2D/3D Model Geometrical �Transformation

​

​

​

Tong-Yee Lee

1

positive rotation

• 3D Graphics Pipeline

Outline

• General Geometry Transform
• Scaling, rotation, translation etc

​

​

3

scale

4

Polygon Mesh

Normal vectors

Overview

• 2D transformations
• Basic 2-D transformations
• Matrix representation
• Matrix composition
• 3D transformations
• Basic 3-D transformation
• Same as 2-D
• Transformation Hierarchies
• Scene graphs

5

Viewing Transformation

Modeling Transform

• Specify transformation for objects
• Allow definitions of objects in own local coordinate systems
• Allow use of object definition multiple times in a scene
• Insert each local object into different locations of world coordinate

​

6

https://www.programmersought.com/article/32674581372/

7

Its own local coordinate

Modeling Transform

Move each local coordinate to a world coordinate�

8

9

View Transform

Local coordinate systems v.s. global coordinate system

Camera is defined at global coordinate system

2-D Transformations

10

Move each local coordinate to a world coordinate

Model is defined in a local

2-D Transformations

11

2-D Transformations

12

Usually, (0,0) point

is used to align (對位)

local and world coordinates

first

2-D Transformations

13

2-D Transformations

14

2-D Transformations

15

Basic 2D Transformations

16

Basic 2D Transformations

17

Sx>1

Sy>1

Scaling Around A Point

18

Scaling

S = S(s_x, s_y, s_z) =

19

Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

x’=s_xx

y’=s_yy

z’=s_zz

p’=Sp

Expand or contract along each axis (fixed point of origin)

Sx=Sy=Sz=0.5

Sx=0.5

Sy=1.5

Sz=1.0

Reflection

corresponds to negative scale factors

20

Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

original

s_x = -1 s_y = 1

s_x = -1 s_y = -1

s_x = 1 s_y = -1

April 2010

21

Basic 2D/3D Transformations

22

Rotation around the origin (2-D)

23

Ex: polar coordinate (極座標)

Rotation around the origin (2-D)

24

Counterclockwise i.e., 逆時針方向 (positive)

Rotation around the origin (2-D)

25

Matrix(矩陣)

Rotation (3-D)

26

Rotation (3-D)

Counterclockwise

i.e., 逆時針方向

(positive)

28

Basic 2D Transformations

29

2D Rotation at any pivot

30

樞軸；支點

Basic 2D Transformations

31

Translation

Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way

32

Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

object

translation: every point displaced

by same vector

Basic 2D Transformations

33

General 2D Rotation at (Xr,Yr)

April 2010

34

Matrix Representation

35

Matrix Representation

36

2x2 Matrix

37

(i.e., to initialize matrix)

2x2 Matrix

38

Shear (2-D)

39

Shear (3-D)

40

2x2 Matrix

41

2x2 Matrix

42

2D Reflections

April 2010

43

2x2 Matrix

44

2D Translation

45

Ex: (x,y) is represented by (x,y,1) in homogenous coordinate

Basic 2D Transformations

46

Homogeneous Coordinates

47

i.e., vector

(x1,y1,1) – (x2,y2,1)

= (x1-x2.y1-y2,0)

i.e., projection, w is related

to depth from eye

i.e. depth in w coordinate

Matrix Composition

48

w=1

Matrix Composition

49

i.e. 1 point is OK

i.e. if many points are used, matries are composed first

Matrix Composition

50

(交換律)

51

T*R*P != RT*P

Matrix Composition

52

i.e., M= T(a,b)*R(Q)*T(-a,-b)

P’=M*P

i.e., M= T(a,b)*S(Q)*T(-a,-b)

P’=M*P

3D Transformations

53

w=1

Basic 3D Transformations

54

w=1

Basic 3D Transformations

55

General rotation about an axis t�from a point P

56

Developing the General Rotation Matrix�

57

t= <Xv, Yv, Zv>

→

58

59

60

61

62

63

Developing the General Rotation Matrix

• Be careful …………

64

Z

X

(+,+)

(-,-)

In both cases, tan(y/x) are positive.

So, we need to carefully choose

it by checking the signs of x and y

OpenGL transformation Matrices

65

glutWireSphere(0.5, 10, 8);

glutWireSphere(1.0, 10, 8);

Example -1

• planet.c
• Control:
• ‘d’
• ‘y’
• ‘a’
• ‘A’
• ESC

66

Example -3

void GL_display() // GLUT display function

{

// clear t he buffer

glClear(GL_COLOR_BUFFER_BIT);

​

glColor3f(1.0, 1.0, 1.0);

glPushMatrix();

glutWireSphere(1.0, 20, 16); // the Sun

glRotatef(year, 0.0, 1.0, 0.0);

glTranslatef(3.0, 0.0, 0.0);

glRotatef(day, 0.0, 1.0, 0.0);

glutWireSphere(0.5, 10, 8); // the Planet

glPopMatrix();

// swap the front and back buffers

glutSwapBuffers();

}

67

Call GL_ display once

Push (original) i.e., 0

Rotate angles

Pop() to the original, i.e., 0

planet:

from local

to global

Example -4

void GL_idle() // GLUT idle function

{

day += 10.0;

if(day > 360.0) day -= 360.0; (i.e., planet self-rotation +10 degrees, faster)

​

year += 1.0;

if(year > 360.0) year -= 360.0; (i.e., planet rotate sun + 1 degree, slower)

​

// recall GL_display() function

glutPostRedisplay(); (i.e., call run display function)

}

​

68

Example -5

void GL_keyboard(unsigned char key, int x, int y) // GLUT keyboard function

{

switch(key)

{

case 'd': day += 10.0;

if(day > 360.0) day -= 360.0;

glutPostRedisplay();

break;

case 'y': year += 1.0;

if(year > 360.0) year -= 360.0;

glutPostRedisplay();

break;

case 'a': glutIdleFunc(GL_idle); // assign idle function

break;

case 'A': glutIdleFunc(0);

break;

case 27: exit(0);

}

}

69

Example -6

int main(int argc, char** argv)

{

glutInit(&argc, argv);

glutInitWindowSize(500, 500);

glutInitWindowPosition(0, 0);

glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB);

glutCreateWindow("Planet");

init();

glutDisplayFunc(GL_display);

glutReshapeFunc(GL_reshape);

glutKeyboardFunc(GL_keyboard);

glutMainLoop();

return 0;

}

​

70

71

72

Hierarchical Scene

73

child is transformed relative to its parent’s new

position

74

upper

lower

child is transformed relative to its parent

75

3D Example: A robot arm

76

From local to global

OpenGL better implementation

77

From local to global

OpenGL transformation Matrices

78

A More Complex Example: Human Figure

torso

A More Complex Example: Human Figure

What’s the most efficient way to draw this figure?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

91

Torso

Transformation Hierarchies Scene graphs

Root

Transform

Head Rotate first and then translate

Rotate

Rotate

translate

Store first

Recover

A more complex example

92

93

94

3D Skeleton Animation

95

p1

96

p.s.vector will not be changed by translation matrix

97

Inverse Transformation

98

OpenGL transformation Matrices

99

100

101

102

103

104

105

106

107

108

109

110