To those of you who hate analysis, engineering and
especially maths, but prefer honest sweat and shavings,
I say: Move along, nothing to see here.

Hmmm. Gone kinda quiet. Anyway...

I write concerning jointers, lengths and planing straight.
To start with, some breathtakingly poor ASCII art.

        ___----^^^^^^----___
    ..--         |h         --..
   /_____________|______________
  A      L      |X
                |
                |
                |
                |
        
       |r
                |
                |
                |
                |
                |
                |
                |
                |
                 O

This is intended to show an arc of a circle of radius r,
with a chord of length l. The distance from the centre of
the chord to the apex of the arc is h.
There are radii of the arc at the end and centre of the chord.

This diagram (and the analysis reliant on it)
represents planing under the DODGY assumptions that:
* planes are perfectly flat
* do not deflect under load
* have blades in the middle of their soles
* make contact at toe, heel and blade whilst cutting a circular arc
(the diagram is also upside down as far as planing concerned)

On with the show (from bad ASCII diagrams, to ASCII maths - yech)

Examining the right angled triangle O, X, A
we have:
(r - h)^2 + (l/2)^2 = r^2 (pythagoras)
expanding, and simplifying in stages:
r^2 -2hr + h^2 + (l^2)/4 = r^2
-2hr + h^2 + (l^2)/4 = 0
h^2 + (l^2)/4 = 2hr
r = (h^2 + (l^2)/4)/2h
r = h/2 + (l^2)/8h
Assuming h is much less than l (and we're comparing blade setting
with plane length here)
r = (l^2)/8h

This equation is also the one need to work out radius versus centre camber
for scrub and jack irons. Example: for a 2" iron, desirous of a camber of
1/32 we have radius = 2^2/(8(1/32)) = 16"

Breaking from algebra into numbers for a moment:
A 14" Jack plane taking 4 thou shavings gives a radius of 49000 inches. Which
sounds a lot,
until you move to a well tuned jointer (#7 jeff)
22", 2 thou shavings, 242000 inches radius. Wow!

But what do these radii mean? Well, we have a formula for moving between arc
heights,
chord lengths and radii, so let use it. More algebra.
(you might want to transcribe this to paper, and use subscripts - it's
much clearer)
h(board) means h on the board. l(plane) means length of plane etc.

h(board) = (l(board)^2)/8r     (1)
r = (l(plane)^2)/8h(plane)     (2)
substituting (2) into (1)
h(board) = (l(board)^2)/8((l(plane)^2)/8h(plane))

(take my word for it, or use paper, but this boils down to...)
h(board) = (l(board)^2)/(l(plane)^2)) * h(plane)

switching symbols, where P is length of plane, B is length of board, S is set
of plane
and G is gap between the board and a straight line we have the "jointing
equation"

G = S(B^2/P^2)

Which is almost interesting. Feeding in numbers, and our 2 planes:

4 foot board:
jack plane: .041 inch
jointer .01 inch

8 foot board:
jack plane: .164 inch
jointer .038 inch

I actually worked all this out for my own curiousity, which may speak volumes
about me. If it interests, intrigues or amuses fellow galoots, my cup
will be full indeed.

        BugBear.