Natural Science Forum / Physics / Research / October 2007

Last night I picked up a book regarding the theory of gearing.  That is,
a mathematical formalism for the action of gear mechanisms was
developed.  The author introduced an formalism that at first seemed very
odd, but upon reflection seems quite elegant.  I am wondering if this
has found other applications.

The first couple of chapters were a straightforward introduction to
[x,y,z] coordinate transforms, and presented in the usual 3 x 3 matrix
formalism.  Then the author discussed an envelope of coordinate
transformations.  That is, a coordinate system that moves along a path
parameterized by a quantity 'm'. One could think of this parameter as
'time', but one can also parameterize the sequence of transformations
for a helicoid or some constrained path in space.  Then the coordinate
transformation relating the original "m_0" coordinate system to an
arbitrary coordinate system "m" is given by a 3 x 4 matrix (which is
then turned into a 4 x 4 matrix with the coordinate specification given
as [x,y,z,m]).

To me this was an interesting way to think about a "fourth dimension":
as an envelope of 3-D spatial coordinates, not nessessarily 'time'.  In
continuum mechanics, one speaks of deformed and undeformed coordinate
system, but I have not seen the deformation process parameterized before.