Crown radius = how much arch in a segment ?

[Clark]

Yes.

Offhand I can think of two ways to give theoretical
figures for ideal, unloaded boards (OK, just one).

If you think of triangles, it's easy. [n] is the span, [r] is the
radius, [e] is the elevation at the center:

e=r-sqrt(r^2-[n/2]^2)

For a circle it's the same. The general equation is

(y-y_0)^2+(x-x_0)^2=r^2

If the span, or chord is [-n/2<=x<=n/2)], you might want its ends to be
at [y=0]:

y_0=-sqrt(r^2-(n/2)^2)

(note the minus sign) The equation for the circle becomes

y=sqrt(r^2-x^2)-sqrt(r^2-(n/2)^2);

it follows that at the center, for [x=0],

y=r-sqrt(r^2-(n/2)^2)=e

(identical to the solution with triangles)

For arc length - in this case the width of the curved panel where it is
attached to a crowned rib with a chord length [n]:

l_a=r*asin(n/r)

Of course, none of this applies to real materials. ;)


Clark