Figure 1. Mount Gay 30 Courtesy
of Geoff Van Gorkum
Renderings may look good in a brochure,
but the 3D model used to generate the rendering may not
be accurate enough for construction purposes. Most rendering
programs hide unfairness in curved surfaces because they
process the surfaces into triangles and the shading routines
smooth the triangles to hide their edges.
This article discusses some of the flexibility
you give up and some of the difficulties you encounter to
take advantage of these benefits. These are the "dirty little
secrets" that most designers learn about only after they
buy and start to use a hull fairing program. Designers should
also know more about the tools they use and understand what
restrictions are placed on them by the program's user interface
and what restrictions are placed on them by the underlying
NURB hull geometry.
**Surface Modeling**
The field of geometric modeling is huge.
Two-dimensional vs. three-dimensional geometry, wireframe
vs. surface vs. solid modeling, relational vs. parametric
vs. variational geometry, parametric vs. implicit equations,
the list goes on. This article, however, will talk only
about smooth curved surface design using NURB (Non-Uniform
Rational B-spline) surfaces, since that is the dominant
mathematical technique used by most hull design and fairing
programs, such as AutoShip, FastShip, MaxSurf, Nautilus,
and ProLines. It is also the dominant technique used by
general-purpose CAD programs, such as Autodesk's Mechanical
Designer, Intergraph's Solid Edge, Trispective's 3D Eye
(all ACIS modelers), and McNeel's Rhino. If you wish to
transfer exact 3D hull geometry to other CAD programs, you
will have to use NURB surfaces (or a surface type which
is mathematically compatible with NURBs). For example, there
are several companies (MolyCAM, Lite System (1995), Formglas)
that will take your computer-generated hull shape and numerically
cut a plug for the hull out of foam using a very large milling
machine. These machines are controlled by one of several
surface NC programs like SmartCAM and MasterCAM using NURB
surfaces. You will have to provide them with a file containing
the NURB surface definition in a standard transfer file
format, such as IGES (Initial Graphics Exchange Specification).
NURBs are the best techniques available
for curve and surface design, but you have to understand
their capabilities to get the most out of your boat design
and CAD software.
**What is a NURB?**
NURBs are Non-Uniform, Rational, B-splines.
They are equations used to define curves or surfaces that
simulate the designer's batten in terms of stiffness and
continuity. In the early 1970's, 'B' or Basis splines were
created (There are an infinite number of splines.) as a
convenient way to define smooth curves and surfaces interactively
on the computer screen. Rather than have a curve pass through
a set of points, like ducks on a batten, a B-spline creates
a curve which is "influenced" by the positions of defining
points called vertices (see figure 2). The designer changes
the shape of the curve by changing the positions of the
defining vertex points. A curve influenced by B-spline vertex
points is smoother than a curve passing through the defining
vertex points, but the curve is by no means automatically
fair. You can create some pretty ugly curves and surfaces
with B-splines. In general, all flexible curve and surface
modeling programs need some form of fairing tool.
Figure 2. NURB Curve Showing
Defining Vertex Points
A NURB surface uniquely defines every square
inch of a rectangular-like spline membrane. For editing
purposes, it is usually represented on the screen by drawing
a grid of lines on the surface called rows and columns.
Like the NURB curve, the surface is defined by vertex points
which do not lie on the surface, but are organized as a
rectangular matrix or grid of vertex points.
Figure 3. NURB Surface Showing
Defining Vertex Points
To understand and use NURB surfaces effectively,
you need to know what controls you have over their shape:
**vertices, degree, knots, and weights**. Programs might
hide this terminology from you, but you need to know how
each of these variables affect the shape of a curve and
surface.
For completeness, here is the NURB surface
equation:
P(i,j) Matrix of vertices: nrows=(k1+1)
ncols=(k2+1)
W(i,j) Matrix of vertex weights: one per
vertex point
b_{i}(s) Row-direction basis or
blending polynomial functions of degree M1
b_{j}(t) Column-direction basis
or blending polynomial functions of degree M2
s Parameter array of row-direction knots
t Parameter array of column-direction knots
**Vertices** - The basic shape of a NURB
surface is defined by its rectangular grid of vertex or
control points which do not necessarily lie on the surface.
When you move one of the vertex points, the surface in that
local area is pulled in the direction of the vertex movement
as if it was attached to the point by a spring. (see figure
4) The surface does not move as much as the vertex point
and the change only affects a small area near the moved
point. The range of influence depends on the degree of the
surface: a smaller degree (see the next sub-section) means
a smaller area of influence and a larger degree means a
larger area of influence. The main drawbacks are:
- It is more difficult to get the surface
to be positioned exactly where you want.
- The vertex mesh hides the shape of the
surface.
Figure 4. NURB Curve Example
of Moving a Vertex Point
Another approach is to use edit points which
lie on the surface. For every defining vertex point "floating
in space" there is an associated point which lies on the
surface. Some programs allow you to control the shape of
the surface using these points instead of the vertex points.
The two common ways to implement shape control using points
which lie on the surface are local control and global control.
Local surface control using points that lie on the surface
means that when you move a point on the surface, it is guaranteed
that the change affects the surface only in a small area
surrounding that point; the same as moving one vertex point.
(see figure 5) The problem is that when you move one of
the points on the surface using this technique, some of
the surrounding defining points on the surface also move!
This technique is the same as moving a vertex point, except
that you are moving the associated point on the surface.
Figure 5. NURB Curve Example
of Points on the Surface - LOCAL
Global surface control means that when you
move one of the defining points on the surface, none of
the other points on the surface move. This is called a full
NURB surface interpolation technique. (see figure 6) As
the selected defining point moves, the entire surface shape
is affected, although the shape change decreases dramatically
as you move away from the changed surface point.
Figure 6. NURB Curve Exampe
of Points on the Surface - GLOBAL
Although the surface interpolation approach
is global, it gives you the most accurate, direct control
over the surface. This is because you control the shape
directly and only the selected defining point moves.
Vertices or surface points can be positioned
anywhere in space and are the major shape control factor.
For fairing and detailed shape control, you need to learn
more about the following variables.
**Degree** - A NURB curve or surface
is represented mathematically by piecewise polynomial blending
functions of the form:
The highest exponent used in the polynomial
(3 in this case) is the degree of the polynomial. NURB curves
use one type of polynomial and NURB surfaces use two different
polynomial types (one for each grid direction - row/column).
The higher the degree of the polynomial, the more flexibility
there is in the curve or surface, but the more wiggles it
might have. Cubic, or degree 3 polynomials are considered
to have the best trade-off in flexibility and stiffness.
It is the degree used most often in CAD programs. Quadratic
or degree 2 curves and surfaces will be stiffer and will
be automatically more fair than cubic curves, but the range
of shapes they can achieve will be much more limited. Many
CAD programs fix the degree to always be 3.
**Knots (Non-Uniform)**
NURB curves and surfaces are polynomials
pieced together using knots or a knot vector. Curves use
one knot vector and surfaces use two; one for each grid
direction. The knots define how the polynomial pieces are
blended together with the proper smoothness. Most programs
give users only indirect control over this shape factor.
It is also the cause of many curve and surface shaping and
fairing problems.
A **uniform** knot vector (with constant
spacing between the knots) is the most common technique
and works best for most situations. A program may automatically
adjust the spacing of the knots in the knot vector (**non-uniform**
spacing) to account for uneven spacing of the rows and columns
of the surface, but this may cause problems with adjoining
surfaces. Some programs also use knots to define tension
or knuckle control over the surface.
The problem with knots is that they have
a semi-global effect on surface shape. If you adjust the
knot spacing in the row direction, then the surface shape
is changed for the full length of the surface, affecting
several columns. In addition, if there is a surface attached
to the end of the rows, then it must use the same knot spacing
to maintain exactly the same edge shape. This is a major
problem with knots. Not only do they have a global affect
on the surface, they can also force changes in adjoining
surfaces! Many programs avoid these problems by fixing the
knot spacing to be uniform for all surfaces. This is a reasonable
solution, but it means that the user should create surfaces
with evenly spaced rows and columns.
Figure 7. Difference Between
Uniform and Centripetal Knot Spacing
**Weights** - For NURB curves and surfaces,
each vertex point has an associated weight value that affects
the shape locally near that point. This weight value is
the result of the **rational** aspect of NURBs, which
means that the equation is defined as a fraction or *ratio*
of polynomials. The purposes of the weights are to allow
for exact descriptions of conic shapes and to add more user
control over the shape of the surface. Although each vertex
point has its own weight and although each weight has a
local effect on the surface, a change in a vertex weight
(1.0 is the default weight) does not necessarily change
the surface the way you want. In practice, changing a vertex
weight value can cause more problems for fairing. The best
approach is to apply weight changes only near the end of
the design process to achieve very specific local effects.
Do not use weight control to create shapes that can be achieved
by moving vertex points. Some CAD programs severely limit
the user's control over weight values because of these problems.
Most hull shapes can be defined and faired without changing
the weight values.
Figure 8. Difference in Curve
Shape Due to Weight Change
Now that you have been introduced to the
NURB shape controls (vertices, degree, knots, and weights),
the next section will discuss many of the hull design problems
that you may encounter and some of their solutions.
Some of the more insightful readers might
note that if a program uses uniform knot spacing and fixes
the weights to 1.0, then this would be a simple uniform
B-spline curve or surface. Then why does a CAD program use
NURBS? That is a good question and I can make a good argument
that NURBs are not necessary, even for conic shapes. I guess
the reason is that NURBs form a superset of many different
spline techniques and that if you wish to transfer surface
geometry between programs, then it is a good solution.
**Dirty Little Secrets**
These are some of problems you can expect
when using one of the hull design programs.
**Rectangular Spline Mesh**
The NURB surface is a rectangular mesh of
rows and columns with four hard corner points. The entire
surface is continuous, except for these four points. You
can stretch and distort the surface to any shape by repositioning
the vertex points, but the more non-rectangular of a shape
you make, the more difficult it will be to fair. It is the
designer's job to determine how to break down a full hull
surface into a collection of rectangular-like pieces, how
to deal with the four hard corner points, and how to maintain
the proper continuity between surfaces. There is no one
way of doing this.
For example, a full keel, wineglass shaped,
sailboat hull is one of the more difficult shapes to define.
The hull is not rectangular and is difficult to model with
one surface. Figure 9 shows how the J Boat Endeavor I was
modeled by one NURB surface because the aft end of the keel
made a smooth transition up to the transom. Even so, there
are many subtle shape problems that have to be dealt with
due to the extreme distortion of the one rectangular surface.
Figure 9. J Boat Endeavor I
Modeled With One NURB Surface
Another approach is to pretend that the
keel is not there and describe the whole canoe body as one
NURB surface. You would then create a second surface to
define the keel, but you would have to attach the upper
edge of the keel surface to the canoe hull with the appropriate
continuity. This technique works best for fin keels, rather
than for full keels.
Figure 10. Fin Keel Sailboat
With Separate Surface for Keel
Still another approach is to define the
hull and keel with one surface from the bow to the back
edge of the keel. Then you would add another surface to
define the counter of the hull from the aft end of the keel
to the transom. You could even overlap and attach the surfaces
to try and maintain some level of continuity between surfaces.
Figure 11. Overlapped Counter
Defined With Separate Surface
These fitting and shaping problems are encountered
in all types and sizes of boats.
**Clutter of Vertex Mesh and Hull Shape**
Since the defining vertex points (the ones
that you have to grab and move) do not lie on the NURB surface,
most programs allow you to connect the vertices with straight
lines to make them more visible. This grid of straight lines
is often called a vertex mesh. If you do not connect the
vertices with a mesh, then the vertices are floating in
space and it becomes very difficult to know which part of
the hull they influence, especially in section view. If
you display the vertex mesh, however, it clutters up the
screen and hides the shape of the hull.
Figure 12. Hull Showing Clutter
of Vertex Lines
To avoid this problem, some programs use
points which lie on the surface of the hull. This cleans
up the clutter and makes the hull easier to see and manipulate.
Figure 13. Hull Showing On-Surface
Edit Points
**Waterlines, Buttocks, and Diagonals Cannot
be Controlled Directly**
NURB surfaces are defined using a grid of
row and column vertex points that can be moved anywhere.
You cannot directly grab and alter the shape of any traditional
lines of the boat, such as waterlines, buttocks, and diagonals.
These lines are considered to be derived lines or surface
cuts which can be automatically calculated from the surface
definition. Some programs, however allow you to display
these lines while dragging a control point so that you can
immediately see what happens when a vertex point is moved.
To change the shape of a waterline, for example, you have
to move the vertex point which lies closest to the point
on the waterline you wish to change. This is the biggest
adjustment required by the experienced hand draftsmen.
**Adding and Deleting Rows and Columns
**
One can start with a very simple NURB surface
consisting of two rows and two columns, and then add in
additional rows and columns to increase the flexibility
and control of the surface. In doing so, the program must
add the row or column in over the entire length or width
of the surface. There is no allowance for partial rows or
columns. This means that if you want to add rows and columns
near the bow to define a bulb, then these rows and columns
must extend over the entire length and width of the surface,
whether you want them to or not. Remember that the only
way to add shape control is by adding in control vertices,
and the only way to add in control vertices is by adding
them in as a complete row or column.
**Spacing of Rows and Columns**
The smoothest surface shape depends on a
relatively even spacing of the rows and columns. If there
is more than a 2 to 1 ratio of consecutive row or column
spacing, you will begin to see some very strange anomalies,
such as hard bumps and loops. A program could adjust the
knot vector to minimize the bumps and loops of uneven spacing,
but this adjustment can cause other difficulties. For example,
if two surfaces are to be joined exactly, edge-to-edge,
then the two surfaces must have the exact same knot vector
at that edge. If you have several surfaces attached together
with common edges (like most multi-chine vessels), then
a change in the knot vector in one surface can affect the
shape of all of the connected surfaces. One technique in
dealing with attached surfaces is to use uniform knot spacing
and always add in rows or columns by splitting the difference
between two existing rows or columns. This will always maintain
a reasonable 2 to 1 ratio in row and column spacing acceptable
for uniform spacing.
Figure 14. Uneven Spacing of
Columns
Figure 15. Add Column to Create
Even Spacing
**Piecing Together Surfaces**
Rarely can you use one NURB surface to define
a whole boat. If chines and deck edges form the edges of
the NURB surfaces, you need to understand the requirements
of attaching two surfaces together. There are two types
of attachment: exact and approximate. To form an exact attachment
between two surfaces, at a chine, for example, the two surfaces
must match up exactly along their entire length. The common
surface edges must have the exact same edge vertex points,
they must use the exact same polynomial degree at the edge,
they must use the exact same knot vector at the edge, and
they must use the exact same weight values at the edge.
Otherwise, the mathematical definition and therefore, the
shape of each surface at the chine will be different. If
the two surfaces do not agree exactly at the common edge,
then the attachment is considered to be approximate. It
is then up to the user to manually make sure that the two
approximate edges are within building tolerances of each
other. If two surfaces match exactly and the user adds a
column which runs into the attached surface, then the program
must add a column to the attached surface or the two surfaces
will not be joined exactly anymore.
**Matching up existing designs**
The original goal of B-splines was to create
smooth curves and surfaces interactively on the computer
screen. That is why they use vertices which do not lie on
the curve or surface. But these floating vertices make it
difficult to recreate shapes that currently exist. For example,
if you have a table of offsets and wish to create a NURB
surface representation of that shape, you will have difficulty
trying to position the vertices to achieve that shape. Some
programs allow you to digitize the sections of an existing
boat and to use that as an interactive guide for moving
vertex points. You move the vertex points until the surface
section shape matches the digitized station shape. This
is a very tedious process. A better approach is to automatically
fit the NURB surfaces to the offset data or the digitized
stations so that the surface passes through the offset points.
You will still have to perform the final shaping and fairing,
but the matching-up time will be greatly reduced.
For those interested in scanning in a bodyplan
of a boat, it is not quite that simple. Most bodyplans consist
of overlapping stations which cause a problem when the scanner
program tries to convert the scanned-in black dots to form
polylines which represent unique stations. You end up spending
a lot of time in your CAD program fixing up the mistakes.
**Fairing**
For example, one of the tricks in fairing
a manual spline is to lift certain ducks for a moment to
"relax" the batten. This smooths the curve, but it doesn't
necessarily maintain the shape that you want. With the computer,
you have the same problem of getting the shape you want
and making sure that the shape is fair. Automatic computer
fairing techniques may smooth a surface, but they do not
have the human judgement to make this shape vs. fairness
trade-off.
Some programs say they guarantee fairness
of a curve or surface, but this is only an opinion and is
done by severely limiting the shape of the surface. For
example, if someone gave you a batten and three ducks, it
would be difficult to create a curve which was not considered
to be fair. On the other hand, you would be severely limited
in the range of shapes you could draw.
Most hull design and fairing programs provide
ways to check the fairness of a hull by using some form
of second derivative or curvature curve, and by using some
form of surface curvature color mapping. Designers don't
need to know the mathematics behind these techniques, they
just need to see how the technique magnifies all bumps,
wiggles, and inflection points. The designer interactively
changes the shape of the hull to obtain a smooth derivative
or curvature curve. As you can see, the designer is actively
involved with the fairing process.
Figure 16. NURB Curve With
Unfair Overlaid Curvature Curve
Figure 17. NURB Curve With
Faired Overlaid Curvature Curve
The problem with fairing NURBs is that the
surfaces are so sensitive to the knot values. If a program
uses uniform knot spacing, then the NURB surfaces need to
have fairly evenly spaced rows and columns. When the knot
spacing does not match the spacing of the rows and columns,
you will have fairing problems. When a NURB surface is stretched
into a very non-rectangular shape, even non-uniformly spaced
knots do not help. In addition, non-uniform knots cause
other problems, as mentioned before.
Another problem is that you cannot control
the shape of the lines (waterlines, buttocks, etc.) directly.
You can only change the shape of the rows and columns of
the surface. In practice, it is relatively easy to obtain
the station and buttock shapes you want, but the waterlines
will always be a problem. The reason for this is that a
waterline cut through the surface does not typically relate
to either a row or a column on the surface. This means that
to shape and fair a waterline, you need to change the shape
of both the rows and the columns. Depending on the shape
of the boat and how much the NURB surface has been stretched
to fit the hull, you will have a greater or lesser problem
with the waterlines.
Part of this problem has to do with how
hand draftsmen cheat when they draw the lines of a boat.
The goal of traditional hand drafting is to match up the
different views of the lines and to fair the lines at the
same time. For most cases, fairing the lines wins out over
making sure that the three views of the lines match up exactly.
For good draftsmen, the errors in matching views are within
building tolerances. For NURB surfaces, however, all three
views always match exactly. You cannot cheat just to get
a waterline or buttock to look good. In addition, the programs
can "cut" any waterline or buttock, so it is easy to evaluate
the full 3D shape of the boat. You cannot gloss over the
very subtle shape problems commonly found in waterlines.
What makes it worse is that the surface curvature display
tools can magnify every last little bump and wiggle, so
it is impossible to fair a surface perfectly. Your goal
should be to fair the boat to within building tolerances
and not to worry about how perfect the lines look.
**Vertex Weight Values**
Although weights have a local influence
on the shape of the surface, manipulating their values can
be difficult. A common approach is to try to obtain the
desired surface shape by only adding and manipulating rows
and columns of vertices. After you have roughed out the
hull shape, you can then change the weight values to achieve
specific local effects. Most boats, however, can be defined
without changing the vertex weight values. Only experienced
users should try manipulating these values.
**Relational vs. Trimmed Surfaces **
There has been some discussion between two
ways of attaching or connecting surfaces: trimmed surfaces
vs. relational or attached surfaces. For trimmed surfaces,
one surface is pushed through another and the program automatically
trims off the overlap. An example of this is to push the
top of a fin keel through the bottom of a canoe body of
a sailboat. For relational or attached surfaces, the top
edge of the keel surface is attached to the canoe body of
the hull. If the top of the keel is changed, it will always
stay attached to the hull, and if the hull changes shape,
the top of the keel will stay attached.
The obvious advantage of attaching one surface
to another is that you do not have to worry about whether
the surfaces intersect if one or the other surface changes
shape. The downside is that if the attached-to surface changes
shape, then the attached surface changes shape. For the
keel and canoe-body attachment example, if the hull changes
shape near the keel, then the attached keel changes shape.
For airfoil-shaped keels, you do not want a change in hull
shape to change the shape of the keel. Likewise, if you
attach a cabin top to a deck and then decide to shift the
cabin fore or aft while still attached, then the cabin top
will change shape. For trimmed surfaces, you would just
have to make sure that the cabin sides penetrate the deck
surface by a little bit. The shape of the cabin top is independent
of the shape of the deck. You can "pick up" the cabin, shift
its position, and set it down on the deck (penetrate the
deck) without changing its shape.
There are benefits and disadvantages to
both approaches. Attachment or relational geometry works
best when the two surfaces or geometry shapes relate and
are dependent on one another, such as when you attach longitudinal
stringers to the inside of a hull. Trimmed surfaces work
best when the two surfaces must be joined, but their shapes
are independent of each other. Trimmed NURB surfaces also
work best if you need to convert the surface model to industry
standard solid modelers. Relational attaching of one surface
to another can leave very small gaps between the surfaces
which are very difficult to deal with in solid modelers.
**Calculations**
All of the hull design programs use open
surface design geometry. None of the programs use solid
modeling, which requires a complete, closed solid geometry
at all times. Solid modeling has some advantages when it
comes to calculations, but it makes the process of modeling
the geometry much more difficult. This means that the boat
design software has to do a lot of work to calculate the
hydrostatics and stability of a vessel.
The first step is to cut all of the surfaces
at a series of station positions along the length of the
hull. The next step is to somehow combine these surface
cuts into realistic, closed station shapes. This can be
very difficult for intersecting and open surfaces, and for
surfaces that do not quite match up exactly along common
edges. Once legal station shapes have been formed, the program
can use the desired waterplane to clip off their above-water
portions. Finally, some form of longitudinal integration
can be performed to produce the final results. For all of
these programs, the key to obtaining accurate results is
to verify the final shapes of the cut stations before the
calculations are done. If the stations look good (and you
have enough of them covering the length of the vessel),
then the results should be accurate. Be aware that many
problems can arise during the station cutting and reconstruction
phase.
Figure 18. Portion of SWATH
Showing Trimmed Surfaces to Create Stations
**Conclusion**
NURBs are the dominant mathematical technique
used for designing curves and surfaces. In spite of this,
there are many things you need to know to get the most out
of your hull design or CAD program. This article discussed
many of these hidden problems. Since all NURB-based programs
vary greatly, the best advice is to get the demos of all
of the boat/ship design programs and spend some time with
them trying to design and fair a boat. The user interface
of a program can have as much of an impact on your ability
to get the job done as which mathematical technique is used
to define the hull. Don't wait until after you buy the program
to learn the truth. |