Graphics Reference
In-Depth Information
they are easy to compute. For that reason it is worthwhile to collect together in one
place some facts about them. The emphasis in this section will be on their basic matrix
representation and how it can be used to analyze the curves. Additional matrix rep-
resentations will be encountered in the sections on Bézier curves and B-splines. We
should point out though that matrix representations are not always the fastest or the
most numerically stable representations. See Section 11.15.
To begin with, it is easy to see, by collecting together terms with the same power
of u, that every cubic curve in
R
m
can be written in the form
()
=+ +
2
3
,
pu
aa a
u
u
+
a
u
(11.34)
0
1
2
3
where the
a
i
are vectors in
R
m
. For example,
(
)
()
=
2
3
3
2
pu
23 375
u
+
u
-
u
+
,
u
+
u
+
u
+
2
can be written as
()
=
(
)
+-
(
)
+
(
)
2
+
(
)
3
pu
372
,,
0 31
, ,
u
201
,,
u
015
,,
u
.
Suppose that one wants to use a polynomial as in (11.34) to design curves. The
a
i
are then the unknowns and in this representation of the function they are what has
to be determined. However, the same function can be specified in many different ways.
The most convenient way to specify the parameterization depends on what one is
doing. Specifying the
a
i
directly is usually the least convenient. Hermite interpolation
was basically a case where one wanted to specify the curve by means of its endpoints
and its tangent vectors at those points. This is a more geometric approach but there
are others. Given that curves can be represented in different ways it is desirable to be
able to switch between representations. We show in this and later sections that matri-
ces can be used effectively for this task.
We shall abbreviate p(c) and p¢(c) to
p
c
and
p
c
, respectively.
Notation.
From now on, unless stated otherwise, the domain of our cubic curve is assumed
to be [0,1]. This assumption leads to simplified formulas but the results in this case
translate easily into corresponding results for other domains. See the comments at
the end of this section.
Define matrices
a
a
a
a
p
p
p
p
Ê
ˆ
Ê
ˆ
0
3
Á
Á
Á
˜
˜
˜
Á
Á
Á
˜
˜
˜
=
(
)
1
2
32
U
u
u
u
1,
A
=
,
and
B
=
.
(11.35)
h
u
1
0
Ë
¯
Ë
¯
u
0
1
Definition.
The vectors
a
i
are called the
algebraic coefficients
of the cubic curve p(u).
The elements of
B
h
are called its
geometric
or
Hermite coefficients
.
