I think I'm getting the hang of this complex curve weirdness
So I made a paper pattern for a quarter sphere. I'll spare you my clumsy, picture-less explanation and just point to a globe instead. Notice the petal shape formed by the equator and two lines of longitude meeting at the pole? That's the sort of piece I drew. Knowing the circumference of the sphere, how many pieces you want to use (I used four because it's a good compromise between a smooth curve and too much fussy work), and that the points on the pieces all have to add up to 180 degrees, it was straightforward to draw pieces of the correct size and with the proper angle at the top.
I then kept enlarging the pattern until I wound up with a prototype that fits over my shoulder well, and I messed around trying to add the outward-curving flange at the bottom to make it look more like Zhang He's pauldrons.
I now have a paper pith helmet.
It really is not too far from the result I'm going for.
And it is a vast improvement over a deformed cone.
Math is fun.
I'll be sure to take a picture of the final paper prototype, assloads of masking tape and all. And I will share my knowledge with the rest of the class when I get it all figured out. Sharing is caring.