The Geometry of Ships

THE GEOMETRY OF SHIPS 19
Fig. 13
Patches with positive, zero, and negative Gaussian curvature.
• The directions of the two principal curvatures are orthogonal,
and are called the principal directions.
• The product 1 2 of the two principal curvatures is
called Gaussian curvature K (dimensions 1/length 2 ).
• The average ( 1 2 )/2 of the two principal curvatures
is called mean curvature H (dimensions 1/length).
Normal curvature has important applications in the
fairing of free-form surfaces. Gaussian curvature is a
quantitative measure of the degree of compound curvature
or double curvature of a surface and has important
relevance to forming curved plates from flat material.
Color displays of Gaussian curvature are sometimes
used as an indication of surface fairness. Mean curvature
displays are useful for judging fairness of developable
surfaces, for which K 0 identically. Figure 13 shows
example surface patches having positive, zero, and negative
Gaussian curvature.
4.4 Continuity Between Surfaces. A major consideration
in assembling different surface entities to build a
composite surface model is the degree of continuity required
between the various surfaces. Levels of geometric
continuity are defined as follows:
G 0 : Surfaces that join with an angle or knuckle (different
normal directions) at the junction have G 0 continuity.
G 1 : Surfaces that join with the same normal direction at
the junction have G 1 continuity.
G 2 : Surfaces that join with the same normal direction
and the same normal curvatures in any direction that
crosses the junction have G 2 continuity.
The higher the degree of continuity, the smoother the
junction will appear. G 0 continuity is relatively easy to
achieve and is often used in “industrial” contexts when a
sharp corner does not interfere with function (for example,
the longitudinal chines of a typical metal workboat).
G 1 continuity is more trouble to achieve, and is widely
used in industrial design when rounded corners and fillets
are functionally required (for example, a rounding between
two perpendicular planes achieved by welding in a
quarter-section of cylindrical pipe). G 2 continuity, still
more difficult to attain, is required for the highest levels of
aesthetic design, as in automobile and yacht exteriors.
4.5 Fairness of Surfaces. As with curves, the concept
of fairness of surfaces is a subjective one. It is closely related
to the fairness of normal sections as curves.
Fairness is best described as the absence of certain
kinds of features that would be considered unfair:
• surface slope discontinuities (creases, knuckles)
• local regions of high curvature (e.g., bumps and
dimples)
• flat spots (local low curvature)
• abrupt change of curvature (adjoining regions with
less than G 2 continuity)
• unnecessary inflection points
On a vessel, because of the principally longitudinal
flow of water, fairness in the longitudinal direction receives
more emphasis than in the transverse direction.
Thus, for example, longitudinal chines are tolerated for
ease of construction, but transverse chines are very
much avoided (except as steps in a high speed planing
hull, where the flow deliberately separates from the surface).
Most surface modeling design programs provide
forms of color-coded rendered display in which each region
of the surface has a color indicating its curvature.
This can include displays of Gaussian, mean, and normal
curvature.
A sensitive way to reveal unfairness of physical surfaces
is to view the reflections that occur at low or grazing
angles (assuming a polished, reflective surface).
Reflection lines, e.g., of a regular grid, can be computed
and presented in computer displays to simulate this
process using the visualization technology known as ray
tracing. A simpler and somewhat less sensitive alternative
is to display so-called “highlight lines,” i.e., contours
of equal “slope” s on the surface; for example, s ŵ
n(u, v), where ŵ is a selectable constant unit vector and
n is the unit normal vector.
4.6 Spline Surfaces. Various methods are known to
generate parametric surfaces based on piecewise polynomials.
These include the dominant surface representations
used in most CAD programs today. Some may be
viewed as a composition of splines in the two parametric
directions (u and v), others as an extension of spline
curve concepts to a higher level of dimensionality.
From their roots in spline curves, spline surfaces inherit
the advantages of being made up of relatively simple
functions (polynomials) which are easy to evaluate,
differentiate, and integrate. A spline surface is typically
divided along certain parameter lines (its knotlines) into
subsurfaces or spans, each of which is a parametric
polynomial (or rational polynomial) surface in u and v.
Within each span, the surface is analytic, i.e., it has