The head on my forthcoming Viking model is connected to the body so that it can nod back and forth and turn left to right. I have used a rotating tube to connect the neck to the head. To make the horizontal tube a good fit in what is effectively a vertical tube that is the head is going to require a little maths.

This 35mm circle represents the top view of the head. The shaded pink area represents the 20mm diameter horizontal tube. Note that the ends of the horizontal tube need to be curved for a good fit in the head, the curves are actually sine waves.

Luckily I have a useful pdf with a single sine wave which I keep on file for just such occasions. I took the file, lined up three waves in a row and trimmed them so that there were two full cycles up and down.

Within Illustrator I fitted two copies of the sine wave, one of them inverted, into the 3mm deep rectangles at the top and bottom of a suitable size template.

To finish off the net I’ve added tabs for gluing.

Once the part is printed out I cut it out and glued it together. Note the smoothly curved ends.

The finished piece fits nicely into a head sized tube. Next step – complete the layout of the head.