Keyframe Animation: Understanding Keyframe Channels, Tangents, and Cubic Coefficients, Slides of Computer Graphics and Animation

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Keyframe Channels

Keyframe Channel

 A channel can be stored as a sequence of

keyframes

 Each keyframe has a time and a value

and usually some information describing

the tangents at that location

 The curves of the individual spans

between the keys are defined by 1-D

interpolation (usually piecewise Hermite)

Keyframe

Time

Value

Tangent In

Tangent Out

Keyframe (time,value)

Keyframe Tangents

 Keyframes are usually drawn so that the incoming tangent points to the left (earlier in time)  The arrow drawn is just for visual representation and it should be remembered that if the two arrows are exactly opposite, that actually means the tangents are the same!  Also remember that we are only dealing with 1D curves now, so the tangent really just a slope

Animating with Keyframes

 Keyframed channels form the foundation

for animating properties (DOFs) in many

commercial animation systems

 Different systems use different variations

on the exact math but most are based on

some sort of cubic Hermite curves

Curve Fitting

 Keyframes can be generated automatically from sampled data such as motion capture  This process is called ‘curve fitting’, as it involves finding curves that fit the data reasonably well  Fitting algorithms allow the user to specify tolerances that define the acceptable quality of the fit  This allows two way conversion between keyframe and raw formats, although the data might get slightly distorted with each translation

Tangent Rules

 Rather than store explicit numbers for tangents, it is often more convenient to store a ‘rule’ and recompute the actual tangent as necessary  Usually, separate rules are stored for the incoming and outgoing tangents  Common rules for Hermite tangents include:  Flat (tangent = 0)  Linear (tangent points to next/last key)  Smooth (automatically adjust tangent for smooth results)  Fixed (user can arbitrarily specify a value)  Remember that the tangent equals the rate of change of the DOF (or the velocity)  Note: I use ‘v’ for tangents (velocity) instead of ‘t’ which is used for time

Flat Tangents

 Flat tangents are particularly useful for making ‘slow in’ and ‘slow out’ motions (acceleration from a stop and deceleration to a stop)

v = 0

Smooth Tangents

(p 1 ,t 1 ) v

1

out

v 1 in

2 0

2 0 1 1 t t

p p v v

in out −

− = =

(p 2 ,t 2 )

(p 0 ,t 0 )

 Keep in mind that this won’t work on the first or last tangent (just use the linear rule)

Step Tangent

 Occasionally, one comes across the ‘step’ tangent rule  This is a special case that just forces the entire span to a constant  This requires hacking the cubic coefficients (a=b=c=0, d=p 0 )  It can only be specified on the outgoing tangent and it nullifies whatever rule is on the next incoming tangent

Cubic Equation (1 dimensional)

f ( ) t = at^3 + bt^2 + ct + d at bt c

dt

df

( ) [ ]

d

c

b

a

f t t^3 t^2 t 1 [^ ]

d

c

b

a

t t

dt

df

Hermite Curve (1D)

v 1

p 1 p 0

v 0

t 0 =0 t 1 =

Hermite Curves

d

c

b

a

v

v

p

p

v a b c

v c

p a b c d

p d

1

0

1

0

1

0

1

0

Matrix Form of Hermite Curve

−

1

0

1

0

1

0

1

0

1

v

v

p

p

d

c

b

a

v

v

p

p

d

c

b

a