Umm.... Ah! You're setting X_2 = -(sun radius) so that I is all ones, rather than [1, -1; 1, ..., 1], yes? On 08/06/15 09:42, Fred Lunnon wrote:
With planets placed in cyclic order around the sun, and axis indicated by semicolon, the vector
X = [ r, s; p, ..., q; u, ..., v ]
of gear radii becomes essentially unique; and combined with sunset (offset) h , it specifies the train. While in a realisable train the radii (more properly, tooth counts) are positive integers, it becomes convenient to generalise them to be rationals. In particular, my sun radius s is normally negative!
Vectors X, Y of different trains are related by some sequence of concentric (coaxial) increments and rescalings just when
a X + b I = c Y + d I , where I = [1, 1; 1, ..., 1]
with a,b,c,d integers. X can be reduced to a unique canonical form via