Calculate Pincushion

Try this in Excel:

1. First work out your throw ratio (TR):
Where H = Image height: TR = THROW * 0.5625/H

2. Then calculate your pincushion as a decimal fraction of your screen height (using what you asked for, a formula):
Pincushion Percentage (PP) = 4.21/(100*{TR}^1.910963)

(if you multiply this by 100, you get Pincushion Percentage)

3. Then the result you want, the Pincushion... the actual amount your image corners will be higher (or lower) than the image center:
P = PP x H

In your case...
This all works out to

• Throw Ratio (TR) = 1.75
• PP (Pincushion Percentage) = 1.45%
• Pincushion (P) = 0.84"
• BONUS FACTOID: Needs a 38' curvature screen to cancel P (40' standard size is OK).

What's the formula based on?
Some research I did into pincushion when designing my Xeitoptics cylinder lens.

I found that pincushion was a straightforward function of throw Ratio (TR) for my lens design. The formula was derived from establishing a whole bunch of accurately calculated pincushion results in a spreadsheet table and then performing a statistical regression on them. The formula above in (2) gave a "best fit" regression.

I wondered whether it worked for other designs, even prism systems.

So I then tried it with models of other designs that I had, including prism lenses and old-fashioned designs like Isco. The results from all types of designs conformed pretty well (within 1% of the modelling results).

I decided the formula was generally usable provided the throw ratio was not lower than 1.5 (yours is OK, at 1.75).

Ideally the best way to model these things is with optical design software. That's precise to thousandths of millimetres.

Next best is something like the spreadsheet I authored. That's precise to hundreths of millimetres.

Last best is the formula above. It's only precise to tenths of millimetres, and It tails off at really small Throw Ratios a little (still pretty accurate, though).

But be reassured, I modelled your setup using all three... and the results were all within 1/50th of an inch of each other.

I provided the formula because you asked if there was one. Given the caveats above, there is.
Edited by Aussie Bob II - 10/31/12 at 6:14pm

Good to see you posting Aussie Bob. Thanks for helping the users out.

I'm "Aussie Bob II".

There was someone called "Aussie Bob"?

Was he really an Aussie, was he?

DOH!! Presumptuous Americans, what can I say?

For myself, I like to be thought of as a "Citizen Of The World".

Well if I just gave you a weird formula you'd ask the question anyway: "How did you figure that out? How do I know it works? 'Why is it 4.21 times whatsitname?'"***

Anticipation is the name of the game.

*** Answer: I don't know. That's just the way it arrived.... "4.21". 10% of "42".

"42" is of course the answer to "Life, The Universe and Everything" (it's actually "42.1", but rounding claimed the decimal point, "42" is neater).

So you've got 10% of mankind's accumulated knowledge right there.

Not a bad effort for a lazy morning. Imagine what I could do if I was trying.