Graphics Reference
In-Depth Information
p
p
p
p
Ê
ˆ
0
Á
Á
Á
˜
˜
˜
1
P
=
.
2
Ë
¯
3
Then the so-called four-point matrix form of a cubic curve is
p
()
=
UM P
,
(11.42)
4
where the
four-point matrix
M
4
is defined by
9
2
27
2
27
2
9
2
Ê
-
-
ˆ
Á
Á
Á
Á
˜
˜
˜
˜
45
2
9
2
9 18
9 1
1000
-
-
M
4
=
.
(11.43)
11
2
9
2
-
-
Ë
¯
The geometric matrix
B
h
and
P
are related by the equation
B
h
=
LP
, where
1000
0001
9
Ê
ˆ
Á
Á
Á
˜
˜
˜
-
h
1
LMM
=
=
.
(11.44)
4
11
2
9
2
-
-
1
9
2
11
2
Ë
-
1
-
9
¯
The discussion up to here has centered on cubic curves with domain [0,1]. What
happens in the case of a different domain [a,b]? One can give a similar analysis, except
that the “geometric coefficients” for such a cubic curve would have to be based on the
values and tangents at a and b, rather than at 0 and 1. Other than that one could
proceed pretty much as before. Note that the Hermite matrix M
h
can no longer be
used, but there would be a matrix that plays the same role but based on a and b.
Exercise 11.3.4 asks the reader to work out one example of such a change.
11.4
Bézier Curves
This section and the next will deal with curves that are defined by control points but
do not interpolate them in general. We shall return to the interpolation problem in
Section 11.5.5.
Although the geometric coefficients approach to defining curves is a big improve-
ment over having to specify the algebraic coefficients, specifying tangent vectors in
an interactive computer graphics environment is still somewhat technical. A better
way allows a user to specify these vectors implicitly by simply picking points that
suggest the desired shape of the curve at the same time. Figure 11.8 shows a cubic
curve p(u) which starts at
p
0
and ends at
p
3
. It is very easy to specify the tangent lines
