maths equation for a boat hull

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peterAustralia

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I am designing a boat hull. Firstly I did the shape as hard chine (flat sides), and then I decided that a curved hull would be better and lighter. (vertical plywood strip for those that care).

The shape I want is a section of a circle, with the y axis stretched approximately 140 percent
Thus the formula looks something like 1.4(x^2) + (y^2) = a
I have two points that I know of exactly. Can I calculate the equation of the curve from those points if I know the curve is a narrowed circle

One point (gunnel) is 522mm from the centerline (x axis), and 1220mm high (y axis)
A second point is 329mm from the centerline (x axis) and 441mm high

For those trying to envisage this, it is an upside down gothic arch, though narrowed a bit

In terms of maths I get this
b(x^2)+(y^2)=a
 
I have a problem. I am trying to work out the coordinates for a boat hull.
One way, might be just to do it visually, however I was wondering if I could find an equation.

The shape I want is a circle arc, however I want the vertical axis stretched to 140 percent.

Addtionally I dont want the bottom of the hull to be the bottom of the arc, I mean that at the bottom of the circle, the slope is zero (horizontal), instead the slope would be roughly (not exactly) 45 percent.

Basically the hull at amidships, would look like a wine glass. Inverted gothic arch, but narrowed in the horizontal axis

I have some data points
At the keel height is zero, centerline offset is zero
At the gunnel, height is 1220mm, centerline offset is 522mm
At a height of 441 mm, centerline offset is 329mm (could be a mm of two out, but best estimate is 329mm)

I know the formula goes something like this
1.4(x^2)+(y^2)=a
I have to find a

reflection, because the arc is lowered from where the maths would put it, maybe the centerpoint of this curve (elipse) is not at 0,0 but elsewhere on the cartesian axes


Note that I think this is a section of arc, but I am not using all of it. Imagine a narrowed circle (is that an elipse???? dont know) but I draw vertical lines, I discard the outer 5 percent, dont want that, I discard the inner 50 percent. I just want between 50 and 95 percent from the centerline. Though I realise that equation should give me all points.

My brain is starting to hurt. Is there a way of finding this equation.... just keep at it?
 
maths equation for a boat hull - part 2

Update.

I was able to superimpose an ellipse that is 1500mm high by 1275mm wide onto the endview of the boat section.
The 1.4 ratio I used earlier was wrong.

I worked out this ellipse size via trial and error. I tried circle arcs,,, made them bigger and smaller, and then tried narrowing the horizontal axis to be 85% of the former. Basically I just tried curve after curve until I got if correct.

So, the question is
If I have an ellipse (assuming that an ellipse is a circle skewed in one axis), which is 1500mm high via 1275mm wide, is there a way of getting the formula for that ellipse so that I can get a full set of coordinates.
 
Assuming I'm reading your posts correctly, you want to find an equation for an ellipse, of which you know the width and height. If that's the case, then you'd be best served by using the general equation of an ellipse:

\(\displaystyle \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\)

Where a is the radius in the x-direction, and b is the radius in the y-direction.
 
ksdhart2 said:
Assuming I'm reading your posts correctly, you want to find an equation for an ellipse, of which you know the width and height. If that's the case, then you'd be best served by using the general equation of an ellipse:

\(\displaystyle \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\)

Where a is the radius in the x-direction, and b is the radius in the y-direction.
Click to expand...



Thanks for that, I eventually worked it out. Took me four hours, should have taken 15 minutes. Been 30 years since I studied ellipses.


I used the formula

y^2/638^2 + x^2/750^2=1

then used algebra and got this

y = (750^2 - x^2 * 750^2/638^2)^0.5

then I was able to put all this into excel, and plot a set of coordinates all the way from x = 0 to x = 638

Resulting values ranged from 0 to 750,,, so that looks good.


Note that these values represent differences from the axis to the data point, not from the keel up. To get that I just use 750 - (the earlier number)

There may be a mathematical way of working out which is the best elliptical curve for three given x,y coordiantes, however since the maths looks very difficult, I will just use the graphical method, draw circles, narrow them down by approx 85% and superimpose the elliptical curve onto my 3 x,y coordinates. Not good in purist mathematical terms, but good enough for everyday real life examples
 
Just a quick little follow up
Possibly a fraction off topic, but will risk it

Using elliptical curves seemed to work well
I tried using cubic polynomial curves and they did not look right, too much change in radius though the hull section
The elliptical curve gives a similar radius throughout the entire hull section.
A circular curve would give consistent radius, at expense of too narrow hull section at the sleeping berth level, berth could be made wider, giving a deeper narrower hull, but this is at the expense of shoal draught (how many inches of water the boat needs to sail in - eg 20 inches, 24 inches etc etc)

If I was using strip planking, then areas of small curvature and areas of large curvature would not matter, however using cold molding (two layers of thin plywood joined to make a thicker layer), then curve radius becomes important. Too tight a radius and the plywood wont bend. Too slack a radius and the plywood is too flat and lacks the curve, and it is the curve that gives it strength. Thus a curved hull needs less internal framing, a hull with flat sides (hard chine construction) gives an easier to build hull at expense of more drag, additionally the hard chine hull needs a lot of internal framing to give the hull strength, example flat sheet of paper is weak, curve it into an arch and it is stronger

I made a typo, the ellipse size was 3000mm x 2500mm not 1500mm x 1275

This is the result of using elliptical curves to define a multi-hull boat shape.

A little question. If the gunnel was straight when viewed side on (the gunnal is the top of the hull, where it meets the deck), and from topview the gunnel was a section of pure circular arc, and the hull sides all used the same elliptical section, does that mean that each individual section of boat hull has the same curvature in two dimensions, and if so does that allow for one mold/mould to be used for the entire boat hull and thus allowing the boat to be built in smaller identical sections.

This is what the hull lines look like. I have them in exact detail, and although these images dont do it justice, that is more about the drawing program (need one that can draw large elliptical curves), than the hull shape. Cubic polynomial is on the left, Elliptical curve is on the right

http://postimg.org/image/qey963kez/
 
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