstances*. Take P on the lens line. The line through the center of the lens is just that very same lens line! There's no horizontal line to extend from P because it is already on the lens line. So the other line is the one from P to the focus point F ... well it started on the lens line, so it intersects the lens line (line through the center of the lens, too, in this case) at P. That intersection is the *definition* of Q, so P=Q. ANYWHERE the sensor line intersects the lens line is in focus! I always thought Scheimpflug was hard. It's a freaking tautology! Note: All theorems are tautologies - the trick is setting up the definitions correctly.

How about the Hinge? Here we get super sneaky. The universe is the same on both sides of the lens. Light rays travel the same in both directions. So if P is any point on the line H parallel to the lens and distance f in FRONT of it, then all the Q's will be infinitely far back. It's the same as focusing on the sun, but the sun is now BEHIND you. So why is the Hinge in focus? Take a laser pointer and move it along H but aim it at the lens center. There's a vertical sensor back there, but not being infinitely far away, the laser pointer is never in focus on the sensor. But now move the laser pointer down towards the Hinge. The spot on the sensor plane gets higher and higher. When the laser is directly below the center of the lens, the beam goes up to infinity, as it is now parallel to the sensor. If you think about vanishing points of railroad tracks, you can believe that parallel lines are said to meet at infinity. Indeed,, the laser pointer is focused at infinity "on" the sensor plane. Focused on the sensor plane is what it MEANS to be in focus, and so the Hinge is in focus, too.

So that's it. Scheimpflug in focus, Hinge in focus, so line between them is in focus. Now forget all this, read Dave again, and go take pictures.

Matt

*This is not cheating. Newton's law of gravitation stemmed from applying a rule that made sense for planets orbiting the sun and applying it to a rock orbiting the earth at head height. The acceleration on the rock turned out to be the same as the acceleration of dropping a rock. BANG! One of the best ideas anyone has ever had EVER.
 
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dchew

Well-known member
Matt,
One thing I cannot grasp: What is going on when the lens is tilted and focused at infinity? Something in the math is set in this situation that keeps the plane of focus perpendicular to the film plane. I.e. the Scheimpflug line and the hinge line are the same vertical distance below the camera, and they move up and down at the same rate with changes in tilt. This makes my head hurt.

Dave
 

anwarp

Active member
Thank you Matt!
I need to draw out you explanation to grasp it properly. The principle of the tilted lens/sensor and the tilted plane of focus made sense for me from the thin lens equation, but I have never given S&H much thought in terms of the “why”.

Anwar
 

MGrayson

Subscriber and Workshop Member
Matt,
One thing I cannot grasp: What is going on when the lens is tilted and focused at infinity? Something in the math is set in this situation that keeps the plane of focus perpendicular to the film plane. I.e. the Scheimpflug line and the hinge line are the same vertical distance below the camera, and they move up and down at the same rate with changes in tilt. This makes my head hurt.

Dave
Dave,

This is true when the tilt angle is relatively small. The focus plane starts to tilt at large tilt angles because the plane H gets further away from the lens HORIZONTALLY. It is still one focal length away, but that is measured perpendicular to the lens.
Some (self-explanatory.. Hah!) pictures! The lens line and the H line move up the Sensor line and the vertical lens center line in tandem. Looking at the pictures, it's clear that the focal plane will tilt downward exactly one half the lens tilt angle. You have to focus back a bit from infinity to get a truly level focal plane.

Untitled by Matthew Grayson, on Flickr

Untitled by Matthew Grayson, on Flickr

Untitled by Matthew Grayson, on Flickr

Untitled by Matthew Grayson, on Flickr


Matt
 

MGrayson

Subscriber and Workshop Member
Thank you Matt!
I need to draw out you explanation to grasp it properly. The principle of the tilted lens/sensor and the tilted plane of focus made sense for me from the thin lens equation, but I have never given S&H much thought in terms of the “why”.

Anwar
Anwar,
I tried to make pictures that were clearer than the words and failed. I'll give it another go!
Matt
 

MGrayson

Subscriber and Workshop Member
OK! Here is a lens. It has a center and a focal point located distance f from the center:



It focuses horizontal rays



Because it does not deflect a line through its center, we can find the location Q behind the lens where a point P is in focus. This is our DEFINITION of points P and Q being in focus. A fun (Kometani - it is not fun) exercise is to show that the construction works backwards. Start at Q and end up at P? I mean it has to, right? RIGHT?



If the sensor passes through Q, then the point P will be in focus in the final image. Repeat that sentence to yourself 5 times. It's important.

That is ALL we need.

So here is the setup as defined everywhere:



The Scheimpflug point, P, is its own focus companion Q! What is the rule? Extend parallel to the lens axis until you hit the lens line. It's already ON the lens line, so nothing to do. Now extend a line to the point F. The point Q is now the intersection of that line with the line from P to the center of the lens C. But P is already on THAT line, too, so the intersection Q is at P. It is on the sensor, so it is in focus in the final image. Damn, that still amazes me.



What about the Hinge? The quadralateral A-F-C-Hinge is a parallelogram. That, BTW, was the reason to take the line H one focal length in front of the lens, so the distance from F to C would be the same as from A to Hinge. So the line AF is parallel to Hinge-C, and THAT was vertical by construction. These lines being parallel, they "intersect" at infinity along with the vertical sensor plane (all parallel lines meet at the same point at infinity. No, really.) We call that good enough and declare the Hinge in focus, too.



We're done!



And not an equation in sight!

Matt
 
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I just re- read Dave’s excellent article on tilt and swing adjustments; thanks Dave-very helpful. On thing I realized is that my widest technical camera lens, at 28mm, is more than wide enough if I stitch with it; the lens effectively becomes a 21mm lens. And I was thinking of buying a wider lens…
 

MGrayson

Subscriber and Workshop Member
It's one thing to get the focal plane to hit the ground in front of you, but getting the slope right to hit the top of that mountain is a bit trickier.

There's a much quicker way of getting the DIRECTION of the focal plane. Once you get more than 50 feet away, the error between this approximate plane and the real one gets small.

The point B is where the infinity focus plane (parallel to the lens and passing through F) hits the sensor. Anything below that on the sensor (above that in the image!) is too close to the lens to be in focus. Not surprisingly, the line from B through the lens center C is parallel to the true focal plane. It emanates from the lens, so it may be 5 or more feet above the real plane. It will work fine for mountain tops!

It works because the B-Sceimpflug distance is the same as the C-Hinge distance. They're both cutting across strips of width f at the same angle!



Matt
 

vieri

Well-known member
That would have been my only question about all that. With non-bellows cameras, we (almost always) always have the tilting mechanism between the nodal point and the sensor, correct? While I am very comfortable in using tilt in practice, I am afraid the theory of that particular case might be messy and elude me - no wait, I am not afraid it might, I am pretty sure it does 😂 But perhaps it's better if I won't ever know it :eek::unsure:😂

Seriously now, thank you so much Matt for adding this to Dave's wonderful article. As a mathematician's son, and one who dabbled with numbers pretty much since I can remember, this was a great read. Again, that doesn't necessarily mean I understood all of it though 😂

Best regards,

Vieri
 

MGrayson

Subscriber and Workshop Member
That would have been my only question about all that. With non-bellows cameras, we (almost always) always have the tilting mechanism between the nodal point and the sensor, correct? While I am very comfortable in using tilt in practice, I am afraid the theory of that particular case might be messy and elude me - no wait, I am not afraid it might, I am pretty sure it does 😂 But perhaps it's better if I won't ever know it :eek::unsure:😂

Seriously now, thank you so much Matt for adding this to Dave's wonderful article. As a mathematician's son, and one who dabbled with numbers pretty much since I can remember, this was a great read. Again, that doesn't necessarily mean I understood all of it though 😂

Best regards,

Vieri
Vieri,

Thank you! I'm quite serious about the title of this thread. This is really not helpful to a photographer in the field, nor is it meant to be! Every treatment I can find of this stuff is full of equations, which are great (and necessary) if you want to calculate a number. But I am never happy until I can see a picture. So doing all this without any calculations was a long time goal for me.

As for tilting about a non-nodal point, it will affect framing, just as it would with a panorama, and it would affect infinity focus very slightly, but the theory then goes through as before.

We are so fortunate with digital that we can use trial and error in the field. Large format film photography was so frustrating that I gave up photography completely for 20 years.

Best,

Matt
 

ThdeDude

Active member
As I wrote in another thread, I agree with Dave that for field work an empirical, iterative approach is more suitable than using a theoretical approach trying to compute the required tilt.

But perhaps soon there will be a iPhone and app capable of generating a depth map accurate enough to compute the optimal combination of tilt, point of focus, and aperture required!
 

MGrayson

Subscriber and Workshop Member
As I wrote in another thread, I agree with Dave that for field work an empirical, iterative approach is more suitable than using a theoretical approach trying to compute the required tilt.
I couldn’t agree more.
 

MGrayson

Subscriber and Workshop Member
Dave pointed out a paradox to me and I want to talk about it. It's a very disturbing observation: Focus at infinity, the plane of focus is infinitely far in FRONT of you. Now tilt the lens ANY amount as small as you want. Suddenly, the focal plane is horizontal and very far below you. It's an honest-to-goodness discontinuity, and it's real.

So what gives? Two ideas are in play, and they both contribute to the paradox.

The first is that a lens focused at infinity is at the very edge of not focusing on anything at all. If any part of the lens gets any closer to the sensor, that part cannot be in focus. So the tiniest forward tilt puts the bottom half of the lens out of commission (and thus the top half of the image). All that's left is the bottom half of the image very far away. That sure doesn't sound like a horizontal plane far below us.

The second notion is that there is no photographic difference between a large distance and infinity. We live inside a photographic ball maybe 10 miles in diameter. Everything outside that may as well be an image projected on the boundary sphere. So the lower half of a plane infinitely far in front of us is really the quarter of this distant sphere below and in front of us - elevation 0 (the horizon) down to negative 90 degrees. But what is a plane 100 miles below us? We can't see behind the lens, so all that is visible is .. you guessed it -90 degrees up to the horizon. As far as we can tell, exactly the same parts of the world are in focus.

A third observation helps us with the seeming discontinuity in the upper half of the plane in front of us. Good Enough Focus (GEF) doesn't happen on a plane. It happens on a slab or wedge. That corresponds to a thin zone around the sensor. The thickness of the zone depends on how wide the cone of light focusing slightly in front of or behind the sensor. Another reason large apertures make for shallow DoF. Actually, the only reason. Never mind. So what we really see is what happens to the near plane of GEF as we tilt the lens.

This shows the depth of field (beginning of Good Enough Focus) with no tilt. The plane of focus is, unfortunately, infinitely far away.


THIS is what happens in the same setup with 2 degrees of tilt. Any smaller tilt and the picture would be a mile high. In front of the camera, things have not changed too much!



Matt
 
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ThdeDude

Active member
"It's a very disturbing observation: Focus at infinity, the plane of focus is infinitely far in FRONT of you. Now tilt the lens ANY amount as small as you want. Suddenly, the focal plane is horizontal and very far below you. It's an honest-to-goodness discontinuity, and it's real."

The apps Dave used for showing this uses a computational approach, i.e. it calculates. Since infinity is not a number but rather a concept, the apps may use a number as an approximation or make other assumptions as for infinity. This could explain any computational anomalies in regard to infinity.
 

MGrayson

Subscriber and Workshop Member
"It's a very disturbing observation: Focus at infinity, the plane of focus is infinitely far in FRONT of you. Now tilt the lens ANY amount as small as you want. Suddenly, the focal plane is horizontal and very far below you. It's an honest-to-goodness discontinuity, and it's real."

The apps Dave used for showing this uses a computational approach, i.e. it calculates. Since infinity is not a number but rather a concept, the apps may use a number as an approximation or make other assumptions as for infinity. This could explain any computational anomalies in regard to infinity.
That's the wonder of the real world. Infinities don't actually happen. See above for what happens slightly behind the infinity focus plane.
 
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