An Alternative Interpretation of the Cone

It is possible to view the cone as an altered plane.
Simply consider a plane with less than a 2pi angle:

a cone

An arbitrary angle has been subtracted out of the plane and the plane rejoined so that its continuous nature is preserved.

The same process can be done with an angular increase:

inverse cone

This saddle type shape has a definite vertex. Informationally this structure contains more than the usual plane.
It does not appear to have an upper limit of angular increase. Here is a biplane:

biPlane

The biplane can be folded flat without any need for surgery.
The folding exposes how it was constructed:

foldedBiPlane

There does not appear to be any upper limit on how much angle may be added. Here is a triplane:

triPlane

The triplane can also be folded flat. It is capable of lots of rotation.
Informationally the triplane contains three times the information of the standard plane. These instances lay upon a continuum of angle.

According to the definition of curvature these surfaces all have zero curvature away from the vertex. At the vertex the standard cone has undefined positive curvature. The inverted forms have undefined negative curvature at the vertex.

The vertex is a special position on this structure.
Curvature is a concept nearby to the generalized cone yet they seem to be two different concepts.
The saddle is readily apparent in both yet I have not found any saddle with a vertex in existing constructions. If you are aware of such a construction I would greatly appreciate your feedback.