Development (flattened version) of a cylinder cut by a curved surface

[Swamp Thing]

This Instructable shows how to make a mouthpiece and reed for a toy wind instrument. https://www.instructables.com/Tiny-Tootophone/ (Warning: The article starts with a closeup profile of a middle-aged man's mouth and chin, with a raggedy unkempt mustache and beard that some viewers may find disturbing )

I would like to create the side view of the cut cylinder in Mathematica, and then print out the developed version on to some stiff card or onto transparency paper, which I would then roll into a mountpiece that has the right cut. How would one go about this?

Also, the reed itself should ideally be cut so that its outline matches the cylinder's cut edge when pressed over it. I could just trace it physically and then cut it, but it would be nicer to have Mathematica do it.

So given a function (or table) of the cut profile, how can I generate the shapes for cutting the cylinder sheet and the reed?

 

[jedishrfu]

While using Mathematica is commendable, I think you’re overthinking this. Eyeball it to make a reasonably similar cut and it should be good enough.

 

[Swamp Thing]

It's not that I want to copy the diagram from Instructables, rather I will cut various different profiles just to see how their voice varies. But once I have a nice one, I would like to replicate it consistently.

It is true that I may be overthinking it, but in a hobby I think it is OK to get into rabbit holes if they are fun and instructional.

Edit: The main thing is that printing, cutting and rolling up is much easier and faster than filing.

 

[jedishrfu]

You get a good cut place it on fine graph paper and take photo then get data points from that to make a curve.

 

[Swamp Thing]

Figured it out. A spreadsheet is handier than Mathematica in this case.

Col. 1: x along the tube in steps of say 0.5 mm

Col. 2: f(x) = cutting profile function as seen in side view (zero at center line)

Col.3: phi = arccos( f(x) / tube_radius ) = angular position of each point along cut line, as viewed from the end of tube

Col. 4: y = phi / (2*Pi) * tube_periphery = y coordinate on flattened sheet

Col. 5: = x

Col. 6: = y

Col. 7: = "L " & col_5 & " " & col_6

Insert column 7 into a boilerplate SVG file and import into InkScape. This will give one half of the shape to be cut; one can copy and flip in InkScape or concatenate column 7 with a reverse-ordered version of itself. If one wants to roll several times over, of course just copy and concatenate the first roll's data.