Interpolating Values - Computer Animation: Algorithms and Techniques - page 90

3.2.5 Curve fitting to position-time pairs
If the animator has specific positional constraints that must be met at specific times, then the time-
parameterized space curve can be determined directly. Position-time pairs can be specified by the ani-
mator, as in Figure 3.22 , and the control points of the curve that produce an interpolating curve can be
computed from this information [ 6 ] .
For example, consider the case of fitting a B-spline 3 curve to values of the form ( P i , t i ), i ¼

1, , j.

...

Given the form of B-spline curves shown in Equation 3.23 of degree k with n þ 1 defining control

vertices, and expanding in terms of the j given constraints (2 j ), Equation 3.24 results.

Put in matrix form, it becomes Equation 3.25 , in which the given points are in the column matrix P , the
unknown defining control vertices are in the column matrix B , and N is the matrix of basis functions
evaluated at the given times ( t 1 ,

k n þ 1

t 2 , , t j ).

...

nþ 1

PðtÞ¼

1 B i N i;k ðtÞ

(3.23)

i¼

P 1 ¼ N 1 ;k ðt 1 ÞB 1 þ N 2 ;k ðt 1 ÞB 2 þ ... þ N nþ 1 ;k ðt 1 ÞB nþ 1

P 2 ¼ N 1 ;k ðt 2 ÞB 1 þ N 2 ;k ðt 2 ÞB 2 þ ... þ N nþ 1 ;k ðt 2 ÞB nþ 1

(3.24)

...
P j ¼ N 1 ;k ðt j ÞB 1 þ N 2 ;k ðt j ÞB 2 þ ... þ N nþ 1 ;k ðt j ÞB nþ 1

P ¼ NB (3.25)

¼
j ), then N is square and the defining control vertices can be solved by inverting the matrix, as in
Equation 3.26 .

If there are the same number of given data points as there are unknown control points (2 knþ 1

B ¼ N 1

P (3.26)

P 2

P 4

P 1

time 10

P 5

time 50

time 0

time 55

P 3

P 6

time 35

time 60

FIGURE 3.22
Position-time constraints.

3 Refer to Appendix B.5.12 for more information on B-spline curves.