**Hull Shape**

**Approach**

My approach to defining the hull geometry starts with the overall characteristics,
like the length, displacement, center of buoyancy, and prismatic coefficient.
These are the most important parameters in defining the hull's carrying
capacity and its wave resistance.

The underwater cross section shape is defined by dimensionless shape
parameters that control the beam/depth ratio, the angle of the sides near
the waterline, and the slackness of the bilge. I vary these parameters
smoothly along the length of the hull to produce a fair shape. The wetted
area is the biggest influence on the hull's low speed resistance, and this
is largely determined by the cross sectional shape. The wave drag is also
influenced by the cross sectional shape, but this is not as strong an influence
as the manner in which the area of each cross section changes along the
length of the hull - the cross sectional area distribution.

The underwater hull shape is determined by combining the section shape
with the cross sectional area distribution. I have used Fourier series
to define how the area changes along the length of the hull. Once I have
the cross sectional area distribution, I size the cross section shapes
at each station to match the desired cross sectional area at that station.

This approach has the advantage that I can change the section shapes
without affecting the overall hull characteristics. It also makes it easy
to shrink or stretch the hull while maintaining a similar shape.

**Cross Sectional Area Distribution**

I've chosen to use some simple Fourier series for defining the cross
sectional area distribution because the Fourier series produce a smooth,
pleasing variation of the areas, and because they can be integrated analytically
to get the overall characteristics. Similar functions have been used for
tank test models that are reported in the literature, so I can use this
information to help estimate the hull's resistance. It also meets the criteria
above, since the free parameters are the length of the waterline, displacement,
center of buoyancy, and prismatic coefficient. The relationships are:

X is the longitudinal distance along the hull, and S is the cross sectional
area distribution. D is the design displacement, Lwl the design length
of the waterline, and Cp the prismatic coefficient.

As x varies from p
to 0, it sweeps the sections from bow to to stern. When x
= p/2, the cross sectional area is at its maximum. This section will be
located twice as far aft of midships as is the center of buoyancy

This constant-length, constant-volume approach also leads to a different
interpretation for the prismatic coefficient, Cp. Traditionally, Cp has
been considered a measure of how full the ends of the hull are. Cp is defined
as:

But when volume and length are kept constant, the choice of Cp controls
the size of the maximum cross sectional area:

Volume and length are largely determined by the purpose of the boat
and its load carrying capacity. The choice of Cp, because of its effect
on the maximum cross sectional area, largely determines the maximum beam.

This is why there is an optimum Cp for different design speeds (Froude
numbers). A larger Cp will result in blunter ends, which would be expected
to cause more wave drag. However, it also narrows the beam, improving the
length-to-beam ratio, which lowers wave drag. For Cp's below the optimum,
the hull is too beamy, and for Cp's above the optimum, the ends are not
fine enough.

Values of optimum Cp for monohulls can be found in "Principles of Yacht
Design". The values should generally lie between 0.5 and 0.6, with higher
values used for higher design speeds. For these functions, values of Cp
less than p/7 (0.45) will result in non-physical
shapes, with negative areas near the ends. Setting Cp greaer than 15p/64
(0.73) will result in necking in the waist of the hull, forming two bulges
instead of a single maximum beam.

Here's a bit of trivia from the strange world department: if you set
Cp to 3p/16 (0.589), lcb to zero, and use circular
cross sections, the resulting shape is known as a Sears-Haack body. This
is the shape that produces the minimum wave drag in supersonic flight.
Coincidentally, this Cp also produces the minimum drag for a boat that
is operated at hull speed (Froude number = 0.40)!

**Conic Section Lofting**

Now that I have the cross sectional area distribution, the next job is to define the shapes of the cross sections. One way to do this is by piecing together curves that are parts of ellipses, parabolas, and hyperbolas. If one has two intersecting line segments forming a "V", one can find a conic section that has these properties:

a) It passes through the end points of the V.

b) It is tangent to the lines at the end points.

c) It can be made to pass through any third point which is located inside the area enclosed by the "V".

By joining together a number of such curves, one can approximate nearly
any shape. The points where the curves join together are fixed in space,
and straight lines are drawn through these points to form an outer skeleton.
The points where the lines through the fixed points intersect are called
corner points. The degree of curvature between the fixed points is controlled
by either defining a third fixed point inside the V, or by a parameter
that represents how much the curve is attracted to the corner point.

This approach to lofting is very intuitive, since the shape can be sketched
out using the straight line skeleton. The curves are smooth and continuous,
and it is easy to see what the effect will be of changing any of the defining
points. Although I will be presenting the algebraic equations that define
these curves, one doesn't need a computer to draw them. They can be quickly
created using just a compass and straight edge. For more details, see Dan
Raymer's "Aircraft Design: A Conceptual Approach".

More aeronautical trivia: the legendary P-51 Mustang was one of the
first aircraft to be designed using conic lofting. Part of its remarkable
performance has been attributed to the fairness of the resulting lines.

Analytically, the conic curve is defined by six coefficients. The parameter, t, is used to sweep the curve from one end to the other. When t = -1 or +1, the curve is at the end points. When t = 0, the curve is at the fixed interior point. This point is defined by the parameter, r, which is the relative distance from the corner point to the midpoint between the fixed ends. When r = 0, the curve goes through the corner point, and has a sharp break. When r = 1, the curve is a straight line between the end points. Separate curves are used for all three coordinate directions (X, Y, Z) so that there is no problem with multiple values. The basic relationships are:

Equation for the curve:

Midpoint between the ends:

The third fixed point in the "throat":

Taking the derivative with respect to t:

By applying the conditions at t = -1, 0, 1, five equations result that
can be used to solve for the coefficients:

@ t = -1, x = x1, x' = x2-x1

@ t = 0, x = x5

@ t = 1, x = x3, x' = x3-x2

Some special cases have to be considered. If x5 = 0 then c6 = 0, and
c5 = 1. Otherwise, c6 = 1.

Putting the equations in matrix form:

If x5 = 0:

These can be solved using standard numerical methods, such as Gauss-Jordan
elimination.

Once the coefficients are in hand, other points on the curve can be obtained, given values for t. If c1 = 0, then the solution is straightforward:

If, in addition to c1 = 0, (c2)t = -c4, there is no solution.

If c1 is not zero, there are two solutions:

Which solution is correct is somewhat problematic. I compute both, and
pick the one that lies inside the "V" formed by the defining points. This
means that the coefficients alone are not sufficient to evaluate the curve
- one must also have the original defining points. This means that I have
to store nine or ten quantities for each segment: the two endpoints, corner
point, six coefficients, and (optionally) the slackness parameter, r. And
this has to be repeated for both of the two spatial dimensions, Y and Z.
This bulkiness is probably the biggest drawback to the method.

In addition to the case of c1 = 0, one also has to be careful of the
case where

This also means there is no solution, and the coefficients are probably
invalid.

**Cross Section Shapes**

The actual hull form is finally determined through the choice of section
shape. I have used the conic section lofting to define a generic shape
that can be adapted to different hull forms by simply varying the parameters.
One segment is used to define the underwater shape, and three more segments
define the shape of the topsides and deck. Dimensionless parameters are
used to define the underwater portion, which makes it easier to size it
to the cross sectional area distribution. The topsides are sized relative
to the underwater portion, but the vertical distances are kept fixed.

Twelve parameters define the section shape. The first five, along with
the cross sectional area, define the underwater portion:

BDR beam/depth ratio = b/d

tan(qD) tangent of deadrise angle

r1 bilge slackness parameter

hM2 height of moldline M2 above the design waterline

tan(qF) slope of the hull near the waterline

BDR is the most important parameter, as it controls the depth of the
hull and has the most effect on the wetted area. A semicircle has a beam/depth
ratio of 2.0, so BDR should be approximately two if minimum wetted area
is the object. Decreasing BDR toward the bow will reduce the amount of
rocker that would otherwise result from the shape of the cross sectional
area distribution, and will make the bow finer. Increasing BDR toward the
stern will give broader, flatter sections, but will also steepen the curve
of the buttlines.

The deadrise angle controls how much "V" there is to the bottom. Setting
the deadrise to zero results in a round bottom. The parameter r1
determines how hard the bilge is. A value of zero results in a sharp chine
at C1, and a value of 1 results in a straight line between M1 and M2. Values
in the neighborhood of 0.3 to 0.5 result in smoothly rounded shapes.

The mold line M2 is intended to shape the design waterline. Raising
it above the waterline maybe necessary to improve the numerical characteristics
at the ends, where the cross sectional area goes to zero. It may also be
desireable to raise the M2 moldline for hulls with overhangs. The slope,
tan(qF), at M2 is defined relative to vertical.
A zero value results in a vertical exit of the topsides from the water.

I originally structured the shape of the topsides to form a bell shape,
similar to that used in Shuttleworth catamaran designs. However, this same
structure can represent the flared shapes of many Newick designs, as well
as a conventional hull shape, with or without tumblehome. The parameters
defining the topsides are:

hC2 height of C2 above design waterline

r2 curvature parameter for segment M2-M3

hM3 height of M3 above design waterline

r3 curvature parameter for segment M3-M4

hM4 height of M4 above design waterline

wM3 width of hull flare, relative to the extended slope at M2

tan(qT) slope of topsides from vertical

wM3 controls how much the topsides below M3 deviate from a straight
line. Positive values will push M3 outboard, forming a knuckle or flare.
Negative values will produce tumblehome. A zero value will result in a
straight line between M2 and M3, regardless of the slope or the position
of C2. hC2 will generally be less than or equal to hM3. If they are equal,
the slope of the hull at the knuckle will be parallel to the waterplane.

The moldline M3 forms a sharp chine. In the bell shaped section, with
its rounded topsides, M3 locates the knuckle. On the other shapes, M3 is
the shear. How this line varies along the hull has a major influence on
the appearance of the boat. The final mold line, M4, determines the height
of the cabin, and must be designed in concert with M3.

A zero value for tan(qT) will result in a
vertical topside starting at the knuckle in the bell shape. A comparatively
large negative value is required for the more conventional shapes. In the
latter two cases, this parameter will control the slope of the deck at
the hull/deck joint. Setting tan(qT) equal to
tan(qF) and wM3 equal to 0 will result in a
smooth join at M3, with no crease.

The degree of curvature in the topsides is controlled by the parameters
r2
and r3. In most cases, the shape is not very
sensitive to these parameters, due to the shallow angles of the skeleton
at the corner points. For the bell shape, however, r3
has a major effect on the shape. A small value will result in drawing the
hull toward C3, and this can be used to create a straighter topsides and
a more conventional, sharp edged, shear line.

I have chosen to define symmetrical hull shapes, largely for ease of
manufacture and because I don't have good tools for analyzing the hydrodynamics
of asymmetrical shapes. For asymmetrical hulls, the parameters above would
have to be doubled, with a set for each side, and an additional parameter
defined that contolled the lateral position of the section according to
the camber in the hull centerline.

**Adjusting Section Shapes to the Cross Sectional
Area**

Finally, I put all this together and determine the width and remaining
dimensions of the section shape. This is an iterative process, but it converges
very rapidly. It uses the numerical integration described in the next chapter,
Hydrostatics. So I will skip over the mechanics of calculating the underwater
cross sectional area numerically, and cover how the shape is adjusted to
match.

The first step is to get an approximate value for the section dimensions.
The skeleton of the underwater shape is very useful here, since it is easy
to calculate the area inside the skeleton, and this allows me to calculate
the beam as a function of the area. C1 is located at the intersection of
the skeleton lines through M1 and M2, and the coordinates of M2 can also
be written in terms of the hull depth:

I use the area of the skeleton to get the first approximation of the
cross sectional area, S0:

This is solved for the first guess at the hull depth, d0 using the quadratic
formula, setting S equal to the desired area. Once a definite value for
the depth is known, all the other dimensions of the section can be calculated,
including the offsets.

A simpler first guess, especially good for rounded sections, is to base
the depth on the area of an ellipse:

The next step is to numerically integrate the underwater area of the
section, S0, using the first guess at the offsets, using the techniques
described under Hydrostatics. This will probably be a little smaller than
the design area, S. The area scales as d2 if hM2 is small, so the final
step is to obtain the revised estimate for d, d1:

The section dimensions are recalculated based on d1. These last two
steps may be repeated until the design and actual areas match as closely
as desired.

**Tables of Offsets**

The preceding relationships are sufficient for completely defining the
shape of the hull's canoe body. However, if x is varied in a regular manner,
the spacing of the sections will not be uniform. The stations will actually
be very well placed for numerical purposes, with more stations in locations,
such as the bow and stern, where the lines are changing rapidly. However,
the typical convention is to have stations distributed evenly, forming
a multiple of ten intervals between the perpendiculars at the ends of the
design waterline.

If the longitudinal locations of the stations are known, the following
relationships can be used to find x so that the cross sectional area can
be computed for that location.

If lcb is zero,

Otherwise,

A similar situation exists with respect to creating waterlines and buttlines
at regular intervals. Computing the value for t, given an x, is similar
to solving for x, given t. The same equations can be used, if t and x are
exchanged, along with the corresponding coefficients (c1 for c3 and c4
for c5).

I have chosen to evaluate the shape at even values of t, however, to
create an internal table of offsets. I generate waterlines, buttlines,
and diagonals by interpolating this internal table's intersection with
an arbitrary cutting plane. This procedure is necessary in any event to
compute the waterline plane for arbitrary pitch and roll attitudes, so
it is convenient to use it to generate the hull's lines.