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Graphics II 91.547

AnimationIntroduction and Motion Control

Session 6

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Animation

0 Traditional 2D animation- Origins in late 1920s- Flat shading- Illusion of 3D produced by fluidity of characters, use of

perspective, motion of “virtual camera”- Disney animators pioneered major techniques

=“Squash and stretch”=Secondary action=Appeal

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Advantages of Computer Animation

0 Eliminates requirements of building models0 No restriction on camera movement0 Easy inclusion of shading models0 Can introduce physical models

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Animation Taxonomy

0 Representational animation- Rigid objects

=Single, unchanging model for each object- Articulated objects

=Rigid subobjects, connected at joints=Motions generally revolute

- Soft objects=Model is deformed

0 Procedural animation0 Stochastic animation0 Behavioral animation

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Motion Control: Keyframing

In computer graphics animation, keyframe -> key parameter. Therefore selection of the parameter becomes critical in defining appropriate motion.

Interpolating angle Interpolating endpoints

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Motion control of rigid objectsParameterization of position

Rigid Objects

Articulated Objects

Soft Objects

Position

Orientation

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Spline-driven Position Animation

Q(u)

Equal arc length, s

Equal u

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Arclength Parameterization of Splines

s A u ( )

Eval. Eval.Spline

su A s 1( )

u

s

Evaluate numerically

(arc length along spline)

Spline parameter

s

x,y,z

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Arclength Parameterization of Splines

s A u

Q u Q A s

ds dx dy dz

dsdx

du

dy

du

dz

dudu

s udx

du

dy

du

dz

dudu

u

u

( )

( ) ( ( ))

( )

( )

1

2 2 2

2 2 2

2 2 2

0

12

12

12

Arclength along spline:

Q u( )

Q u du( )

z

x

y

dx

dy

dz

Integrating gives:

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Arclength Parameterization of SplinesQ u au bu cu d

x u a u b u c u d

y u a u b u c u d

z u a u b u c u d

s u Au Bu Cu Du E du

A a a a

B a b a b a b

C a c a c a

x x x x

y y y y

z z z z

u

u

x y x

x x y y z z

x x y y

( )

( )

( )

( )

( ) ( )

( )

( )

(

3 2

3 2

3 2

3 2

4 3

0

2

2 2 2

12

9

12

6 z z x y x

x x y y z z

x y x

c b b b

D b c b c b c

E c c c

) ( )

( )

4

4

2 2 2

2 2 2

General form ofCubic Spline:

Taking the derivativesand integrating:

Where:

This function will not integrate analytically, so integration must bedone numerically.

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Forward Differencing Approach to Evaluating A

ui

ui1

ui2

ui3

di1

di2

di3

s u s n u d

d Q u Q u

ii

n

i i i

( ) ( )

( ) ( )

1

1

Where:

Q s( )

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Ease-in, ease-out motionArc length no longer proportional to time

Q(t)

Equal t

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Velocity Curves

t

s

Velocity Curve

V t u s u( ( ), ( ))

V t u s u

t a u b u c u d

s a u b u c u do

( ( ), ( ))

0

30

20

13

12

1 1

t

u

BisectionSearch

Eval.Cubict

us

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Velocity Curves

s3

Q u( )

Position Splinet

s

1

2 V t u s u( ( ), ( ))

Velocity Curve

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Velocity Curves

Q u( )

Position Spline

t

s

Velocity Curve

Equal timeIntervals

GentleAccelerationfrom Rest

Gentle decelerationto Rest

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General Kinetic Control(Steketee & Badler 1985)

0 “Position Spline”- Let the motion parameter to be interpolated be . is

specified at n key values, . The position spline is constructed by assigning a keyframe number to each key value and interpolating through the resulting tuples:

0 “Kinetic Spline”- Each keyframe number is assigned a time. The kinetic

spline interpolates through the resulting pairs:

0 1 1, ,..., n

( , ),( , ),...,( , ) 0 1 10 1 1n n

( , ), ( , ), ..., ( , )0 1 10 1 1t t n tn

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Parameterization of Orientation:Euler Angles

x

y

z

x

y

z

x

y

z

1 0 0 0

0 0

0 0

0 0 0 1

1 1

1 1

cos sin

sin cos

Transformation:

cos sin

sin cos

2 2

2 2

0 0

0 1 0 0

0 0

0 0 0 1

cos sin

sin cos

3 3

3 3

0 0

0 0

0 0 1 0

0 0 0 1

Transformation: Transformation:

12

3

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Multiple Ways to Define Rotation withEuler Angles

cos cos cos sin sin

sin sin cos cos sin sin sin sin cos cos sin cos

cos sin cos sin sin cos sin sin sin cos cos sin

2 3 2 3 2

1 2 3 1 3 1 2 3 1 3 1 2

1 2 3 1 3 1 2 3 1 3 1 2

0

0

0

0 0 0 1

Rx -> Ry -> Rz

Rx -> Rz -> RyRy -> Rx -> RzRy -> Rz -> RxRz -> Rx -> RyRz -> Ry -> Rx

Possible Orderings:

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Problems with Euler Angle Parameterization:Gimbal Lock

x

y

z

1

x

y

z

2 2

x

y

z

x’

3 1 now has same effect as

3

Selecting arotation about y removes a degree of freedom.

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Problems with Euler Angle Parameterization:Interpolation

y

x

z

y

x

z

X roll

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Problems with Euler Angle Parameterization:Interpolation

y

x

z

y

x

z

y

x

z

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Problems with Euler Angle Parameterization:Interpolation

R R t R t

R R t t R

( , , ),..., ( , , ),..., ( , , ) [ , ]

( , , ),..., ( , , ),..., ( , , )

0 0 0 0 0 0 0 0 1

0 0 0 0 0

Picture from p. 359

Case 1:

Case 2:

Generating Interpolated Orientations:

Case 1 Case 2

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Alternate Approach to ParameterizingArbitrary Orientation

R R( , , ) ( , ) 1 2 3 n

Replace Euler angles with a single angle of rotation about

an axial direction defined by the unit vector, n.

r Rr

n

Rr r n(n r) n r (cos ) ( cos ) (sin ) 1

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Quaternions

i

j

k

A quaternion is made up of a scalarplus a vector:

q a b c d i j k

We use the notation: q s

s a

b c d

q q s s s s

q s

qq q s

( , )

( , )

( , )

v

v i j k

v v v v v v

v

v

where:

1 2 1 2 1 2 1 2 2 1 1 2

2 2 2

Multiplication isdefined:

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Quaternions

Take a pure quaternion (one that has no scalar part): p ( , )0 r

And a unit quaternion: q (cos ,(sin ) ) n n where: 1

Define: R p qpq qpq qq ( ) 1 (for of unit magnitude)

R p

R p

q

q

( ) ( ,(cos sin ) sin ) cos sin

( ) ( , (cos ) ( cos ) ) (sin ) )

0 2 2

0 2 1 2 2

2 2 2

r n(n r n r

r n(n r n r

(cos , sin ) ( , ) (cos , sin ) ( , ') 2 2 2 20 0n r n r

r n r' ( , )R r

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Moving in and out of Quaternion Space

q

Z XY WZ XZ WY

XY WZ X Z YZ WX

XZ WY YZ WX X Y

(cos ,(sin ) ) n (W,(X,Y,Z))

1- 2Y2 2 2 2 2 2 0

2 2 1 2 2 2 2 0

2 2 2 2 1 2 2 0

0 0 0 1

2

2 2

2 2

Converts to the transformation matrix:

The quaternion:

M M M

M M M

M M M

00 01 02

10 11 12

20 21 22

0

0

0

0 0 0 1

W

MMZ

W

MMY

W

MMX

MMMW

4

4

4

)1(2

1

0110

2002

1221

21

221100

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Interpolating Between Two Quaternions

interp( , , )sin( )

sin

sin

sinq q u q

uq

u1 2 1 2

1

q q1 2 cosWhere:

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Interpolating Examples

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Interpolating Examples, contd.

Pictures from page 367