The Geometry of Ships

THE GEOMETRY OF SHIPS 25
A typical computational method might take the following
steps:
• Intersect two meshes to find candidate starting
locations.
• Use Newton-Raphson iteration to refine such a start,
finding one accurate point on a candidate intersection.
• “Tracking”: Use further Newton-Raphson steps to find
a series of intersection points stepping along the intersection.
Be prepared to take smaller steps if the curvature
of the intersection increases.
• Terminate tracking when you come to an edge of either
surface, or return to the starting point of a closed loop.
• Assemble the two directions into a single curve and
select a suitable parameterization for it.
• Substitute a spline approximation for the intersection
as a 3-D curve, and two other spline approximations as
2-D parametric curves in each of the surfaces.
However, you can see that this simplified procedure
does not deal with the majority of the difficulties mentioned
in the preceding paragraph.
An obvious conclusion from this list of difficulties is
to avoid surface-surface intersections as much as possible.
Nevertheless, most CAD systems are heavily dependent
on such intersections. Users are encouraged to
generate oversize surfaces that deliberately intersect,
solve for intersections, and trim off the excess. This one
problem explains the bulk of the slow performance and
unstable behavior that is so common in solid modeling
software.
Relational geometry provides construction methods
for durable “watertight” junctions that can frequently
avoid surface-surface intersection. These often take the
form of designing the intersection as an explicit curve,
then building the surfaces to meet it. Two transfinite surfaces
that share a common edge curve will join accurately
and durably along that edge. A transfinite surface
that meets a snake on another surface will make a
durable, watertight join. An intersection of a surface
with a plane, circular cylinder, or sphere can be cut
much more efficiently by an implicit surface. Intersections
with general cylinders and cones are performed
much more efficiently as projections.
Nevertheless, there are situations where surface-surface
intersections are unavoidable, so there is an
Intersection snake (IntSnake) that encapsulates this
process. The IntSnake is supported by a magnet on the
host surface, which is used as a starting location for the
initial search; this helps select the desired intersection
curve when there are two or more intersections, and
also specifies the desired parametric orientation.
4.16 Trimmed Surfaces. A general limitation of parametric
surfaces is that they are basically four-sided
objects. This characteristic arises fundamentally from
the rectangular domain in the u, v parameter space. If
we look around us at the world of manufactured goods,
we see a lot of surfaces that are four-sided, but there
are a lot of other surfaces that are not. Parametric surfaces
with three sides are generally supported in CAD
by allowing one edge of a four-sided patch to be degenerate,
but this requires a coordinate singularity (pole)
at one of the three corners (Fig. 11). Parametric surfaces
with more than four sides are also possible (e.g.,
a Coons patch with a knuckle in one or more of its
sides), but such a surface will have awkward slope discontinuities
in its interior. A parametric surface with a
smooth (e.g., circular or oval) outline, with no corners,
is also possible, but involves either a pole singularity
somewhere in the interior, two poles on the boundary,
or “squash” singularities at two or four places on the
boundary. Surface slopes and curvatures are likely to
be irregular at any of these coordinate singularities or
discontinuities.
The use of trimmed surfaces is the predominant way
to gain the flexibility in shape or outline that parametric
surfaces lack. A trimmed surface is a portion of a base
surface, delineated by one or more loops of trimming
curves drawn on, or near, the surface (Fig. 22).
The base surface is frequently a parametric surface,
but in many solid modeling CAD systems it can be an implicit
surface such as a sphere, cylinder, or torus. In general,
the trimming curves can be arbitrarily complex as
long as they link up into closed loops and do not intersect
themselves or other loops. One loop is designated
as the “outer” loop; any other loops enclosed by the
outer loop will represent holes.
4.17 Composite Surfaces. A composite surface is the
result of assembling a set of individual trimmed or
untrimmed surfaces into a single 2-D manifold. Besides
the geometries of the individual component surfaces, a
composite surface stores the topological connections
between them — which edges of which surfaces adjoin.
The most common application of composite surfaces
is for the outer and inner boundaries of B-rep solids. In
this case, the composite surface is required to be topologically
closed. However, there are definite applications
Fig. 22 A trimmed surface is the portion of a base surface bounded by
trimming curves. In this case, the base surface for the transom is an
inclined circular cylinder.