projecting a cylinder acutely onto a cylinder

[neilbourjaily]

I am building a cylindrical core for a metal-melting furnace. It will be gas fired and eventually may get a waste oil burner. In forming for the burner I need to project a small cylinder at less than 90* onto the inner cylinder form of the core and onto the outer cylinder form of the core. Common sense, web research, as well as Error and Trial efforts, clearly demonstrate the projection to the inner cylinder is an ellipse and the outer cylinder is also an ellipse, though less ovoid. I could continue with a hack attack approach and eventually get it right. I’d rather do this with some finesse. The burner must enter the core acutely so as to cause swirling of the flame.
I have visited Wolfram and some other internet sites, however, I cannot find clear and workable direction for projecting the ellipses. They all want me to start with the core forming cylinders, that are to be cut, laid flat, which I cannot do with the Sona tube forms I am using. Please forgive me for not coming to the masters of Homemade Tools sooner. I was trying to figure it out for myself.
I come now, hat in hand. I ask for clear and actionable instruction on projecting the acute intersection of a 2” diameter cylinder into nested cylinders, the outer being 10” and the inner being 8”, their obviously being separated by 1".
For those who wish to know, the refractory material is Harbison Walker International Kast-O-Lite 30 LI Plus. The dimensions are for an A8 crucible.
Thank you in advance to you who answer.
Neil

[mklotz]

A typical method of laying out the cuts is to use a program to generate a paper pattern. Said pattern is then wrapped around the pipe(s) and used to mark the cut. There are numerous versions of these programs available on the internet. Two examples are here...

https://www.blocklayer.com/pipe-miter.aspx

Tube Joint Pattern Generator

but, by all means do a search yourself and see what is best for you. Google the search term "tube angle cutting templates" and have fun.

[neilbourjaily]

Thank you, mklotz. I am a reflective learner, meaning the obvious comes to me after I have found myself stumped. Not long after I posted, it dawned on me the easiest thing to do is to produce a template to lay over my cylinders. I particularly like the tube joint generator.

[hemmjo]

Found this thread while looking for another thread I remembered seeing here a while ago. Someone showed a simplified way to make a paper pattern, was not as precise as the links listed above, but was close.