1. CSCE 441: Computer Graphics: Hierarchical Models
Jinxiang Chai

2. Summary: 3D Geometry Pipeline
Object space
World space
View space
Normalized project space
Image space 2

3. Complex Models

4. Outline Hierarchical Models
Reading: HB 9-8&9-9, HB chapter 11, OpenGL Programming Guide, chapter 3, and course slides

5. Symbols and Instances Most graphics API supports a few primitives:
- sphere
- cube
- cylinders
These symbols are instanced using instance/model transformation:

6. What’s the matrix for the instance transformation above?
Symbols and Instances
Most graphics API supports a few primitives:
- sphere
- cube
- cylinders
These symbols are instanced using instance transformation:
What’s the matrix for the instance transformation above?

7. Symbols and Instances Most graphics API supports a few primitives:
- sphere
- cube
- cylinders
These symbols are instanced using instance transformation:

8. Sample Instance Trans.
In opengl, instance/model transformation is created by modifying the Model-view matrix:
glMatrixMode(GL_MODELVIEW);
glLoadIdentity(…);// set current matrix to the identity
glTranslate(…); // translate
glRotate(…); //rotate
glScale(…);//scale
house();

9. Sample Instance Trans.
In opengl, instance transformation is created by modifying the Model-view matrix:
glMatrixMode(GL_MODELVIEW);
glLoadIdentity(…);
glTranslate(…);
glRotate(…);
glScale(…);
house();
Does the transform seem to be backward?

10. Composite Transformation: Opengl Implementation
Opengl postmultiplies transformation matrices as they are called
Each subsequent transformation call concatenates the designated transformation matrix on the right of the composite matrix
We must invoke the transformation in the opposite order from which they are applied.
glMatrixMode(GL_MODELVIEW);
glLoadIdentity(…);
M4;
M3;
M2;
M1; …

11. Lamp
What’s the current coordinate A ? 11

12. Lamp
Suppose we know how to draw each bone in local reference frames

13. Lamp
How can we draw the character under a particular pose

14. Lamp How can we draw the character under a particular pose base();
upper_arm();
middel_arm();
lower_arm();

15. Lamp
What’s the current coordinate A ? 15

16. Lamp
What’s the current coordinate A ? 16

17. Lamp
What’s the current coordinate A ? 17

18. Lamp
What’s the current coordinate A ? 18

19. Lamp Implementation
The lamp can be displayed by computing a global matrix and computing it at each step
lamp() {
M_model =
base();
upper_arm();
middel_arm();
lower_arm(); }
Matrix M_model;
Main()
{ …
M_model=Identity()
lamp(); } 19

20. Lamp Implementation
The lamp can be displayed by computing a global matrix and computing it at each step
lamp() {
M_model =
base();
upper_arm();
middel_arm();
lower_arm(); }
Matrix M_model;
Main()
{ …
M_model=Identity()
lamp(); }
Can we make it more efficiently? 20

21. Better Implementation
Instead of recalculating the global matrix each time, we can just update it in place
lamp() {
M_model *=
base();
upper_arm();
middel_arm();
lower_arm(); }
Matrix M_model;
Main()
{ …
M_model=Identity()
lamp(); } 21

22. Opengl Implementation
Opengl maintains a global state matrix called model-view matrix
Main() { …
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
2D_lamp(a,b,c,d,e,f) }
2D_lamp(x, y, θ0, θ1, θ2, θ3) {
glTranslatef(x,y,0)
glRotatef(θ0,0,0,1,);
base();
glTranslatef(0,l0,0)
glRotatef(θ1,0,0,1,);
upper_arm();
glTranslatef(0,l1,0)
glRotatef(θ2,0,0,1,);
middel_arm();
glTranslatef(0,l2,0)
glRotatef(θ3,0,0,1,);
lower_arm(); }
//set current matrix
to identity

23. More Complex Objects

24. Hierarchical Modeling
Consider a model of a car
– how many symbols/primitives?
– how many instances?

25. Hierarchical Modeling
Consider a model of a car
– 2 primitives :
– 5 instances : 25

26. Hierarchical Modeling
Chasis system
Consider a model of a car
– 2 primitives : chassis + wheel
– 5 instances : 1 chassis + 4 wheel
World system
front-left wheel system 26

27. Hierarchical Modeling
Consider a model of a car
– 2 primitives : chassis + wheel
– 5 instances : 1 chassis + 4 wheel
We can represent our car as a tree to show the relationship between the parts
- tree nodes store subparts (i.e., instances)
- transition edges represent relationship between the parts

28. Hierarchical Modeling
Consider a model of a car
– 2 symbols : chassis + wheel
– 5 instances : 1 chassis + 4 wheel
We can represent our car as a tree to show the relationship between the parts
However, since all 4 wheels are instances of the same model, we’d like to only have that model appear once

29. Hierarchical Modeling
Hierarchical model can be composed of instances using trees or directed acyclic graphs (DAGs)
- edges contains geometric transformations
- nodes contains geometry of subparts (i.e., primitives)
This allows us to decompose geometry representation of a complex object into a set of geometric primitives and the geometric transformations between the primitives.

30. Hierarchical Modeling
Hierarchical model can be composed of instances using trees or directed acyclic graphs (DAGs)
- edges contains geometric transformations
- nodes contains geometry of subparts (i.e., primitives)
What might we draw the tree for the lamp?

31. Hierarchical Modeling
Hierarchical model can be composed of instances using trees or directed acyclic graphs (DAGs)
- edges contains geometric transformations
- nodes contains geometry

32. Hierarchical Modeling
Hierarchical model can be composed of instances using trees or directed acyclic graphs (DAGs)
- edges contains geometric transformations
- nodes contains geometry
world
base
Upper arm
middle arm
lower arm 32

33. Different geometric models for Lamp

34. More Complex Objects

35. A More Complex Example: Human Figure
torso

36. A More Complex Example: Human Figure
torso 36

37. A More Complex Example: Human Figure
torso 37

38. A More Complex Example: Human Figure
torso
What’s the most efficient way to draw this figure?
Suppose we know how to draw each primitive in local reference frames

39. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree? 39

40. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

41. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

42. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

43. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

44. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

45. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

46. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

47. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

48. A More Complex Example: Human Figure
torso
What’s the most sensible way to traverse this tree?

49. Opengl Implementation: Human Figure
torso Mh
Mlua
Mlla 49

50. Opengl Implementation: Human Figure
torso Mh
Mlua
Mlla
Is this correct? 50

51. Opengl Implementation: Human Figure
torso Mh
Mh*Mlua should be Mlua
Mh*Mlua*Mlla should be Mlua*Mlla
Is this correct? No 51

52. Matrix Stack glMatrixMode(GL_MODELVIEW)
- Initially, each of the stacks contains one matrix, an identity matrix.
- The top matrix in the stack is “current matrix”
- The depth is at least 32 Mc … I

53. Push and Pop the Current Matrix
glPushMatrix() copy the current matrix at the top of the stack Mc Mc … Mc I … I

54. Push and Pop the Current Matrix
glPopMatrix() destroys the matrix at the top of stack
Mtop
Mtop-1
Mtop-1 … … I I

55. Opengl Implementation: Human Figure

56. Opengl Implementation: Human Figure56

57. Opengl Implementation: Human Figure57

58. Opengl Implementation: Human FigureMh I 58

59. Opengl Implementation: Human Figure59

60. Opengl Implementation: Human Figure60

61. Opengl Implementation: Human Figure
Mlua I 61

62. Opengl Implementation: Human Figure
Mlua
Mlua I 62

63. Opengl Implementation: Human Figure
Mlua*Mlla
Mlua I 63

64. Opengl Implementation: Human Figure
Mlua I 64

65. Opengl Implementation: Human Figure65

66. Opengl Implementation: Human FigureMh
Mlua
Mlua Mlla 66

67. Articulated Models You can draw these models as long as
- you know how to draw each primitive in local reference frames
- you know how to call transformation matrices to model the relationship between the primitives