Graphics Reference
In-Depth Information
Figure 11.6.
Possible and impossible cubic curves.
and
¢¢
()
=
(
)
¢¢
uu
pu
UM
B
=
UM
B
=
F B
,
¢¢
(11.40)
b
h
h
h
where
0000
0000
12 12 6 6
6 642
Ê
ˆ
Á
Á
Á
˜
˜
˜
M
uu
=
.
(11.41)
-
Ë
¯
-
-
-
Before moving on to other matrix descriptions of a cubic curve we pause to show
how just the geometric matrix by itself already tells us a lot about its shape. A more
thorough discussion of the shape of curves can be found in Section 11.10. First of all,
we need to realize that only a limited number of shapes are possible here because
cubic polynomials have the property that their slope can change sign at most twice
and they can have only one inflection point. For example, Figure 11.6(a) shows pos-
sible shapes and Figure 11.6(b) shows impossible ones. Secondly, although there may
be many ways to specify a cubic curve, it is uniquely defined once one knows its geo-
metric coefficients. To put it another way, if one can come up with a cubic curve that
has the same geometric coefficients as some other cubic curve, then this will be the
same
curve as the other one, no matter how the other one was defined. Having said
that we shall now show how looking at the x-, y-, and z-coordinates of a cubic curve
separately and then combining the analysis can tell us a lot about its shape and
whether it has loops or cusps.
11.3.1
Example.
Consider the following four geometric coefficient matrices
B
h
:
131
731
600
600
13 1
73 1
60 10
60 10
13
1
13 1
73 1
6010
6010
Ê
ˆ
Ê
ˆ
Ê
ˆ
Ê
ˆ
Á
Á
Á
˜
˜
˜
Á
Á
Á
˜
˜
˜
Á
Á
Á
˜
˜
˜
Á
Á
Á
˜
˜
˜
73
1
()
()
()
()
a
b
c
d
20
0
40
Ë
¯
Ë
¯
Ë
¯
Ë
¯
-
20
0
-
40
