Those that have been following my DOFIS and TiltSim developments will be aware that this has been a cathartic journey for me: as I tried to create a Magic Lantern focus tool for my normal and tilted lenses.
On the way I've mentioned others that have helped me in my understanding, but, to date, I haven't mentioned Tim Parkin over at OnLandscape.
I subscribe to OnLandscape and can thoroughly recommend it if you are interested in landscape photography: BTW I have no affiliation to OnLandscape, ie I pay my subscriptions like others.
Tim has a great video introduction to Tilt/Shift lenses: https://youtu.be/SEo-zXOzEBQ
The only thing I think Tim over simplifies, IMHO gets wrong, is the placement of the lens plane in a DSLR T/S lens.
As we know from Scheimpflug's original patent, the so-called Scheimpflug line is split between the 'rear' and 'front' principal nodes:
Which in DOFIS and TiltSim 2 looks like this:
In other words, the Hinge line is located relative to the front principal, ie we need to know the lens hiatus between the front and rear principal nodes. Note, the above sketch, and the DOFIS model below, illustrates a retro focus lens arrangement.
Some may say: why bother, surely a thin lens approximation is OK?
Using our split lens model, we know the front principal is located as follows:
In other words, according to the DLSR T/S design, the front (or object side) principal can vary considerably, ie with the focal length, and relative to the rear (or image side) principal.
Using a thin lens model will lead you astray ;-)
As was discussed in the last post, the position of the front principal, from where we measure the hinge height (J), parallel to the sensor plane, will vary between a retro or a tele designed T/S lens.
Because of 'lens breathing', ie focusing, the front principal will also move, according to whether we are focused at infinity or at the closest point of focus.
As an example, let's take my 24mm TS-E, with an M of 0.34 at an X of 230mm.
The hiatus between the two principal nodes is found from solving the equation for t above. Which gives a hiatus of just over 103mm. That is the front principal is positioned 103mm in front of the rear principal.
The rear principal, of course, is positioned at (1+m)f; where m is the magnification and f the focal length.
As for lens breathing, this is 24*0.34, ie some 8mm. That is, the front principal is positioned at 24+103 = 127mm, from the sensor, when focused at infinity, and at 24+8+103 = 135mm, when focused at the minimum focus.
Whereas, on a Canon 135mm TS-E, with an X of 486mm and an M of 0.5, the hiatus is about -121mm!
That is the front principal can just be in front of the sensor plane when focused at infinity, ie at 135-121= 14mm in front of the sensor. Whereas focused at the minimum focus, the top of the hinge plane (the J height, parallel to the sensor plane) will be positioned, along the tilted optical axis, at 1.5*135-121 = 81.5mm from the sensor.
Bottom line: first, if you are interested in landscape photography, then think about subscribing to OnLandscape. Second, it is easy to estimate the position of your front principal, and you only need to do this once!
As usual I welcome any feedback on this post.