[Example images](https://i.stack.imgur.com/32U9C.jpg)

I need a real world solution to a reconstructive surgery problem!

A fairly commonly encountered clinical scenario results in the need to create a tube (from skin) in the shape of an inverted truncated right cone to reconstruct a defect in the pharynx after cancer resection.

The lengths of the defects are somewhat variable (8-12cm) as are the radii of the base of the truncated cones (upper pharynx, 2-3cm) and the radius where the cone has been truncated (lower pharyngeal defect, 1-2cm).
Obviously cutting the truncated cone along its perpendicular slant length would give a very simple template. However this is unusable in the real world as the width of the template means that the edges of the skin defect cannot be sutured back together after harvesting the skin. The maximum width of any template would have to be approximately 8cm.

I have created paper templates of the tubed reconstructions and then cut them in an approximate spiral fashion with a helical angle of between 30 and 40 degrees. This results in a template that can be transferred to the donor area (usually the thigh) from which the skin can then be harvested and the resulting defect closed with sutures without undue tension in the remaining skin. The harvested skin is then sutured along the spiral to recreate the truncated conic tube which is then transferred to the pharynx, sutured in at the upper and lower openings and the blood supply to the tube restored with microvascular techniques.

To my mind, this is an untidy method of planning and I would like to look for a general solution where I can plan a template based on the 2 radii and the length of the defect, and based on a conical spiral which gives a relatively long and narrow template pattern.

I have trawled the internet and have found the attached images (for calculating the length of Christmas tree lights) but no explanation of how the curves were drawn. There are many sites giving the formulae for conical helices but none that I can find showing how these can be constructed on an unfolded two-dimensional conical plan.

I am well aware that this is a verysimple problem for this site but I would be grateful if you would bear with me. I am unfamiliar with coordinate systems other than Cartesian but am prepared to put in some groundwork if necessary.