Lecture 5b

Transformation

RefresherTransformation matrices [4 x 4]:

the fourth coordinate is homogenous coordinate.

Rotation Transformation:

Axis of rotation must through origin (0,0,0). If not, translation is applied before and after.

Composite Transformation:

must be in sequence.

Principle of Transformation in OpenGLPushing the transformation to the stack and after

application, discard the transformation

glPushMatrix(), which copies the current matrix and adds the copy to the top of the stack

glPopMatrix(), which discards the top matrix on the stack,

Procedure in OPenGL

glPushMatrix( );

… intended transformation;

… apply the transformation to geometrical properties.

glPopMatrix();

Transformation in OpenGL

void glRotated(GLdouble ang, GLdouble x, GLdouble y, GLdouble z)

void glRotatef( GLfloat ang, GLfloat x, GLfloat y, GLfloat z )

computes a matrix that performs a counterclockwise rotation of angle degrees about the vector from the origin through the point (x, y, z).

*Take note: x,y,z is not axis.

Transformation in OpenGL

void glTranslated( GLdouble x, GLdouble y, GLdouble z )

void glTranslatef( GLfloat x, GLfloat y, GLfloat z )

x, y, z Specify the x, y, and z coordinates of a translation vector

Rotation at axis y at point (x1, y1, z1)

Mat1= translation (-x1, -y1, -z1);

Mat2= rotation (ang, 0.0, 1.0, 0.0);

Mat3= translation (x1, y1, z1);

Transformation in OpenGL

void glScaled( GLdouble x, GLdouble y, GLdouble z )

void glScalef( GLfloat x, GLfloat y, GLfloat z )

x, y, z Specify scale factors along the x, y, and z axes, respectively.

glScalef (1.0, 1.0 3.0);

glutWireCube (1.0);

Composite Matrices in OpenGL

In OpenGL, the order of the transformation appears in reverse order

first will appear last

Intended transformation :[M] = [M1][M2][M3]… [Mi]

In OpenGL, [M] = [Mi]…[M3][M2][M1]

i.e: rotation at different point will

glTranslatef(x1, y1, z1); MAT3

glRotatef(ang, x, y, z); MAT2

glTranslatef(-x1, -y1, -z1); MAT1

glPushMatrix();

glTranslatef(4.0, 0.0, 0.0);

glRotatef (45, 0.0, 0.0, 1.0);

glTranslatef(-4.0, 0.0, 0.0);

glBegin(GL_POLYGON);

…

glEnd();

glPopMatrix();

Composite matrix in reverse order

Start modeling

User Defined Matrices

◊ The constructed matrices must be 4 x 4 matrix in single array form a[16]

]15[a]11[a]7[a]3[a

]14[a]10[a]6[a]2[a

]13[a]9[a]5[a]1[a

]12[a]8[a]4[a]0[a

M

Glfloat m[16];

for (int i = 0; i < 16; i++)

m[i] = 0.0;

Loading and multiplying self defined matricesLoading:

void glLoadMatrixd( const GLdouble *m )

void glLoadMatrixf( const GLfloat *m )

Multiplying

void glMultMatrixd( const GLdouble *m )

void glMultMatrixf( const GLfloat *m )

i.e:glPushMatrix();

glTranslatef(x1, y1,z1);

glMulMatrix(m);

glTranslatef(-x1, -y1, -z1);

glPopMatrix();

Rotating square by theta

glBegin( GL_LINE_LOOP );

for (int i = 0; i < 360 ; i= i+90 )

{

v1x = 0.8*cos ( (i+theta)* pi/180);

v1y = 0.8*sin ( (i+theta)* pi/180);

glVertex2f( v1x, v1y);

}

glEnd();

glPushMatrix();glRotatef (theta, 0.0, 0.0, 1.0);

glBegin( GL_LINE_LOOP );

for (int i = 0; i < 360 ; i= i+90 ){

v1x = 0.8*cos ( i* pi/180);v1y = 0.8*sin ( i* pi/180);

glVertex2f( v1x, v1y);

}glEnd();

glPopMatrix();

Previous Method Transformation Method

Rotate by theta at Z axis

Advantage: Transformation method does not interfere with the modeling formula.

Source Code: transformation.cpp

xx1= 0.5* sin (deg_to_rad * theta)+4.0; yy1 = 0.5* cos(deg_to_rad * theta);

xx2= 0.5* sin (deg_to_rad * (theta+90))+4.0; yy2 = 0.5* cos(deg_to_rad * (theta+90));

xx3= 0.5* sin (deg_to_rad * (theta+180))+4.0; yy3 = 0.5* cos(deg_to_rad * (theta+180));

xx4= 0.5* sin (deg_to_rad * (theta+270))+4.0; yy4 = 0.5* cos(deg_to_rad * (theta+270));

glBegin(GL_POLYGON);

glVertex2f(xx1, yy1);

glVertex2f(xx2, yy2);

glVertex2f(xx3, yy3);

glVertex2f(xx4, yy4);

glEnd()

(4,0) (r*sin(0)+4 , r*cos (0))

(r*sin(90)+4 , r*cos (90))

(r*sin(180)+4 , r*cos (180))

(r*sin(270)+4 , r*cos (270))

*Rotation at (4,0) using variablesOrbital rotation using transformation

Procedure to rotate the square (orbital rotation)

1. Rotate the square at (4,0) with increment of 10

(theta)

Translate (-4, 0 , 0)

Rotate (theta, Z axis)

Translate ( 4,0,0)

2. When square rotates full 3600, the square will be rotated by 10 at (0,0)

Rotate (theta2, Z axis) OpenGLglRotatef (theta2, 0.0, 0.0, 1.0);glTranslatef (4.0, 0.0, 0.0);glRotatef (theta, 0.0, 0.0, 1.0); glTranslatef (-4.0, 0.0, 0.0);

Source Code: transformation.cppglPushMatrix();

glRotatef (theta2, 0.0, 0.0, 1.0);

glTranslatef (4.0, 0.0, 0.0);

glRotatef (theta, 0.0, 0.0, 1.0);

glTranslatef (-4.0, 0.0, 0.0);

glBegin(GL_POLYGON);

xx1= 0.5* sin (deg_to_rad * 0)+4; yy1 = 0.5* cos(deg_to_rad * 0);

xx2= 0.5* sin (deg_to_rad * (90))+4; yy2 = 0.5* cos(deg_to_rad * (90));

xx3= 0.5* sin (deg_to_rad * (180))+4; yy3 = 0.5* cos(deg_to_rad * (180));

xx4= 0.5* sin (deg_to_rad * (270))+4; yy4 = 0.5* cos(deg_to_rad * (270));

glVertex2f(xx1, yy1);

glVertex2f(xx2, yy2);

glVertex2f(xx3, yy3);

glVertex2f(xx4, yy4);

glEnd();

glPopMatrix();

Modeling and animation of 2D robot arm)

1. Model the robot arm with all the arm horizontallyTransformation required: translation

2. Rotate all the arm using the arm1 rotation angleTransformation required: rotation

3. Rotate the arm2 using the rotation angle arm2.Transformation required: translation before after, rotation

Initial modelingglBegin( GL_LINES );

glVertex2f( 0, 0);

glVertex2f( 5, 0);

glEnd();

glPushMatrix();

glTranslatef(5,0,0);

glBegin( GL_LINES );

glColor3f(0.0,1.0,0.0);

glVertex2f( 0, 0);

glVertex2f( 2, 0);

glEnd();

glPopMatrix();

L1: (0,0) to (5,0)

L2: (0,0) to (2,0)Translate the line by 5,0,0

glPushMatrix();

glRotatef(45,0,0, 1.0);

glBegin( GL_LINES );

glVertex2f( 0, 0);

glVertex2f( 5, 0);

glEnd();

glPushMatrix();

glTranslatef(5,0,0);

glBegin( GL_LINES );

glColor3f(0.0,1.0,0.0);

glVertex2f( 0, 0);

glVertex2f( 2, 0);

glEnd();

glPopMatrix();

glPopMatrix();

Rotation by 45o

Rotate L1 and L2 by 45o (Z axis)

glPushMatrix();glRotatef(45,0,0, 1.0);

glBegin( GL_LINES );glVertex2f( 0, 0);glVertex2f( 5, 0);glEnd();

glPushMatrix();glTranslatef(5,0,0);glRotatef(-15,0,0, 1.0);glTranslatef(-5,0,0);

glPushMatrix();glTranslatef(5,0,0);glBegin( GL_LINES );

glColor3f(0.0,1.0,0.0);glVertex2f( 0, 0);glVertex2f( 2, 0);

glEnd();glPopMatrix();

glPopMatrix();glPopMatrix();

Rotate L2 by -15o

Step1. Translate by -5,0,02. Rotate by -15 at Z axis3. Translate by 5,0,0