1. Hierarchical Modeling
Model: Representation of a system
Modeling: Creation and manipulation of a system representation
Geometric Model: Graphical representation of a system using geometric shapes
Symbol: Representation of system component
Instance: Each occurrence of a symbol within a model

2. Hierarchical Tree Structure
robot
base
body
torso
head
left arm
right arm
base(1)
head(1)
torso(1)
arm(2)
basic symbols (number of instances) lb lh la lt wb wh wa wt

3. Ingredients of a geometric model:
Geometric shapes and other attributes of components
Interconnection between components
Other properties of components (physical properties, text description, etc.)
Interconnections
shoulders   1 2
neck
waist
base and body
torso and head
torso and arms

4. Local Coordinates and World Coordinates
World Coordinates (Global Coordinates): A global reference frame that contains all the models.
Local Coordinates (Modeling Coordinates, Master Coordinates): Coordinate system that is attached to an individual component. y
World coordinate system x

5. Local coordinate systemsx x x
robot
base
body y y y x x x
torso
head
arm

6. Modeling Transformation
Modeling Transformation: Transform model components from local coordinate system to world coordinate system.
Two different views of modeling transformation:
Global View: Transform object in world coordinate system
Local View: Transform local coordinate system of an object

7. Global View:
Rz(45)
T(10, 0, 0)
Local View:
T(10, 0, 0)
Rz(45)

8. Global View:
T(10, 0, 0)
Rz(45)
Local View:
Rz(45)
T(10, 0, 0)

9. The following two views are equivalent for composite transformation:
M = Mn  Mn-1  ... M2  M1
Global View:
Transform the object with M1.
Transform the object with M2.
. . .
Transform the object with Mn.
Draw the object in world coordinate system.
Local View:
Transform the local coordinate system with Mn.
Transform the local coordinate system with Mn-1.
. . .
Transform the local coordinate system with M1.
Draw the object in local coordinate system.
Transformation of local coordinate system is relative to itself

10. In hierarchical tree structure, each tree node stores transformation matrix from its parent’s local coordinate system to its own local coordinate system. y x

11. waist  
neck

12. left shoulder 1
right shoulder 2

13. Hierarchical Tree Traversal
Tree_traverse( NODE n, MATRIX M ) {
M1 = local transformation matrix of n;
M = M  M1;
if (n is a leaf)
Set current transformation matrix to M;
Draw object associated with leaf n; }
else
for (each child node n1 of n)
Tree_traverse( n1, M );
NODE: Data structure for hierarchical tree node
MATRIX: Data structure for 4 by 4 transformation matrix

14. Example for the robot: Draw_robot( MATRIX M ) { Draw_base( M );
Draw_body( M ); }
Draw_base( MATRIX M ) {
M = M  Mbase;
Set current transformation matrix to M;
Draw geometric structure for base; }
Draw_body( MATRIX M ) {
M = M  Mbody;
Draw_torso( M );
Draw_head( M );
Draw_arm( M  Mleftarm );
Draw_arm( M  Mrightarm ); }

15. Draw_torso( MATRIX M ) {
M = M  Mtorso;
Set current transformation matrix to M;
Draw geometric structure for torso; }
Draw_head( MATRIX M ) {
M = M  Mhead;
Set current transformation matrix to M;
Draw geometric structure for head; }
Draw_arm( MATRIX M ) {
Set current transformation matrix to M;
Draw geometric structure for arm; }

16. Physics Based Modeling
Particle Systems
Collection of particles moving under external forces. Can be used to model clouds, smoke, fire, fireworks, explosion, waterfalls, fountain, etc.
Particle Properties:
position
velocity
mass
age
life span
shape, color, texture, etc.
Particle System
Particle generation and annihilation
External force field construction
Particle movement control
Particle rendering

17. Particle generation and annihilation
Controlled randomization on initial properties of particles
Fireworks
Fountain
Waterfall
while (…) {
Get elapsed time dt;
n = generation rate * dt;
Generate n new particles;
for (each particle)
Update particle velocity and position;
age += dt;
if (age > life span) delete particle; }

18. External force field construction
Gravity
Buoyant force
Air friction
Sine wave
Add randomization to the force field
Particle movement control
Newtonian law
Numerical time integration

19. Particle rendering
Render particle as a texture mapped quad
The quad should always face the viewer (billboard effect) C U D P B R F A
F: Camera Forward Direction
U: Camera Up Direction
R: Camera Right Direction
P: Particle Position
a: Particle Size

20. Cloth simulation using Mass-Spring model
A number of particles with mass interconnected by springs
(i, j+2)
(i+1, j+1)
(i, j+1)
(i, j)
(i+1, j)
(i+2, j)

21. Types of springs Tensile spring Shear spring Bending spring (i, j+1)

22. Spring forces vj vi xj xi
mass particle
original position
position
velocity i
x0i xi vi j
x0j xj vj
Elastic force
Damping force

23. Dynamic equations for mass particles
Numerical time integration
Collision detection and collision response

24. Examples

25. Skeletal Animation Hierarchical Bone Structure Joint tree neck
right shoulder
left shoulder
hip
right thigh
left thigh
chest
right elbow
left elbow
waist
right knee
Left knee
waist
right wrist
left wrist
right ankle
left ankle
hip
chest
right thigh
left thigh
right shoulder
left shoulder
neck
right knee
Left knee
right elbow
left elbow
right ankle
left ankle
right wrist
left wrist

26. Joint Transformations
For joint i:
Tilocal: Position relative to its parent joint
Rilocal: Rotation transformation relative to its parent joint
Tiworld: Position in world coordinate system
Riworld: Rotation transformation in world coordinate system
Assume joint i’s parent is joint j, then:

27. Vertex Skinning Each vertex is attached to one or more joints
For vertex i:
wij: Weight for joint j
Pijlocal: Position in joint j’s local coordinate system
Piworld: Position in world coordinate system

28. Key Frame Animation ...... Key frames Frame 0 Frame 1 Frame 2 ...... t
Frame interval: t
Frame rate: 1/t

29. Key Frame Interpolation
Intermediate Frame
Frame k
Frame k+1 t h t
Position interpolation

30. Slerp (Spherical Linear Interpolation) for rotation
Slerp in 2D P A B  

31. Quaternion Slerp
When cos > 0
When cos < 0