The Geometry of Ships

30 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
Curves of intersection arising from the intersections
between two surfaces can be recognized as snakes residing
on both of the surfaces. The difficulties that can be
present in computing such intersections have been discussed
above in Section 4.15.
6.2 Applications of Curves on Surfaces. Curves on surfaces
can play several roles in definition of ship geometry:
• As decorative lines; e.g., cove stripe, boot stripe, hull
decorations
• As boundaries of subsurfaces and trimmed surfaces;
e.g., delineating subdivision of the hull surface into shell
plates for fabrication
• As a junction between surfaces; e.g., the deck-at-side
curve drawn on the hull and used as an edge curve for
the weather deck surface
• As a trace for a linear feature to be constructed on another
surface; e.g., a guard, strake, or bilge keel
• As alignment marks to be carried through a plate expansion
process.
Section 7
Geometry of Solids
The history of geometric modeling in engineering
design has progressed from “wireframe” models representing
curves only, to surface modeling, to solid modeling.
Along with the increase in dimensionality, there
is a concomitant increase in the level of complexity of
representation. Wireframe and surface models have
gone a long way toward systematizing and automating
design and manufacturing, but ultimately most articles
that are manufactured, including ships and their components,
are 3-D solids, and there are fundamental
benefits in treating them as such. Wireframe representations
were the dominant technology of the 1970s; surface
modeling became well developed during the 1980s;
during the 1990s the focus shifted to solid models as
computer speed and storage improved to handle the
higher level of complexity, and as the underlying mathematical,
algorithmic, and computational tools required
to support solids were further developed.
We will first briefly review a number of alternative
representations of solids, each of which has some advantages
and some limited applications. Of these, boundary
representation or B-rep solids have emerged as the most
successful and versatile solid modeling technology, and
they will therefore be the focus of this section.
7.1 Various Solid Representations.
7.1.1 Volume Elements (Voxels). A conceptually
simple solid representation is to divide space into a 3-D
rectangular array (lattice) of individual cubic volume elements
or voxels, and then characterize the contents of
each voxel within a domain of interest. This is a 3-D extension
of the way 2-D images are represented as arrays
of picture elements or “pixels.” For a homogeneous
solid, the voxel information can be as little as one bit,
i.e., is this voxel occupied by material, or is it empty? Or,
if a complex inhomogeneous solid is being described,
numerous attributes can be attached to each voxel; e.g.,
density, temperature, concentration of various chemical
species, etc.
Voxels are most useful for medium-resolution descriptions
of inhomogeneous solids with significant internal
structure. The storage requirements and processing
effort are high, and increase as the cube of the
resolution. For example, a voxel description of the
human body at a resolution of 1 mm requires on the
order of 10 8 voxels (and of course, 1 mm is still a very
coarse resolution for describing most tissues and
anatomical structures).
7.1.2 Contours. Contours or level sets on surfaces
were described in Section 4.19, and were related to the
description of an object as a solid. In naval architecture,
transverse sections (contours of the longitudinal coordinate
X) are the standard representation of the envelope
of a vessel for purposes of hydrostatic analysis. The individual
sections are represented as closed polylines.
Contours are also used within a hydrostatic model to describe
tanks, voids, or compartments inside the vessel.
Fig. 27
Offsets representation of a ship as a solid cut by contours (X constant).