Shapes that can be made with a bandsaw Is anything known about surfaces immersed in $\mathbb{R}^3$ with the following property? 

$S$ has a definable interior and exterior, and any point on $S$ has a tangent line which can be continuously deformed (rotation, displacement) away to infinity while never passing through $S$

 A: The set of objects that have no blind holes or blind furrows?
By the way, the line being a tangent is not strictly necessary.  It is not necessary for the surface to be continuous, as the theoretical band saw can be retracted and plunged in for each surface point separately.
It is sufficient that every point on the surface has at least one infinitely long line intersecting the surface at that point, such that the line be continuously moved to infinitely far from the surface without intersecting the volume enclosed by the surface.
Obviously, this line is the position the theoretical band saw could be used to cut that surface point (and possibly others) from a sufficiently larger volume.
Points on the surface that have a well-defined normal vector do have an interesting property: the local curvature is such that in at least one direction, the curvature is either parallel or away from that normal vector.
While this is may not be useful for mathematicians, for people considering whether some specific object can be wholly cut with a bandsaw-like tool, that rule gives a necessary but not sufficient condition for bandsawability.
That means a computer aided design tool (like anything from Fusion 360 to SketchUp) could easily provide a warning for objects that cannot be produced using bandsaw-like tool alone.
