Graphics Reference
In-Depth Information
where the a
i,j
are defined recursively by
1
a
=
1
0
,
if t s t
elsewhere
£
£
,
i
j
i
+
1
ij
,
=
,
.
s
-
-
t
t
-
s
jk
+-
1
i
ik
+
jk
+
-
1
r
r
-
1
r
+
-
1
a
=
a
+
a
,
r
>
1
.
ij
,
ij
,
i
1
,
j
t
t
t
-
t
ik
+-
1
i
ik
+
i
+
1
Proof.
See [CoLR80].
We finish this section with several other theorems that follow easily from the
multiaffine map approach to splines. First of all, an important fact that drops out of
the formalism is the differentiability of the functions N
i,k
(u), which is not totally
obvious from their recursive definitions.
11.5.2.15 Theorem.
If t = t
j+1
= t
j+2
= ...= t
j+m
is a knot of multiplicity m £ k for
N
i,k
(u), then N
i,k
(u) is C
k-1-m
at t.
Proof.
See [Seid89].
11.5.2.16
Theorem.
(Curry-Schoenberg Theorem) All splines are linear combina-
tions of B-splines.
Proof.
See [Seid89].
One can use Theorems 11.5.2.12 and 11.5.2.15 to find the Bézier control points of
a spline of order k. Simply keep inserting knots until all have multiplicity k - 1. At
that point the de Boor points reduce to the Bézier points.
11.5.2.17 Theorem.
(Variation diminishing property) A plane (line in planar case)
intersects a B-spline in no more points than it intersects the control polygon.
Proof.
See [LanR83], [Seid89], or [PieT95]. In particular, this theorem applies to
Bézier curves.
One important point about
all
the results in this section is that the proofs are very
short and straightforward. The reader should have little trouble filling in those that
are omitted.
11.5.3
Rational B-spline Curves
Although B-splines curves represent a very large class of curves, they are unable to
represent some very simple curves
exactly
. It is easy to show that conics like circles
and ellipses cannot be represented by polynomial curves and so B-spline curves can
only approximate them. This is a drawback because conics are curves that one often
wants to represent.
Fortunately, all is not lost. Conics can be represented by rational curves via a
simple trick. We show how this works in the case of a circle. Consider the circle of
