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Geometrical �Transformation

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positive rotation

Tong-Yee Lee

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3D Graphics Pipeline

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Outline

General Geometry Transform

Scaling, rotation, translation etc

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scale

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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
Insert each local object into different locations of world coordinate

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Basic Concept

Local coordinate systems v.s. global coordinate system

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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

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Viewing Transformation

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Rotate C1: will globally

change locations but relatively not

change local relationship C2 and C3

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2-D Transformations

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Move each local coordinate to a world coordinate

Model is defined in a local coordinate

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2-D Transformations

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2-D Transformations

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a. Usually, (0,0) point

is used to align

local and world coordinates

First

b. (0,0) will not changed

for scaling and rotation

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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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Sx>1

Sy>1

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Scaling Around A Point

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Scaling

S = S(s_x, s_y, s_z) =

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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

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Reflection

corresponds to negative scale factors

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original

s_x = -1 s_y = 1

s_x = -1 s_y = -1

s_x = 1 s_y = -1

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April 2010

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Move to origin

Scale

Move back

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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)

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Rotation around the origin (2-D)

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

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Rotation around the origin (2-D)

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Rotation (3-D)

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Rotation (3-D)

Counterclockwise

i.e., 逆時針方向

(positive)

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Basic 2D Transformations

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2D Rotation at any pivot

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樞軸；支點

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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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object

translation: every point displaced

by same vector

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Basic 2D Transformations

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General 2D Rotation at (Xr,Yr)

April 2010

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Move to origin

Rotate

Move back

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Matrix Representation

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Matrix Representation

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2x2 Matrix

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

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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

i.e. depth in w coordinate

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Matrix Composition

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w=1

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Matrix Composition

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i.e. 1 point is OK

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

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Matrix Composition

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(交換律)

T*R*P != RT*P

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Matrix Composition

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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

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3D Transformations

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w=1

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Basic 3D Transformations

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w=1

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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

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OpenGL transformation Matrices

object

glutSolidSphere(1.0, 50, 50);

(4,5,6)

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Example -1

planet.c

Control:

‘d’day ;自轉
‘y’year ; 公轉 along sun
‘a’; 自轉 & 公轉
‘A’
ESC

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Example

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

Adding more ……………………………..

glPopMatrix();

// swap the front and back buffers

glutSwapBuffers();

}

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Call GL_ display once

Push (original) i.e., 0

Rotate angles

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

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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)

}

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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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Example

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

Adding more ……………………………..

glPopMatrix();

// swap the front and back buffers

glutSwapBuffers();

}

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Call GL_ display once

Push (original) i.e., 0

Rotate angles

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

Add more

like moon

Add more

like 土星

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upper

lower

child is transformed relative to its parent

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child is transformed relative to its parent’s new

position

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3D Example: A robot arm

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OpenGL better implementation

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changed

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