Graphics Reference
In-Depth Information
t
.
−
a
γ
(
t
)=
ζ
(22.3)
b
−
a
Inline Exercise 22.2:
Verify that in Equation 22.3,
γ
(
a
)=
P
,
γ
(
b
)=
γ
(
a
)=
v
, and
γ
(
b
)=
w
.
Q
,
This kind of substitution works in great generality: If we find a function of
t
on
[
0, 1
]
with nice properties, we can transform it to a function of
s
on
[
a
,
b
]
with
the substitution
s
=
a
+
t
(
b
a
)
,or
t
=
s
−
a
b
−
a
.
The Hermite basis functions are all cubic polynomials. We can, by a small
change in notation, write the polynomial
a
0
+
a
1
t
+
a
2
t
2
+
a
3
t
3
using matrix
multiplication:
−
⎡
⎣
⎤
⎦
1
t
t
2
t
3
a
0
+
a
1
t
+
a
2
t
2
+
a
3
t
3
=
a
0
a
3
a
1
a
2
.
(22.4)
Letting
t
(
t
)
denote a vector containing powers of
t
,or
t
(
t
)=
1
t
3
T
,
t
2
we can write
⎡
⎤
10
32
00 3
−
⎣
⎦
·
t
(
t
)
.
−
2
γ
(
t
)=[
P
;
Q
;
v
;
w
]
·
(22.5)
01
−
21
00
−
11
The first factor is a matrix, called the
geometry matrix
for the curve, and is
denoted
G
. Its columns are the coordinates of
P
,
Q
,
v
, and
w
, respectively (we'll
use the semicolon notation for this in the future as well). The middle matrix, called
the
basis matrix
and denoted
M
, contains the coefficients of the polynomials for
the Hermite curve, from lowest to highest degree. In effect, it represents the
change from the basis for cubic polynomials consisting of the four Hermite poly-
nomials to the
1,
t
,
t
2
,
t
3
{
}
basis.
Inline Exercise 22.3:
(a) Multiply out, by hand, the second and third factors in
the expression for
(
t
)
; you should get a column vector of four polynomials.
Confirm that these are the Hermite polynomials.
(b) Suppose that we had defined
t
to be the vector
t
3
γ
1
T
instead; how
t
2
t
would the second matrix in the expression for
γ
(
t
)
have to change to make the
formula correct in this case?
Inline Exercise 22.4:
Suppose that
ζ
(
t
)=(
1
−
t
)
P
+
tQ
. Write
ζ
in a matrix
form like that of Equation 22.5. Your vector
t
(
t
)
will be just
1
t
T
.
Thus, in brief, the Hermite curve can be written
γ
(
t
)=
GMT
(
t
)
.
(22.6)
All our subsequent curve formulations will have the form of Equation 22.6,
namely, a geometry matrix
G
(which usually contains four points rather than two
