The formula used to calculate the 'excess area' is A * (2X + Y + 2Z)/6.  Does anyone know how this formula is derived?  It's actually quite hard to work out the precise effect of the X,Y & Z dimensions on the sail area so I wonder if this is just a simple empirical formula that seems to work fairly well.

I have no idea how the formula was derived in the first place.  However in terms of a simple way to work out the effect on sail area calculations.  Try using the measurement and certification spreadsheet which you can download from the IRSA website.  You can enter the measurement figures and then play with the data and the calculation shows you if there is any excess area and what the effect of adjusting a X,Y or Z dimension is instantly.

Thanks but that's not really what I'm asking.  It's easy enough to work out the effect of X, Y & Z on the certified sail area via the formula.  What I'm trying to understand is how the formula was obtained in the first place because it doesn't appear to relate to the actual geometry of the additional area (well not that I can see anyway).

Probably worth contacting one of the class measurers.

Graham Bantock has explained the formula for the excess area of a Marblehead leech roach, as below.

Edited November 3, 20214 yr by Lester Gilbert

Graham's original answer, posted on his behalf.

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The formula is a straightforward use of Simpson's Rule.

Imagine the leech roach represented by a graph with 5 ordinates at equal spacing. Let's call the ordinates z0, z1, z2, z3, and z4. The length of the leech is L.

The area under this curve - the area of the leech roach - given by Simpson's Rule is:

(L/4) * ( z0 + 4 z1 + 2 z2 + 4 z3 + z4 ) / 3

The first term, (L/4), is the spacing of the ordinates, the second term in brackets is the sum of the ordinates each multiplied by the Simpson multiplier (1, 4, 2, 4, 2, 4, 2....... 4, 1), the third term is the divisor 3.  It will be obvious with a little though that Simpson's Rule works only for odd numbers of ordinates. For even numbers the trapezoidal rule is used. For special numbers of ordinates there are 'other' Simpson's multipliers but I'd have to check my uni notes (or Google) to discover those.

For a Marblehead leech roach both z0 and z4 are zero, so this becomes:

L * ( 4 z1 + 2 z2 + 4 z3 ) / 12

In the case of the Marblehead excess sail area calculation we do not have the leech length, L, but we do have the luff length, A. And we are interested in an excess area gained by excesses of the cross widths over the permitted cross widths. Those excesses are called X, Y and Z. So, if we replace L by A and z1, z2 and z3 by X, Y and Z we get the following:

A * ( 2X + Y + 2Z ) / 6

Voila.
 

Thanks for that.  Hadn't thought of Simpson's rule.  There are a couple of approximations in the application here though.  Firstly the quarter widths do not divide the luff into equal parts, in fact the widths of the 'steps' are dependent upon the value of X,Y & Z themselves. Graham's sketch seems to acknowledge this as they are not drawn on the 'step' lines.  Secondly, the relationship between the luff length and leech length changes if the height of the clew varies.   Typically conventional rigs have the tack and clew at the same height i.e. the boom is horizontal.  Swing rigs tend to have the clew higher than the tack: to reduce the risk of it hitting the water.  By using the luff rather than the leech, sails with a high clew attract a higher penalty than those with a parallel foot (because the leech length is reduced but the luff length isn't).

I've been doing some calculations to estimate the actual sail area of a Marblehead (as opposed to the certified area).  I'll publish the conclusions in a separate thread but (assuming I've done the sums right) the X, Y & Z values are pretty fair (less than 1% difference for reasonable values), but raising the clew 80mm  gives a reduction of nearly 2% in real sail area for the same certified area.