The Geometry of Ships

28 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
a list of 3-D points, called vertices or nodes, and
a list of faces, each face being an ordered list of vertices
which form a closed polygon.
The lines connecting adjacent vertices in a face are
called edges or links. An edge can be shared by two adjacent
faces, or it can belong to only one face, when it is
part of the mesh boundary. If no edge is shared by more
than two faces, the mesh is said to have manifold topology.
Figure 24 (a) is a small example of a triangle mesh.
It has five vertices:
1: 1.0, 1.0, 0.0
2: 1.0, 1.0, 0.0
3: 0.0, 0.0, 2.0
4: 1.0, 1.0, 0.0
5: 1.0, 1.0, 0.0
four faces:
1: 1, 2, 3
2: 2, 5, 3
3: 5, 4, 3
4: 4, 1, 3
and eight edges. The four edges connecting to vertex 3
are each shared by two faces. The four edges at the plane
Z 0 each belong to only one face, so they form the
boundary of the mesh.
A polygon mesh, and especially a triangle mesh, is
easy to render for display as either a surface or a solid.
It is also a commonly accepted representation for many
kinds of 3-D analysis, e.g., aerodynamic and hydrodynamic
flows, wave diffraction, radar cross-section, and
finite element methods.
5.2 Subdivision Surfaces. Given a polygon mesh
consisting of triangle and/or quad polygons, it is easy to
generate a finer polygon mesh by the following linear
subdivision rule:
• insert a new vertex at the center of each original edge,
and at the center of any quad polygon; then
• connect the new vertices with new edges, so each
original face is split into four new faces.
This subdivision can be repeated any number of
times, generating successive meshes of smaller and
smaller polygons. However, subdivision alone does not
improve the smoothness of the mesh; each new face
constructed this way would be exactly coincident with a
portion of the original face that it is descended from.
The key idea of subdivision surfaces is to follow (or
combine) such a subdivision step with a smoothing
step that repositions each vertex to a weighted average
of a small set of neighboring vertices. Then the successive
meshes become progressively smoother, approaching
C 2 continuity (comparable to cubic splines)
at almost all points, and C 1 continuity everywhere, in
the limit of infinite subdivision. There are several competing
schemes for choosing the set of neighbors and
assigning weights.
As an example, Fig. 24 (b) and (c) show the original
“coarse” triangle mesh of Fig. 24 (a) following one and
two cycles of Loop subdivision.
The vertices and edges of the coarse mesh can be
interpreted as a “control point net,” similar in effect to
the control net for a B-spline or NURBS parametric
surface. For example, Fig. 24 (d) shows the effect of
moving vertex 3 to (1.0, 0.5, 2.0) and regenerating
the mesh.
Smoothing rules can be modified at specified vertices
or chains of vertices, to allow breakpoints and breaklines
in the resulting surface.
A subdivision surface has the following attractive
properties, similar to B-spline and NURBS surfaces:
• Local support: A given control point affects only a
local portion of the surface.
• Rigid body: The shape of the surface is invariant with
respect to a rigid body displacement or rotation of the
control net.
• Affine stretching: The surface scales affinely in response
to affine scaling of the net.
• Convex hull: The surface does not extend outside the
convex hull of the control points.
Compared with parametric surfaces, subdivision surfaces
are far freer in topology. The surface inherits the
topology of its control net. A subdivision surface can
have holes, any number of sides, or no sides at all. (A
closed initial net produces a closed surface.)
A major disadvantage of subdivision surfaces as of
this writing is a lack of standardization. Because different
CAD systems employ different subdivision and
smoothing algorithms, subdivision surfaces cannot generally
be exchanged between systems in a modifiable
form. In the subdivision world, there is not yet any equivalent
of the IGES file. (Of course, there are many file formats
for exchanging the triangle meshes that result from
subdivision.)