Practical Algorithms for 3D Computer Graphics, 2nd Edition

2.16. Splines 51

equations for the x component of the Ks. Written in matrix form these are:









0 0 0 1

1 1 1 1

0 0 1 0

3 2 1 0

















K

3

x

K

2

x

K

1

x

K

0

x









=









x

i

x

i+1

x

′

i

x

′

i+1









On solution the following expr es s ion s are obtained:

K

3

x

= 2x

i

− 2x

i+1

+ x

′

i

+ x

′

i+1

K

2

x

= −3x

i

+ 3x

i+1

− 2x

′

i

− x

′

i+1

K

1

x

= x

′

i

K

0

x

= x

i

If P

i

and P

i+1

are part of a set of points making up a path, such as, ... P

i−1

,

P

i

, P

i+1

, P

i+2

..., the Ks are obtained using the valu e of p at P

i−1

, P

i

, P

i+1

and P

i+2

as follows:

K

3

x

= −

1

2

x

i−1

+

3

2

x

i

−

3

2

x

i+1

+

1

2

x

i+2

(2.34)

K

2

x

= x

i−1

−

5

2

x

i

+ 2x

i+1

−

1

2

x

i+2

(2.35)

K

1

x

= −

1

2

x

i−1

+

1

2

x

i+1

(2.36)

K

0

x

= x

i

(2.37)

Similar expressions may be written for K

3

y

, K

3

z

, etc., an d thus the