Geometrical Transformation

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Tong-Yee Lee

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Course Outline

Standard Graphics Pipeline

Local coordinate

Outline

• General Transform

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• Scale,rotation, translation etc

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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 (in global coordinate system)

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Insert them

Into different

Locations of

World coordinate

system

Overview

• 2D transformations
• Basic 2-D transformations
• Matrix representation
• Matrix composition
• 3D transformations
• Basic 3-D transformation
• Same as 2-D

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2-D Transformations

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Local coordinate

To world coordinate

2-D Transformations

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2-D Transformations

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Usually, (0,0) point

Is used to align

Local and world coordinates

first

2-D Transformations

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2-D Transformations

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2-D Transformations

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Basic 2D Transformations

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Basic 2D Transformations

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Scaling

S = S(s_x, s_y, s_z) =

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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)

Reflection

corresponds to negative scale factors

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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

General 2D Scaling

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Basic 2D Transformations

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Rotation around the origin (2-D)

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Ex: polar coordinate (極座標)

Rotation around the origin (2-D)

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Rotation around the origin (2-D)

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Rotation (3-D)

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Rotation (3-D)

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Basic 2D Transformations

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Basic 2D Transformations

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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

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

object

translation: every point displaced

by same vector

tx

ty

Basic 2D Transformations

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2D Rotation

April 2010

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General 2D Rotation

April 2010

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Matrix Representation

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Matrix Representation

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2x2 Matrix

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Scaling

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Scaling Around A Point

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2x2 Matrix

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Shear (2-D)

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Shear (3-D)

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2x2 Matrix

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2x2 Matrix

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2D Reflections

April 2010

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2x2 Matrix

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2D Translation

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Basic 2D Transformations

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Homogeneous Coordinates

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i.e., vector

(x1,y1,1) – (x2,y2,1)

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

i.e., projection, w is related

to depth from eye

Matrix Composition

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w=1

Matrix Composition

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i.e. 1 point is OK

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

Matrix Composition

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(交換律)

T*R*P != RT*P

Matrix Composition

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

P’=M*P

3D Transformations

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w=1

Basic 3D Transformations

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w=1

Basic 3D Transformations

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General rotation about an axis

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Developing the General Rotation Matrix�

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t= <Xv, Yv, Zv>

→

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Developing the General Rotation Matrix

• Be careful …………

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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

Example -1

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

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Example -3

void GL_display() // GLUT display function

{

// clear the buffer

glClear(GL_COLOR_BUFFER_BIT);

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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();

}

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Example -4

void GL_idle() // GLUT idle function

{

day += 10.0;

if(day > 360.0) day -= 360.0;

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year += 1.0;

if(year > 360.0) year -= 360.0;

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// recall GL_display() function

glutPostRedisplay();

}

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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);

}

}

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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;

}

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Review of Raster Displays

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p1

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p.s.vector will not be changed

By translation matrix

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Inverse Transformation

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Inverse Transformation

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OpenGL transformation Matrices

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OpenGL transformation Matrices

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Another Example: Robot

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Rotation center is at

(-1,0,0) instead of

(0,0,0)

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shoulder

elbow

glTranslate(2,0,0);

glTranslate(-1,0,0);

OpenGL transformation Matrices

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OpenGL transformation Matrices

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90

(a)

(b)

Step 1:

Separate

x vector

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Note: vector a

is a unit vector

i.e: cos(90-Ɵ) = sin(Ɵ)

��Angular displacement �glRotate(θ, Ax,Ay,Az)�

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The above formula is a matrix form, so we can use Matrix to compute rotation

In above equation, v=(x,y,z)^T and n=(a_x,a_y,a_z)^T

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