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3D Game Engine Design, 2nd Edition
book

3D Game Engine Design, 2nd Edition

by David Eberly
November 2006
Intermediate to advanced content levelIntermediate to advanced
1040 pages
26h 31m
English
CRC Press
Content preview from 3D Game Engine Design, 2nd Edition
11.5 B-Spline Curves 551
The boundary condition S
0
(t
0
) = S
n1
(t
n
) implies
b
0
= b
n1
+ 2c
n1
h
n1
+ 3d
n1
h
2
n1
We also know that
b
0
=
a
1
a
0
h
0
(2c
0
+ c
1
)h
0
3
b
n1
=
a
n
a
n1
h
n1
(2c
n1
+ c
n
)h
n1
3
d
n1
=
c
n
c
n1
3h
n1
Substituting these quantities in the last constraint yields
h
n1
c
n1
+ 2(h
n1
+ h
0
)c
0
+ h
0
c
1
= 3
a
1
a
0
h
0
a
0
a
n1
h
n1
(11.18)
Equations (11.12), (11.17), and (11.18) form a linear system of equations. It is not
tridiagonal, but the cyclical nature of the matrix allows you to solve the system in
O(n) time.
The natural-, clamped-, and closed-spline interpolations were defined for fitting
a sequence of scalar values, but they can be simply extended to curves
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Publisher Resources

ISBN: 9781482267303