radial rafters
I was
reading Fan of the Fan (VI)
and became curious about the mathematical basis for the spacing on the
rafters.
I
developed a
spreadsheet based on his numbers and cobbled together some
formulas to understand what is going on. I eventually developed the
following formulas:
 L is the length of the perpendicular (shortest) rafter
 s_{t} = L×q is the "tangent spacing" of the rafters. It is the length of the short side of the right triangles in his diagram
 θ_{0} = 0
 L_{n} = L / cos(θ_{n})
 θ_{n+1} = θ_{n} + atan(s_{t} / L_{n})
It turns out that the exact value of L does not affect the value of
any θ, and is only relevant when it is time to actually cut the
rafters. Any computer scientist can type up a function f(q,m) which
will iteratively compute θ_{m}.
If you want to figure out the proper q value to make
θ_{m} = 45° for any particular value
of m (the number of rafter spaces) I am somewhat confident
there is no closedform solution, so you'll have to experiment with
various values of q until you get it
right. Newton's
Method is an obvious choice and would allow you to develop the
carpentry tables Chris is referring to.

Chris Hall's diagram of rafter tip spacing 