Interpolating Values - Computer Animation: Algorithms and Techniques - page 103

P 1

Original data point

Average of original data point
and estimated point

P 2

P (1/3)

P 0

P (2/3)

P (0)

P 3

P (1)

New estimate based on parabolic curve fit through the three adjacent points

FIGURE 3.39
Smoothing data by parabolic interpolation.

For the end conditions, a parabolic arc can be fit through the first, third, and fourth points, and an
estimate for the second point from the start of the data set can be computed ( Figure 3.39 ). The coef-
ficients of the parabolic equation,

2

þ bu þ c, can be computed from the constraints in
Equation 3.38 , and the equation can be used to solve for the position P 1 ¼ P

P ( u ) ¼ au

(1/3).

; P 3 ¼ P ð

2
3

P 0 ¼ P ð 0 Þ; P 2 ¼ P 1 Þ (3.38)

This can be rewritten in geometric form and the point can be constructed geometrically from the
three points P

0 1

( Figure 3.40 ) . A similar procedure can be used to estimate the
data point second from the end. The very first and very last data points can be left alone if they represent
hard constraints, or parabolic interpolation can be used to generate estimates for them as well, for

¼ P 2 þð 1 = 3 ÞðP 0 P 3 Þ

New estimate for P 1 based on parabolic curve fit through the three adjacent points

2 P 2

P 0

1

P 3

1. Construct vector from P 3 to P 0
2. Add 1/3 of the vector to P 2
3. (Not shown) Average estimated point with original data point

FIGURE 3.40
Geometric construction of a parabolic estimate for smoothing a data point.