Determining Max Shift with Tilt applied for Ultrawides

O.K. So this is two posts in a row. I went and bought a macro lens (see FS and macro post) and then a 55mm Grandagon at the same time. I know, I know, buy one lens at a time, learn it inside and out, and then move onto the next FL. Sounds good in theory, but when some of these optics come up used for a great price it is hard to resist :D

So here is the Q. Without burning through a pack of film, is it possible to calculate the amount of fall required when using a rear base tilt (knowing the angle of tilt) in order to maximize the already limited image circle? The amount of shift available without tilts has already been tabulated elsewhere, so no worries there, but this all changes when tilt is applied. I find it impossible to tell from the GG when vignetting has occured, so unless there is a handy calculation available, it seems to me either trial and error, or taking a more rigorous approach and for each degree of tilt, increment fall and record an image. Is the relationship linear beween tilt angle and maxium fall, such that only two tilt angles need be evaluated?

Me, I shoot a Polaroid to make sure. Was never good a math.

If you have cut corners on your groundglass, look to see if you can see the full aperture through each corner to check for vignetting. If you can't see any of the aperture, you have vignetting. If you can see part of the aperture in the shape of an oval or an American football, then the corners will be darker than the center, but not completely vignetted. If you can see the lens shade or the filter threads, that would also be a source of vignetting.

If you don't have cut corners on your groundglass, you can sight the corners of your groundglass through the lens the other way around to check for vignetting.

You should not need to use fall (or rise) when using rear base tilts within broad limits. That's the beauty of rear movements with lenses of limited coverage. The middle of the ground glass stays on the optical axis of the lens even when the GG is tilted (or swung). The down-side of back movements is the change in the geometry of the image. The ground-glass should show you if this "distortion" is desirable or not. Image stretching can become rather gross with very short focal length lenses.

The story for front tilts is very different. Forward tilt of the lens aims the optical axis higher on the ground glass and some lens fall may be needed to get it back to centre. This can be checked by looking through the cutoff corners of the ground glass to confirm that the exit pupil of the stopped down lens can be seen from all four corners. If not, more (or less) fall may be needed. An alternative check is to look through the stopped down lens from the front to confirm all four corners of the ground glass are visible.

In exchange for the advantages of front tilt/fall and no distortion you get to fight a compressed and recalcitrant pleated bellows or a saggy bag bellows plus the neck wrenching postures you may have to adopt just to see if you have coverage.

I can calculate what you want using well known geometric principles, but it does require knowing a bit about the mechanics of your camera and how you go about things. Let me first describe geometrically the two situations where the back is tilted and where it is untilted. The situations is more complicated if the front is titled , so let me assume for the moment that you don't do that.

Since the front is not tilted, the image falls within a circular cone centered on the lens axis which initially is centered on the frame and perpendicular to it. The size of the image circle is determined by the distance of the lens to the rear standard. That distance is measured from the rear principal point, so if you want to make all the calculations correctly, you should know where that is for your lens. I will ignore that issue for the moment, and assume you have measured that distance---usually called the bellows extension relatively accurately for the untilted case. (See below for determining where the principal point is for your lens.) With the back untitlted, the amount of allowable rise or fall is determined by how far up or down you can move the frame within the image circle. You can determine that by drawing a circle of the right size, putting a cardboard rectangle the same size as the frame and moving it within the circle. I can also calculate it if I know the size of the frame and the diameter of the image circle. It would depend on whether the frame were in portrait or landscape orientation. But you appear to have determined that by trial and error, so let's try to work with that.

Now what happens when you tilt the back. You say you use base tilt on the back. The frame is initially tilted about its base, but you may also refocus, which will change the position of the frame relative the the lens (rear principal point).
What happens next depends on how you do rise or fall. Let's assume first you are doing the rise/fall on the rear standard., and that occurs not in the plane of the tilted frame but in the original vertical plane. (Whether or not that is true depends on the mechanism of your camera.) Now measure the distance of the bottom of the frame to the lens (rear principal point). Divide that by the initial bellows extension. The image cirlce in a plane perpendicular to the lens axis will be smaller than the initial image circle by that ratio. But the bottom of the frame will still be at the same vertical height. The amount you can drop it should be reduced by that ratio.

If you wanted to calculate how much rise you could use in this situation, the argument would be similar, but with one complication. Instead, you measure the distance from the lens (rear principal point) to the top of the standard, and use that to cacluate the ratio, which will now be larger than one. Multiply the acceptable rise untilted by that ratio. But, because of the tilt, the top of the frame dropped by a certain amount, so you have to add that amount on also. You can calculate how much the top of the frame drops using simple geometry from the measurements you have, but you can also probably measure it accurately enough. Just make sure you noted the vertical poistion of the top of the frame before the tilt and using a scale estimate how much it has dropped after the tilt.

You can instead just use rise on the (untitled) front standard by the amount of allwable drop on the rear stndard or drop on the front standard by the amount of the acceptable rise on the rear standard.

If instead, "rises" and "drops" of your rear standard are movements in the plane of the tilt, the situation is more complicated. That is because when you move the frame in that plane you also shift its position relative to the lens. I could tell you what to do in that case with some more effort, but it would seem simpler just to use rise or fall on the front standard as indicated above.

Finally, let me tell you how to locate the rear principal point for your lens. Let's assume you have access to lens data for the lens. That will include the focal length and the rear flange focal length (which may be described by a different term). For a wide angle lens, the rear flange focal length is usually somewhat larger than the focal length, which means that the principal point is closer to the lens than the front of the lensboard. The distance it is set back is the difference between the flange focal length and the focal length. (For telephoto lenses, the principal point is in front of the lensboard by the difference between the focal length and the rear flange focal length.) So when measuring your bellows extensions, you have to subtract (add for telephoto lenses) that difference to the measured distance to the lensboard.

If you don't have access to lens data, you can do as follows. It assumes you acurately know what the focal length is, but for most lenses it should be very close to the marked focal length. Focus on a distant object, so far away that you are essentially focused at infinity. Now measure distance from the gg to the front of the lensboard. I will assume for your lens it is greater than the focal length. Subtract the focal length from that distance to find the displacement of the principal point from the lensboard.

By the way, one way to check the focal length is as follows. Measuring very accurately the size of the image on the gg, set the distance so the magnification is 1:1 (image size equals subject size). Measure the distance between the standards. Now do the same thing when the magnification is 1:2 (image size equals half subject size), and measure the distance between those standards, The focal length should be twice the difference between those two distances. You can also use this method to determine the position of the rear principal point. When the magnification is 1:1, the distance from the rear standard to the prinicipal point is twice the focal length. So measure the distance between the standards in the 1:1 case, and subtract twice the focal length from that. That is the set back.