3 useful cones, part 2

Related the post "3 useful cones" (part 1). When unrolled, these cones make up 1/4, 2/4 and 3/4 of a circle.

If you look closely, you can see all the lines/squares line up perfectly...

Developable fork

A developable fork is way to connect three developable surfaces with each other. They are joined by a flat triangle, tangent to the three surfaces (and tangent to the three edge curves). The developable fork is very useful when creating volumes from developable surfaces.

Some more images and info here.

Self intersecting developable surfaces

Some sketch models:

The idea is to create one continous, developable, self intersecting surface that can create a useful space (a room) and perhaps extend to become a whole building. The surface is allowed to overlap itself when unrolled.

Wrapping cones

Some more testing with wrapping cones.

This is what the unrolled strip should look like. The yellow lines represent a fixed change of direction.

Collar on the cone

Today I discovered that the loop I have been trying so hard to understand may in fact be a part of a cone!

The loop sits like a collar on the cone.

The loop photographed from the cone focal point.

This is a bit of a breakthrough! It's interesting that such a simple paper model can be so useful. In CAD, how would you constrain a straight strip to follow the shape of a (developable) cone?