The Geometry of Ships

16 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
direct curves, and there are many more things that can
go wrong; for example, a surface and a plane may not intersect
at all, or may intersect in more than one place.
3.12 Relational Curves. In relational geometry,
most curves are constructed through defined relationships
to point entities or to other curves. For example,
a Line is a straight line defined by reference to two control
points X 1 , X 2 . An Arc is a circular arc defined by
reference to three control points X 1 , X 2 , X 3 ; since there
are several useful constructions of an Arc from three
points, the Arc entity has several corresponding types.
A BCurve is a uniform B-spline curve which depends
on two or more control points {X 1 , X 2 ,...X N }. A
SubCurve is the portion of any curve between two
beads, reparameterized to the range [0, 1]. A ProjCurve
is the projected curve described in the preceding section,
equation (29).
One advantage of the relational structure is that a
curve can be automatically updated if any of its supporting
entities changes. For example, a projected
curve (ProjCurve) will be updated if either the basis
curve or the plane of projection changes. Another important
advantage is that curves can be durably joined
(C 0 ) at their endpoints by referencing a given point entity
in common. Relational points used in curve construction
can realize various useful constraints. For
example, making the first control point of a B-spline
curve be a Projected Point, made by projecting the second
control point onto the centerplane, is a simple way
to enforce a requirement that the curve start at the centerplane
and leave it normally, e.g., for durable bow or
stern rounding.
3.13 Points Embedded in Curves. A curve consists of
a one-dimensional continuous 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 curve is called a bead; several ways are provided
to construct such points:
Absolute bead: specified by a curve and a t parameter
value
Relative bead: specified by parameter offset t from another
bead
Arclength bead: specified by an arc-length distance from
another bead or from one end of a curve
Intersection bead: located at the intersection of a curve
with a plane, a surface, or another curve.
A bead has a definite 3-D location, so it can serve any
of the functions of a 3-D point. Specialized uses of beads
include:
• Designating a location on the curve, e.g., to compute a
tangent or location of a fitting
• Endpoints of a subcurve, i.e., a portion of the host
curve between two beads
• End points and control points for other curves.
A surface is a 2-D continuous point set embedded in a 2-
D or (usually) 3-D space. Surfaces have many applications
in the definition of ship geometry:
• as explicit design elements, such as the hull or
weather deck surfaces
• as construction elements, such as a horizontal rectangular
surface locating an interior deck
• as boundaries for solids.
4.1 Mathematical Surface Definitions: Parametric vs.
Explicit vs. Implicit. As in the case of curves, there are
three common ways of defining or describing surfaces
mathematically: implicit, explicit, and parametric.
• Implicit surface definition: A surface is defined in 3-D
as the set of points that satisfy an implicit equation in the
three coordinates: f(x, y, z) 0.
• Explicit surface definition: In 3-D, one coordinate is
expressed as an explicit function of the other two, for
example: z f(x, y).
• Parametric surface definition: In either 2-D or 3-D,
each coordinate is expressed as an explicit function of
two common dimensionless parameters: x f(u, v), y
Section 4
Geometry of Surfaces
g(u, v), [z h(u, v)]. The parametric surface can be described
as a locus in three different ways:
° 1. the locus of a moving point {x, y, z} as the parameters
u, v vary continuously over a specified domain
such as [0, 1] [0, 1], or
° 2, 3. the locus of a moving parametric curve (parameter
u or v) as the other parameter (v or u) varies continuously
over a domain such as [0, 1].
A fourth alternative that has recently emerged is socalled
“subdivision surfaces.” These will be introduced
briefly later in Section 5.
Implicit surfaces are used for some CAD representations,
in particular for “constructive solid geometry”
(CSG) and B-rep solid modeling, especially for simple
shapes. For example, a complete spherical surface is very
compactly defined as the set of points at a given distance
r from a given center point {a, b, c}: f(x, y, z) (x a) 2
(y b) 2 (z c) 2 r 2 0. This implicit representation
is attractively homogeneous and free of the coordinate
singularities that mar any explicit or parametric representations
of a complete sphere. On the other hand, the
lack of any natural surface coordinate system in an im-