Abstract:
We thoroughly analyse the method used by Pogorelov to construct
piecewise-smooth tubular surfaces in $\mathbb R^3$ isometric
to the surface of a right circular cylinder. The properties of the
inverse images of edges of any tubular surface on its planar unfolding
are investigated in detail. We find conditions on plane curves lying
on the unfolding that enable them to be the inverse images of edges
of some tubular surface. We make a refinement concerning the number
of smooth pieces that form a piecewise-smooth tubular surface.
We generalize Pogorelov's method from the surface of a right circular
cylinder to that of a right circular cone.
Keywords:surface theory, surfaces in three-dimensional space.