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Delaunay showed in 1841 that
any surface of revolution of constant mean curvature in ℝ3 has as its profile curve a
roulette — specifically, the curve described by the focus of a quadric rolling on a
line. Here we introduce a notion similar to the roulette that we call the
treadmill sled, and we use it to provide a dynamical interpretation for the
profile curves of twizzlers — helicoidal surfaces of nonzero constant mean
curvature.
The treadmill sled is connected with a change of variables that allows us to solve
the ordinary differential equation that produces twizzlers in a fairly easy way. This
allows us to prove that all twizzlers are isometric to Delaunay surfaces; this is similar
to work done by do Carmo and Dajczer.
We also provide a moduli space for twizzlers and Delaunay surfaces that shows
the connection of each surface with its dynamical interpretation, and we explicitly
show the foliation of our moduli space by curves of locally isometric CMC
“associated surfaces” analogous to the well-known helicoid-to-catenoid deformation.
Our dynamical interpretation for twizzlers also allows us to naturally define
the notion of a fundamental piece of the profile curve of a twizzler, which
yields the fact that, whenever a twizzler is not properly immersed, it is dense
in the region bounded by two concentric cylinders if the twizzler does not
contain the axis of symmetry, or dense in the region bounded by a cylinder
otherwise.
Using the change of coordinates induced by the notion of the treadmill sled, we
also provide a dynamical interpretation for helicoidal surfaces with constant Gauss
curvature, and we find an easy way to describe Delaunay surfaces by a relatively
simple first integral.
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