I have a rather old lamp with a plain conical shade that has fallen apart and two thoughts occurred to me: I could make one of those, and if it's a cone, I could use a conical map projection to decorate it.

So I measured it, and started playing with the idea in Geogebra.

lampshade-map.png

The angles on the right are the standard parallels for an equidistant conic projection. The angles on the left are the parallels where the top and bottom of the shade would be.

Initially I had used the top and bottom of the lamp shade as the standard parallels, but then I realized I could reduce the distortion by moving the standard parallels in a bit. Hence the points F and G. I then tweaked them to get the distance between the projection surface and the sphere about equal at the top, bottom, and middle.

So now I just need to decide if I want to use some real world data or a made up world, and make a map out of it using this projection.