Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 454

P i 1 P (1)

P P (0)

P i 1 P (1)

P i P (0)

P ( u )

FIGURE B.34
Hermite interpolation.

U T ¼ u

3 2

u u 1

2
4

3
5

2 211

33 1

0010
1000

2

M ¼

(B.74)

2
4

3
5

P i
P iþ 1
P iþ 1
P

B ¼

0

i

0

P

iþ 1

Continuity between beginning and ending tangent vectors of connected segments is ensured bymerely
using the ending tangent vector of one segment as the beginning tangent vector of the next. A composite
Hermite curve (piecewise cubic with first-order continuity at the junctions) is shown in Figure B.35 .
Trying to put a Hermite curve through a large number of points, which requires the user to specify all
of the needed tangent vectors, can be a burden. There are several techniques to get around this. One is to
enforce second-degree continuity. This requirement provides enough constraints so that the user does not
have to provide interior tangent vectors; they can be calculated automatically. See Rogers and Adams [ 18 ]
or Mortenson [ 14 ] for alternative formulations. A more common technique is the Catmull-Rom spline.

B.5.6 Catmull-Rom spline
The Catmull-Rom curve can be viewed as a Hermite curve in which the tangents at the interior control
points are automatically generated according to a relatively simple geometric procedure (as opposed to
the more involved numerical techniques referred to above). For each interior point, P i , the tangent at

FIGURE B.35
Composite Hermite curve.