Graphics Reference
In-Depth Information
PðuÞ¼
F
0
ðuÞ
F
1
ðÞ
¼FB
T
½
P
0
P
1
(B.68)
¼ U
T
A
a
1
a
0
PðuÞ¼ u
½
1
(B.69)
P
0
P
1
¼ U
T
MB ¼ FB ¼ U
T
A
11
10
PðuÞ¼ u
½
1
(B.70)
B.5.3
Parameterization by arc length
It should be noted that in general there is not a linear relationship between changes in the parameter
u
and the distance traveled along a curve (its
arc length
). It happens to be true in the previous example
concerning a straight line and the parameter
u.
However, as Mortenson [
14
] points out, there are other
equations that trace out a straight line in space that are fairly convoluted in their relationship between
changes in the parameter and distance traveled. For example, consider
Equation B.71
, which is linear in
P
0
and
P
1
. That is, it traces out a straight line in space between
P
0
and
P
1
. However, it is nonlinear in
u.
As a result, the curve is not traced out in a nice monotonic, constant-velocity manner. The nonlinear
relationship is evident in most parameterized curves unless special care is taken to ensure constant
velocity.
PðÞ¼P
0
þ
ð
ð
1
u
Þu þ u
Þ P
1
P
0
ð
Þ
(B.71)
B.5.4
Computing derivatives
One of the matrix forms for parametric curves, as shown in
Equation B.70
for linear interpolation, is
U
T
MB
. Parametric curves of any polynomial order can be put into this matrix form. Often, it is useful to
compute the derivatives of a parametric curve. This can be done easily by taking the derivative of the
U
vector. For example, the first two derivatives of a cubic curve, shown in
Equation B.72
,
are easily eval-
uated for any value of
u
.
PðuÞ¼U
T
MB ¼ ju
3
2
u
u
1
jMB
0
T
0
P
ðuÞ¼U
MB ¼
j
3
2
u
10
jMB
(B.72)
3
u
00
T
00
P
ðuÞ¼U
MB ¼j
6
u
200
jMB
B.5.5
Hermite interpolation
Hermite interpolation generates a cubic polynomial from one point to another. In addition to specifying
the beginning and ending points (
P
i
,
P
i þ
1
), the user needs to supply beginning and ending tangent
vectors (
P
0
i
,
P
0
i þ
1
) as well (
Figure B.34
)
. The general matrix form for a curve is repeated in
Equation B.73
, and the Hermite matrices are given in
Equation B.74
.
PðÞ¼U
T
MB
(B.73)
Search WWH ::
Custom Search