The Geometry of Ships

THE GEOMETRY OF SHIPS 13
Fig. 6
Curvature profile graph and porcupine display of curvature distribution. Both tools are revealing undesired inflection points in the curve.
However, this explicit definition will be unusable if the
waterline endings include a rounding to centerline at either
end, because dy/dx would be infinite at such an end;
splines are piecewise polynomials, and no polynomial
can have an infinite slope. Because of such limitations,
explicit spline curves are seldom used. A parametric
spline curve x X(t), y Y(t), z Z(t) (where each of
X, Y, and Z is a spline function, usually with the same
knots) can turn in any direction in space, so it has no
such limitations.
3.5 Interpolating Splines. A common form of spline
curve, highly analogous to the drafting spline, is the cubic
interpolating spline. This is a parametric spline in 2-D or
3-D that passes through (interpolates) a sequence of N 2-
D or 3-D data points X i , i 1,...N. Each of the N-1 spans
of such a spline is a parametric cubic curve, and at the
knots the individual spans join with continuous slope and
curvature. It is common to use a knot at each interior
data point, although other knot distributions are possible.
Besides interpolating the data points, two other issues
need to be resolved to specify a cubic spline uniquely:
(a) Parameter values at the knots. One common way
of choosing these is to divide the parameter space uniformly,
i.e., the knot sequence {0, 1/(N 1), 2/(N
1),...(N 2)/(N 1), 1}. This can be satisfactory when
the data points are roughly uniformly spaced, as is sometimes
the case; however, for irregularly spaced data,
especially when some data points are close together,
uniform knots are likely to produce a spline with loops
or kinks. A more satisfactory choice for knot sequence is
often chord-length parameterization: {0, s 1 /S, s 2 /S,...,1},
where s i is the cumulative sum of chord lengths
(Euclidean distance) c i between data points i 1 and i,
and S is the total chord length.
(b) End conditions. Let us count equations and
unknowns for an interpolating cubic spline. First, the unknowns:
there are N 1 cubic spans, each with 4D coefficients,
where D is the number of dimensions (two or
three), making a total of D(4N 4) unknowns.
Interpolating N D-dimensional points provides ND equations,
and there are N 1 knots, each with three continuity
conditions (value, first and second derivatives), for
a total of D(4N 6) equations. Therefore, two more conditions
are needed for each dimension, and it is usual to
impose one condition on each end of the spline. There
are several possibilities:
• “Natural” end condition (zero curvature or second
derivative)
• Slope imposed
• Curvature imposed
• Not-a-knot (zero discontinuity in third derivative at
the penultimate knot).
These can be mixed, i.e., there is no requirement that
the same end condition be applied to both ends or to all
dimensions.
3.6 Approximating or Smoothing Splines. Splines are
also widely applied as approximating and smoothing
functions. In this case, the spline does not pass through
all its data points, but rather is adjusted to pass optimally
“close to” its data points in some defined sense such as
least squares or minmax deviation.