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
- st = L×q is the "tangent spacing" of the rafters. It is the length of the short side of the right triangles in his diagram
- θ0 = 0
- Ln = L / cos(θn)
- θn+1 = θn + atan(st / Ln)
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 θm.
If you want to figure out the proper q value to make
θm = 45° for any particular value
of m (the number of rafter spaces) I am somewhat confident
there is no closed-form 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.
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| Chris Hall's diagram of rafter tip spacing |