So I understand that given a throw, there is fixed curve associated with it, in which the focus is precise.
If for the same throw, a different curve was used (e.g SI Screens make 25ft and 35ft as standard), what would you comment on the focus?
If you wanted to purchase a commercial screen fixed at a "standard" curvature, you could use my spreadsheet and the "Goal Seek" function of Excel. Fix as a constant either "Throw" or "Constant Screen Height" and then use Goal Seek to calculate the other variable.
As to astigmatism correction (or in plain language: focus), the lens will either be designed for best focus on a flat screen or a curved screen. In either case, there will be some small trade-off in sharpness from edge to edge if you vary the curvature of the screen.
Say my spreadsheet calculates a sagitta of 79mm (as in your example). Now project onto a flat screen and focus as best you can for edge-to-edge performance. Now take a piece of white card, walk to the edge of the screen and hold it 79mm in front of the plane of the screen. The pixels will be slightly softer on the white card than they would be on the flat screen at the same extremity. This is because you have in effect two screens: one 79mm (3.1") closer to the projector than the other. You will probably have to adjust your focus, by minutely sacrificing some centre detail to improve the image at the edges as displayed on the white card. Not by much, but a little.
If you can actually do this experiment, do so. It'll put your mind at ease.
No lens can focus perfectly on two different throw distances, except if it's been designed to focus on a curved screen... in which case it'll have slight problems with a fla
The question is "How much difference?".
The answer is "Not much at all". If you can see pixel tiling edge-to-edge on a flat screen focus, you should be able to see it on a curved screen focus. It's just that the curved screen focus point will be slightly different than the flat screen focus point if you really want to be picky about it. More than a foot or two away from the screen the two focus points will appear just as good as each other even if you don't adjust.
My experience is that projector lenses themselves (i.e. no anamorphic in the system) fall off somewhat when not perfectly straight on to the centre of the screen, and even then they can be uneven in focus. This is especially true of 3-chip projectors, as there is too much to line up - prisms, polarizers and imaging chips - to get everything perfect (except if you want to pay $0000s more for a "top of the line" 3-chipper... and I've still
heard some horror stories). Because Home Cinema projectors are generally dimmer than commercial cinema projectors, and because HC projectors are in competition with very bright flat screen TVs, the manufacturers tend to make them as bright as they can by the lazy method of using lower f/-numbers on the projector lenses, which means the image is starting out using too much glass to achieve perfect focus across the screen. The wider the beam angle, hence the lower the throw ratio, the more edge-to-edge performance suffers. Remember: this is without an anamorphic lens, just the raw projector lens.
As a general rule, the smaller in scale anything is, the tighter the tolerances have to be to get near-perfect results that match the expectations of the user. Home Cinema situations test the extremities of optical projection design. The beam angles are large, much larger than in commercial cinemas. The throws are much shorter. And here's the killer: the ambitions of the users are usually much bigger. Add all these together and perfection suffers. It's up to you to decide where your exhaustion point is, where you're prepared to say, "That's great!" and stop worrying.
Having said that, I don't want to scare you. I'm just saying that "perfection" usually boils down to what you're prepared to accept. A well-designed cylindrical lens will certainly and quite noticeably
outperform a prism system, when you take focus adjustability, better image geometry and, say, ease of use into account. The two examples you cite above are both over 2.0 in throw ratio, so you should get a pretty good result whether your screen is flat or