Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 456

( P 2 P 1 )

P 1

P 1

P 2

P 0

P 0

FIGURE B.37
Automatically forming the initial tangent of a Catmull-Rom spline.

FIGURE B.38
Catmull-Rom spline with end conditions using Equation B.77 .

P i

R i

Q i

P i 1

P i 1

FIGURE B.39
Three curve segments, ( P i-1 , P i ,P i þ1 ), ( P i-1 , Q i ,P i þ1 ), and ( P i-1 , R i ,P i þ1 ), using the standard Catmull-Rom form

for computing the internal tangent.

An advantage of Catmull-Rom is that the calculation to compute the internal tangent vectors is
extremely simple and fast. However, for each segment the tangent computation is a one-time-only cost.
It is then used repeatedly in the computation for each new point in that segment. Therefore, it often
makes sense to spend a little more time computing more appropriate internal tangent vectors to obtain
a better set of points along the segment. One alternative is to use a vector perpendicular to the plane that
bisects the angle made by P i -1 P i and P i þ 1 P i ( Figure B.40 ). This can be computed easily by adding
the normalized vector from P i -1 to P i with the normalized vector from P i to P i þ 1 .
Another modification, which can be used with the original Catmull-Rom tangent computation or
with the previously mentioned bisector technique, is to use the relative position of the internal point ( P i )