Interpolating Values - Computer Animation: Algorithms and Techniques - page 102

P i

Average of original data point
and estimated point

Original data point

P i 1

P (1/2)

P i 1

P (3/4)

P (1/4)

P i 2

P i 2 P (1)

P (0)

New estimate for P i based on cubic curve fit through the four adjacent points

FIGURE 3.37
Smoothing data by cubic interpolation.

evaluated at u ¼ 1/2; and the result is averaged with the original data point (see Figure 3.37 ) . Solving

for the coefficients and evaluating the resulting cubic curve is a bit tedious, but the solution needs to be
performed only once and can be put in terms of the original data points, P i -2 , P i -1 , P iþ 1 , and P iþ 2 . This is
illustrated in Figure 3.38 .

3 2

P ðuÞ¼au þ bu þ cu þ d (3.36)

P i 2 ¼ P ð 0 Þ¼d

1
64 þ b

1
16 þ c

1
4 þ d

P i 1 ¼ P ð 1 = 4 Þ¼a

(3.37)

27
64 þ b

9
16 þ c

3
4 þ d

P iþ 1 ¼ P ð 3 = 4 Þ¼a

P iþ 2 ¼ a þ b þ c þ d

New estimate for P i based on cubic curve fit through the four adjacent points

3

2

P i 1

P i 1

1

P i 2

P i 2

1. Average P i 1 and P i 1
2. Add 1/6 of the vector from P i 2 to P i 1
3. Add 1/6 of the vector from P i 2 to P i 1 to get new estimated point
4. (Not shown) Average estimated point with original data point

FIGURE 3.38
Geometric construction of a cubic estimate for smoothing a data point.