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An Introduction to Computer Animation

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An Introduction to Computer Animation

Michiel van de Panne

- Creating Animations
- Representing Rotations

algorithm

(parameterized models,physics, optimization)

user

data

(motion capture)

(keyframes)

- DOF vs time
- cubic polynomial curves

x

smooth

t

fast

in-out

linear

- alternative for motion through space:

y

+

v

apply arc-length

reparameterization

t

x

world

torso

RUarm

RLarm

LUarm

Rhand

LLarm

Lhand

RUleg

RLleg

LUleg

LLleg

Rfoot

Lfoot

head

trans(0.30,0,0) rot(z, )

glTranslate3f(x,y,0);

glRotatef( ,0,0,1);

DrawBody();

glPushMatrix();

glTranslate3f(0,7,0);

DrawHead();

glPopMatrix();

glPushMatrix();

glTranslate(2.5,5.5,0);

glRotatef( ,0,0,1);

DrawUArm();

glTranslate(0,-3.5,0);

glRotatef( ,0,0,1);

DrawLArm();

glPopMatrix();

... (draw other arm)

- Example

y

x

- 2D: 1 DOF
- 3D: 3 DOF
- 4D: 6 DOF

- Rotation

glRotatef(angle,x,y,z);

glRotated(angle,x,y,z);

- 9 elements
- 6 constraints
- renormalization algorithms
- extracting pure rotational component (polar decomp)

... and determinant = 1

- SO(3)
- rotations do not commute
- require at least 4 parameters for a smooth parameterization
- analogy: surface of the earth
- 2D surface, 3 params

- analogy: surface of the earth
- combing the hairy ball
- camera orientation: view object from any dir

- orientation vs rotation?
- how to specify?
- how to interpolate?
- 2D vs 3D

pitch

yaw

roll

- fixed sequence of 3 rotations
- RPY orientation:

- can use many ordering of axes
- Euler angles:

- can always go from one orientation to anotherwith one rotation about a single axis

180 deg about line in xy-plane

where

- review of complex numbers

- quaternions

- rotation of a vector

- two successive rotations

RH rule

- unit quaternions

- addition
- multiplication