Graphics Reference
In-Depth Information
(5) Bézier curves can be considered as a special case of B-spline curves. For
example, the four B-splines from the periodic spline over the knot sequence
0,0,0,0,1,1,1,1 give the cubic Bernstein polynomials.
11.5.2
The Multiaffine Approach to B-Splines
This section describes another geometric approach to Bézier curves and B-splines.
The approach taken in the previous two sections is in a sense rather ad hoc but it has
the advantage that it is easier to follow initially. On the other hand, it gets messy and
complicated to prove anything. The multiaffine approach in contrast may seem totally
confusing initially with all of its sub- and superscripts, but, once one gets over that,
lots of properties of Bézier and B-spline curves become really trivial and very geo-
metric. The core of the material in this section comes from [Seid89]. See also [Seid93]
and [Rock93].
It is an old fact in mathematics that there is a kind of duality between degree
k
polynomials of
one
variable and multivariate polynomials that have degree
1
in
each
variable. To show this we need some definitions. Let
V
and
W
be real vector spaces.
Definition.
A function g :
V
Æ
W
is said to be an
affine
map if
k
k
Ê
Á
ˆ
˜
=
ÂÂ
()
ga
v
ag
v
ii
i
i
i
=
1
i
=
1
k
Â
A function f :
V
d
for all k > 0,
v
i
Œ
V
, and a
i
Œ
R
satisfying
a
i
=
1
.
Æ
W
is said to be
i
=
1
a
multiaffine
map if for all i and
v
1
,
v
2
,...,
v
i-1
,
v
i+1
,...,
v
d
Œ
V
, the map g
i
:
V
Æ
W
defined by
()
=
(
)
g
v
f
v
,
v
,...,
v
, ,
v v
,...,
v
i
12
i
-
1
i
+
1
d
is an affine map.
The definition of an affine map given here agrees with the “usual” definition of an
affine map, namely, that it is a map that sends lines to lines. (In the case of maps from
R
n
to
R
n
this follows from Theorem 2.5.9 in [AgoM05].) In this section the barycen-
tric coordinate preserving property of affine maps is emphasized because that is the
key to everything that we do here. We will be using this property over and over again.
Proofs of results will be trivial as long as one keeps this in mind. Figure 11.21 shows
the critical property of linear maps when using barycentric coordinates. If the linear
map from one simplex to another maps a point
p
to the point
p
¢, then both
p
and
p
¢
have the
same
barycentric coordinates, albeit with respect to different vertices. The
map simply replaced the vertices in the barycentric coordinate representation.
If
V
=
R
and
W
=
R
m
, which is the special case of interest to us, then g(u) =
(g
1
(u),g
2
(u),...,g
m
(u)) and g is an affine map if and only if each g
i
is a polynomial of
degree 1. Furthermore, a multiaffine map is then simply a polynomial of degree 1 in
each variable separately.
