Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 462

1

0.8

0.6

0.4

0.2

0.5 1 1.5 2

Default tension, t 0

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.5 1 1.5 2 0.5 1 1.5

Low tension, t 3/4 High tension, t 3/4

FIGURE B.47
The effect of varying the tension parameter.

The three parameters, tension, continuity, and bias, are combined in Equation B.88 .

ð ð 1 t Þ ð 1 þ c Þ ð 1 þ b Þ Þ ð ð 1 t Þ ð 1 c Þ ð 1 b Þ Þ

T i ¼

ð P i P i 1 Þ þ ð P iþ 1 P i Þ

2 2

(B.88)

ð ð 1 t Þ ð 1 c Þ ð 1 þ b Þ Þ ð ð 1 t Þ ð 1 þ c Þ ð 1 b Þ Þ

T i ¼ ð P i P i 1 Þ þ ð P iþ 1 P i Þ

2 2

B.5.12 B-splines
B-splines are the most flexible and useful type of curve, but they are also more difficult to grasp intu-
itively. The formulation includes Bezier curves as a special case. The formulation for B-spline curves
decouples the number of control points from the degree of the resulting polynomial. It accomplishes
this with additional information contained in the knot vector . An example of a uniform knot vector is
[0,1,2,3,4,5,6,..., n þ k 1], in which the knot values are uniformly spaced apart. In this knot
vector, n is the number of control points and k is the degree of the B-spline curve. The parametric value