Polygons
are used to model many modern-day problems
touching the fields of computer vision, robotics, physics, biology, and
chemistry, among others, and their use in each is motivated by the mathematical
properties and analysis that their structure entails. Computational
geometry methods can be applied to such models in order to derive properties
and solutions to problems in those fields, yielding inisght into problems
that might otherwise be considered unsolvable.
The study of
linkages
represented by polygons whose
vertices act as hinges and whose edges serve as rigid bars has been covered
in numerous publications, from which a
unversality
result has emerged: that every polygon can
be reached by another polygon with the same sequence of edge lengths
through motions that preserve edge lengths. The proof of this result
in papers covering the topic involves proving that every polygon can be
convexified
while preserving edge lengths, and employs the
reconfiguration
of the polygon into a "canonical triangle" with the crossing of edges
allowed.
First, it will be demonstrated that any convex
polygon can be reconfigured into any other polygon with the same counterclockwise
sequence of edge lengths without the use of a "canonical triangle"
intermediate state. The method described here will also be shown
to be the simplest one possible as each vertex angle decreases or increases
monotonically with time as the polygon approaches the desired configuration.
It will then be shown that the reconfiguration
of a polygon into any other with the same sequence of edge lengths is
possible using only
pivots
, the simplest possible move in 3 dimensions, and thus that the restrictions
on motions in 3D do not limit the ability to reconfigure a polygon in the
desired fashion. All proofs, lemmas and propositions that follow are
taken from [1] unless otherwise specified.