Is there an agreed upon ratio of the mouth/window opening where the angled lip is vs the cross section of a flute? For example a theoretical flute with a internal diameter of 2 units would have a cross sectional area of PI units. The area of the square/rectangle mouth/window would be PI/4 units. Thus the ratio would be:
Area(cross section) = 4 * Area(mouth/window opening)
Thanks!
Are you talking about “a low D flute has a set length and bore size, so if I am going to make a High a flute I need to scale all of the dimensions?”
I think not, the embouchure usually is with in a range that is comfortable for the player and moved slightly up and down the bore of the flute to compensate any effects on the tuning. I would not be surprised if a piccolo and a flute of the same maker had similar sized embouchure (allowing for limitations of the piccolo’s bore size), but the ratio to the end being slightly shorter for the piccolo (all other ratios equivalent). The smaller the embouchure, (I believe) the shorter the over all sounding length.
Some flutes have rectangular, round, or elliptical embouchure and I think the over all size of the window, makes a slight difference to whether it is sharp or flat given the same length bore. There are some ball park calculators out there that take try to account for this.
Thanks for the feedback 
I guess in the end it is try and see…
Yes.
There has been some work in regards to your question.
I have no idea to the equation used but you can’t do this without the math.
It ain’t great but it certainly works.
Er… the link was to a robotic flutist is that the link you meant to send?
Thanks 
I’d think so…
If you plan on trying to make one, check out Doug Tipple’s site along with the PVC flute site that I linked above. No sense reinventing the wheel.