


A developable surface is s.t. you can lay it (or roll it) flat on the
plane (it may require you to give it a "cut" though...)

E.g., a cylinder, a cone, a plane (of course!) or any surface or patch
having vanishing Gaussian (intrinsic) curvature (i.e., with singular
Hessian, the matrix of 2nd derivatives for an adequate coordinate patch)
are "developable". In more technical words, a developable surface is
"locally isometric to a plane" at all points.


Think also of the sphere (or the earth) which in a non-developable:
whatever way(s) you cut it, you will not be able to lay flat any pieces
of it... (its intrinsic curvature is nowhere vanishing).

For more details on this look at any book on differential geometry
which treats surfaces (2D manifolds); e.g., M. do Carmo's book:

@Book{Carmo76Differential,
  author =      {do Carmo, Manfredo P.},
  title =       {Differential Geometry of Curves and Surfaces},
  year =        1976,
  publisher =   {Prentice-Hall},
  note =        {503 pages.}}

Enjoy!