The Geometry of Ships

26 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
Fig. 23
A ship hull molded form defined as a composite surface made from five patches. A, B, and C are ruled surfaces; D and E are trimmed surfaces
made from B-spline lofted base surfaces, whose outlines are dashed.
for treating open assemblies of surfaces as a single entity.
Figure 23 is an example of the molded form of a ship
hull assembled from five surface patches. For the layout
of shell plating, it is desirable to ignore internal boundaries
such as the flat-of-side and flat-of-bottom tangency
lines. Treating the shell as a single composite surface
makes this possible.
4.18 Points Embedded in Surfaces. A surface consists
of a 2-D point set embedded in 3-D space. It is often useful
to designate a particular point out of this set. In relational
geometry, a point embedded in a surface is called
a magnet. Several ways are provided to construct such
points:
Absolute magnet: specified by a host surface and u, v parameter
values
Relative magnet: specified by parameter offsets u, v
from another magnet
Intersection magnet: located at the intersection of a line
or curve with a surface
Projected magnet: normal or parallel projection of a
point onto a surface.
A magnet has a definite 3-D location, so it can always
serve as a point. Specialized uses of magnets include:
• Designating a location on the surface, e.g., for a hole
or fastener
• End points and control points for snakes (curves embedded
in a surface).
4.19 Contours on Surfaces. A contour or level set on
a surface is the set of points on that surface where a
given scalar function F(u, v) takes a specified value. The
most familiar use of contours is to describe topography;
in this case, the function is elevation (the Z coordinate),
and the given value is an integer multiple of the contour
interval. Contours of coordinates X, Y, or Z are the familiar
stations, buttocks, and waterlines used by naval architects
to describe, present, and evaluate ship hull
forms. Any plane section such as a diagonal can be computed
as the zero contour of the function F(u, v) û
[X(u, v) p] where û is a normal vector to the plane and
p is any point in the plane.
A contour on a continuous surface normally has the
basic character of a curve, i.e., a one-dimensional continuous
point set. However, under appropriate circumstances
a contour can have any number of disjoint branches,
each of which can be closed or open curves or single
points (e.g., where a mountaintop just comes up to the
elevation of the contour). Or a contour can spread out
into a 2-D point set, e.g., where a level plateau occurs at
the elevation of the contour.
Contours are highly useful as visualization tools. For
this purpose it is usual to generate a family of contours
with equally spaced values of F. Families of contours
also provide a simple representation of a solid volume,
adequate for some analysis purposes. Thus, transverse
sections (contours of the longitudinal coordinate X) are
the most common way of representing the vessel envelope
as a solid, for purposes of hydrostatic analysis.
Computing contours on the tabulated mesh of a parametric
surface is fairly straightforward. First, evaluate F
at each node of the mesh and store the values in a 2-D
array. Then, identify all the links in the mesh (in both u
and v directions) which have opposite signs for F at their
two ends. On each of these links calculate the point
where (by linear interpolation) F 0. This gives a series
of points that can be connected up into chains (polylines)
in either u, v-space or 3-D space. Some chains may
end on boundaries of the surface, and others may form
closed loops.