The curve of a deck

Alan Gilbert

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Mar 23, 2022
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The transverse of curve of a yacht’s deck is traditionally laid out on the boatyard’s floor as follows:

1648024229884.png

Draw XY the length of half the maximum beam.

Draw XZ normal to XY the height of the desired crown.

With the compass centred on X sweep an arc from Z to touch XY at W.

Divide XY into four equal parts, XA, AB, BC and CY.

Divide XW into four equal parts, Xa, ab, bc and cW.

Divide the arc ZW into four equal parts, Zd, de,ef and fW.

Join ad, be and cf.

Draw AD normal to XY and the same length as ad.

Draw BE normal to XY and the same length as be.

Draw CF normal to XY and the same length as cf.

Sweep a fair curve through Z, D, E, F and Y—that is your half-deck curve. (The other half is obviously just a mirror-image, so make sure the curve at Z is exactly parallel to XY.)

I would love to know exactly just WHAT (if anything) is that curve? I have seen a claim that it’s merely an arc of a circle, another that it’s a hyperbola. I’ve seen no proof of either claim, and though I did once study maths at university it was many decades ago, and developing such a proof is way beyond me. Can anyone help, please?
 
I would love to know exactly just WHAT (if anything) is that curve? I have seen a claim that it’s merely an arc of a circle, another that it’s a hyperbola. I’ve seen no proof of either claim, and though I did once study maths at university it was many decades ago, and developing such a proof is way beyond me. Can anyone help, please?
I haven't tried to analyze this algebraically, but I constructed the curve on GeoGebra, along with a parabola, and they look nearly identical:

1648044071907.png
 
The exact formula for the curve, given height XZ = h and width XY = w is

y=h12xwsin(πx2w)+x2w2y=h\sqrt{1 - \frac{2x}{w}\sin\left(\frac{\pi x}{2w}\right) + \frac{x^2}{w^2}}

Here is a graph showing the construction (solid magenta) and this function (dotted cyan):

1648079704973.png

The parabola is

y=h(1(xw)2)y = h\left(1 - \left(\frac{x}{w}\right)^2\right)

I'll let someone else ponder why these are so close in most cases.
 
Many thanks!

For all practical purposes, then, the traditional transverse curve of a deck is a parabola. That's good to know.

Here's a little bridge I built for my garden, with both curves laid out as above. They are certainly more aesthetically pleasing than if they were merely arcs of a circle.

DSCF0318.JPG
 
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