Graphics Reference
In-Depth Information
As we did before, we substitute the end values at t =0 and t =1 to get the
six conditions
NNt
=
(
0
),
==
N Nt
(
1
),
0
1
GGt
=
(
0
),
==
G Gt
(
1
),
=
0
1
CCt
=
(
0
),
==
C Ct
(
1
).
=
0
1
As before, this gives us a system of six equations in six unknowns, and
we can express that in matrix form as
N
N
G
G
C
C
100000
111111
010000
01234
A
B
C
D
E
F
0
1
0
=
.
5
002000
002612 20
1
0
1
As above, we can invert this matrix and gather all the coefficients of N 0  , N 1 , G 0  ,
G 1 , C 0  , and C 1 together. In the end, this lets us express the quintic noise func-
tion as
Nt BN BN BG BG BC BC
N
()=
+ + + + +
00 11 00 11 00 11
N
G
G
C
C
as we did for the cubic case. The coefficients are quintic functions of t :
B
=− +−
=−+=−
=− +−
11056
10
t
3
t
4
t
5
,
N
0
3
4
5
B
t
15 61
68
t
t
B
,
N
1
N
0
3
4
5
Bt
t
t
3
t
,
G
0
3
4
5
B
=− +−
473
1
2
t
t
t
,
G
1
(
)
B
=
t
2
− +−
33
t
3
t
4
t
5
,
C
0
1
2
(
) .
=
3
2
4
+
5
B
t
t
t
C
1
These equations define a quintic function for any combination of value,
gradient, and curvature at the two endpoints of the parameter interval.
As with the cubic case, if we ensure that these six conditions are the same
at the t = 1 end of one interval and at the t = 0 end of the next interval, the
combined function is not only differentiable at the point, but is also C 2 at the
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