The Geometry of Ships

THE GEOMETRY OF SHIPS 11
In 3-D, two implicit equations are required to define a
curve:
f(x, y, z) 0, g(x, y, z) 0 (15)
Each of the two implicit equations defines an implicit
surface, and the implicit curve is the intersection (if any)
of the two implicit surfaces.
Explicit curve definition: In 2-D, one coordinate is expressed
as an explicit function of the other: y f(x), or
x g(y). In 3-D, two coordinates are expressed as explicit
functions of the third coordinate, for example: y
f(x), z g(x).
Parametric curve definition: In either 2-D or 3-D,
each coordinate is expressed as an explicit function of a
common dimensionless parameter:
x f(t), y g(t), [z h(t)] (16)
The curve is described as the locus of a moving point,
as the parameter t varies continuously over a specified
domain such as [0, 1].
Implicit curves have seen little use in CAD, for apparently
good reasons. An implicit curve may have multiple
closed or open loops, or may have no solution at all.
Finding any single point on an implicit curve from an arbitrary
starting point requires an iterative search similar
to an optimization. Tracing an implicit curve (i.e., tabulating
a series of accurate points along it) requires the
numerical solution of one or two (usually nonlinear) simultaneous
equations for each point obtained. These are
serious numerical costs. Furthermore, the relationship
between the shape of an implicit curve and its
formula(s) is generally obscure.
Fig. 4
Typical midship sections.
Fig. 5
Construction of a parametric curve.
Explicit curves were frequently used in early CAD
and CAM systems, especially those developed around a
narrow problem domain. They provide a simple and
efficient formulation that has none of the problems just
cited for implicit curves. However, they tend to prove
limiting when a system is being extended to serve in a
broader design domain. For example, Fig. 4 shows several
typical midship sections for yachts and ships. Some
of these can be described by single-valued explicit equations
y f(z), some by z g(y); but neither of these formulations
is suitable for all the sections, on account of
infinite slopes and multiple values, and neither explicit
formulation will serve for the typical ship section (D)
with flat side and bottom.
Parametric curves avoid all these limitations, and are
widely utilized in CAD systems today. Figure 5 shows how
the “difficult” ship section (Fig. 4D) is produced easily by
parametric functions y g(t), z h(t), 0 t 1, without
any steep slopes or multiple values.
3.2 Analytic Properties of Curves. In the following, we
will denote a parametric curve by x(t), the boldface letter
signifying a vector of two or three components ({x, y} for
2-D curves and {x, y, z} for 3-D curves). Further, we will
assume the range of parameter values is [0, 1].
Differential geometry is the branch of classical
geometry and calculus that studies the analytic properties
of curves and surfaces. We will be briefly presenting
and utilizing various concepts from differential
geometry. The reader can refer to the many available
textbooks for more detail; for example, Kreyszig (1959)
or Pressley (2001).