Interpolating Values - Computer Animation: Algorithms and Techniques - page 83

0
@

0
@

0
@

1
A þ

1
A

1
A =f

0
@

¼ k 1 2
p k 1 2 2

t

sin 1 t k 1

0
@

1
A =f

k 1
p=

easeðtÞ ¼

2 þ t k 1

k 1 t k 2 (3.19)

0 0 1 0 0 1

A 2

1 1

k 1
p=

2
p

t k 2
1

@ @ A sin @ @ A A =f 2 t

¼

2 þ k 2 k 1 þð

1 k 2 Þ

k 2

where f ¼ k 1 ð

=p þ k 2 k 1 þð k 2 Þð =pÞ

2 1 2

Single cubic polynomial ease-in/ease-out
A single polynomial can be used to approximate the sinusoidal ease-in/ease-out control ( Eq. 3.20 ) . It
provides accuracy within a couple of decimal points while avoiding the transcendental function 2 invo-
cations. It passes through the points (0,0) and (1,1) with horizontal beginning and ending tangents of 0.
Its drawback is that there is no intermediate segment of constant speed (see Figure 3.14 ) .

3 2

s ¼

2 t

þ

3 t

(3.20)

1

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1

FIGURE 3.14
Ease-in/ease-out polynomial

3 2 .

s ¼ 2 t þ 3 t

2

A transcendental function is one that cannot be expressed algebraically. That is, it transcends expression in algebra.