Graphics Reference
In-Depth Information
is actually an approximation to the answer but is accurate enough for computer
graphics display purposes. His answer reduces to a simple formula.
The interpolation in this section assumed that the values u
i
at which the interpo-
lating spline took on the value
p
i
were also its knots. One can be more general and
look for an interpolating spline for which not only the parameter values but also the
knots are specified (the two sequences could be different). For a solution to that
problem and interpolation by higher-order B-splines see [PieT95].
11.6
Nonlinear Splines
When we defined splines we pointed out that the reason for cubic splines being so
popular is that they are low-degree polynomials and yet provide very good approxi-
mations to many of the curves needed in CAD and CAGD. Specifically, they are a good
substitute for the physical splines that were used in the past. To make this argument,
however, we need to know what physics tells as about how flexible rods bend. It is
not possible to delve into the justification from physics here, but one can show that
the so-called mechanical and wooden splines defined below are two models for a curve
that describes the shape of a bent rod.
Definition.
Let F(s) be an arc-length parameterized curve with domain [0,L] and
curvature function k(s) that interpolates a fixed set of points. The curve F(s) is called
a
mechanical spline
if it minimizes the energy functional
L
Ú
k
2
ds
.
(11.114)
0
It is called a
wooden spline
if
2
d
ds
k
=
0
.
(11.115)
2
The mechanical and wooden splines are called
nonlinear splines
whereas the poly-
nomial splines defined in Section 11.2.3 would be called “linear” splines. The reason
for this is that a linear combination of two polynomial splines with the same knots
and degree would again be a spline of that type. In fact, such splines form a finite
dimensional linear vector space. On the other hand, a linear combination of mechan-
ical or wooden splines would not have the right curvature and hence not be a spline
of that same type. The rest of this section will point out a few facts about nonlinear
splines that make them interesting in graphics. Good references are [Mehl74],
[Malc77], or [HosL93]. For a more mathematical introduction to nonlinear splines
see [Wern79]. We shall stick to the case of planar curves and use signed curvature in
the discussion below.
First, consider a mechanical spline. If it is the graph of a function y = f(x) over
some interval [a,b], then the integral in (11.114) that defines it turns into
