Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 457

P i

R i

Q i

P i 1

P i 1

FIGURE B.40
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 perpendicular to the angle

bisector for computing the internal tangent.

to independently determine the length of the tangent vector for each segment it is associated with. Thus,
a point P i has an ending tangent vector associated with it for the segment from P i -1 to P i as well as a
beginning tangent vector associated with it for the segment P i to P i þ 1 . These tangents have the same
direction but different lengths. This relaxes the C

1
continuity instead. For example, an initial tangent vector at an interior point is determined as the vector
from P i -1 to P i þ 1 . The ending tangent vector for the segment P i -1 to P i is computed by scaling this
initial tangent vector by the ratio of the distance between the points P i and P i -1 to the distance between
points P i -1 and P i þ 1 . Referring to the segment between P i -1 and P i as P i -1 ( u ) results in Equation B.78 .
A similar calculation for the beginning tangent vector of the segment between P i and P i þ 1 results in
Equation B.79 . These tangents can be seen in Figure B.41 . The computational cost of this approach is
only a little more than the standard Catmull-Rom spline and seems to give more intuitive results.

1 continuity of the Catmull-Rom spline and uses G

Þ¼ jP i P i 1 j
jP iþ 1 P i 1 j

0

P ð 1 : 0 ð P iþ 1 P i 1 Þ (B.78)

i 1

Þ¼ jP iþ 1 P i j
jP iþ 1 P i 1 j

0

P

i ð

0 : 0 ð P iþ 1 P i 1 Þ (B.79)

P i (0.0)

P i

P i 1 (1.0)

P i ( u )

P i 1 ( u )

P i 1

P i 1

FIGURE B.41
Interior tangents based on relative segment lengths.