Why is
the tube rigid?
by Marek Kordos
Take a surface and at some point
cut it with all the planes that contain the straight line orthogonal
to the surface at that point. Among the curves so generated there
is one with the greatest curvature and one with the least. Even
more: if the curvatures of those curves are different, the curves
with extreme curvatures are orthogonal. Gauss has proved (in what
is known as theorema egregium, the "wonderful theorem")
that when a surface is being bent so that no curve on the surface
changes its length, the product of extreme curvatures remains
constant.
Thus if a surface (e.g. a tube)
is made of some flat material, then at each point one of its extreme
curvatures must be equal to 0, to preserve the constant value
of the product, equal to 0 on the plane (where the curvatures
are equal to 0). This implies that in order to curve a tube (i.e.
to bend the generator of the cylinder) you must first stretch
the material it is made of.
