There’s probably a more elegant way to describe this problem. Especially for ∠ψ.
The math and geometry is beyond me, but
has it right.
If you are too cheap for that route, a reasonable substitute template may be constructed this way:
Wrap trace paper around a like diameter object of the larger cylinder. Determine the centerline of intersection for the smaller pipe, and mark it. Take a child’s compass (the kind with a cam-clamp for holding an ordinary pencil) and set it to the radius of the smaller, intersecting pipe. Place the compass fulcrum on that mark and rotate…
Easier said than done, two people are handy for this procedure because: You must maintain a perpendicular orientation of the compass to the larger pipe throughout this operation,* and* you need to allow the pencil to “slide” in order to maintain contact with the curvature of the trace paper.
Remove the home made template and transfer it to the flat stock. You’ll still have some filing to do after you radius (bend) the cylinder before the smaller pipe will fit, because of stretch. Metallurgy often doesn’t agree with simple mathematics.
You also have to take into consideration the metal thickness. The sides of the hole will be angled in from the nominal dimension at the surface of the cylinder with the hole. So your pipe won’t fit in unless you grind this out or calculate an allowance.
Hence the aforementioned necessary filing.
Nice work clarifying the thinking. Easy to follow.
IANA machinist nor sheet metal worker, but I believe you missed one more degree of freedom.
Your first bit “… , if θ < 90°, …” is IMO unneccesary. The offset angle that you labeled ψ applies just as much at θ = 90.
My thought is :
A) the centerline of the small pipe touches the surface of the large pipe at some point. Which may or may not be directly on a radius extending from the large pipe surface to its centerline. Said another way, the small pipe centerline *may *pass through the large pipe centerline. It might, but it doesn’t have to.
B) the small pipe centerline touches that large pipe tangent surface at some 2D angle.
So we need to fully specify both A & B to have a complete solution.
B is easy: As you say, in large-pipe-centric coordinates we need to specify the two angles θ and ψ co-planar and perpendicular to the large pipe long axis.
A could be expressed as an angle in large pipe centric coordinates or as a linear offset in small pipe centric coordinates. But it’s a variable independent of θ and ψ.
When we get done getting this right let’s consider the case where both pipes are curved (or 3D spiraled along) their axis as they come together. :eek: 
Ahh yes, good point, put another way, the two axes do not need to intersect (axis of the pipe, and axis of the cylinder).
As for curved pipes, we will need a bottle of good whiskey while we ponder that.
Plus some cold pizza, a large whiteboard, and 6 colors of markers, one of which is mostly dried out.
Those were the days!! 
Whiteboard? Rookie kid.
Talk to me when you’re breathing chalk dust. 
Well, at least one of the intersecting surfaces should have to be developable (possible to cut open and flatten). This would, however, include non-circular cylinders and cones, as well as some weird things.
I attended college in the late '70s. White boards were just coming into use in the newest buildings. Plenty of chalk dust to be had in the older ones. Overhead projector transparencies were also typical for the younger, more technologically advanced instructors.
I attended college in the early 80s. Good morning to you, sir. 
No whiteboards then at UCSB. Lots of blondes in bikinis, though. Not that I ever noticed.
USC. The supply of babes was similar, but you had to drive to the beach to see the bikinis.
And you had to get through Watts before getting to the beach.
OTOH the shack down the street sold the finest kitty tacos in the whole county.
Yes, this was a relatively elementary exercise in drafting when I was in high school.
With a 3-way orthographic view of the finished product, you would divide the smaller cylinder into an series of angles. draw the intersection points
This - the first diagram part-way down - illustrates the process.
The side view and overhead view/end view give the point locations. The fact that they are laid out in equal angles means equal distances on the flat sheet.
https://en.wikisource.org/wiki/Sheet_metal_drafting/Chapter_4
the more points, the more accurate. We’d fill in the curve using a French curve trying to match several points at a time.
Presumably you can do the whole thing much more simply (and accurately) in a CAD program nowadays.
Awesome link there md2000. Thanks.
Yes md2000, thanks for that!
I agree great link!
Thanks.
Note on the large pipe, which OP is all about, the spacing for the data points on the flat pattern is NOT equidistant. The distances are measured from the diagram’s AB, BC, CD, etc.
Those were the days… you could get some very elegant approximate solutions from 3-view diagrams with T-square, compass, triangles and a sheet of paper.
And wait till sometime maybe next week for a solution to come back from drafting/engineering…
Or, you could get out the kiddie compass and have the pattern laid out in fifteen minutes. Fabed & welded by lunch.
I appreciate the complexity of your Wiki cite, and the feelings of elegance it inspires in draftsmen and nerds, but sometimes in the real world, you just have to make due with what you have at hand to get the job done.
Either way, there will be grinding involved.