Lecture 5b. Transformation

Transcription

1 Lecture 5b Transformation

2 Refresher Transformation 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.

3 Principle of Transformation in OpenGL Pushing 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.

4 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.

5 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);

6 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, ); glutwirecube (1.0);

7 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); glrotatef(ang, x, y, z); gltranslatef(-x1, -y1, -z1); MAT3 MAT2 MAT1

8 glpushmatrix(); gltranslatef(4.0, 0.0, 0.0); glrotatef (45, 0.0, 0.0, 1.0); gltranslatef(-4.0, 0.0, 0.0); Composite matrix in reverse order glbegin(gl_polygon); Start modeling

9 User Defined Matrices The constructed matrices must be 4 x 4 matrix in single array form a[16] M a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11] a[12] a[13] a[14] a[15] Glfloat m[16]; for (int i = 0; i < 16; i++) m[i] = 0.0;

10 Loading and multiplying self defined matrices Loading: 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);

11 Rotating square by theta Previous Method 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); } Rotate by theta at Z axis Transformation Method 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); Advantage: Transformation method does not interfere with the modeling formula.

12 Source Code: transformation.cpp (r*sin(90)+4, r*cos (90)) (r*sin(180)+4, r*cos (180)) (4,0) (r*sin(0)+4, r*cos (0)) xx1= 0.5* sin (deg_to_rad * theta)+4.0; xx2= 0.5* sin (deg_to_rad * (theta+90))+4.0; xx3= 0.5* sin (deg_to_rad * (theta+180))+4.0; xx4= 0.5* sin (deg_to_rad * (theta+270))+4.0; glbegin(gl_polygon); glvertex2f(xx1, yy1); glvertex2f(xx2, yy2); glvertex2f(xx3, yy3); glvertex2f(xx4, yy4); glend() (r*sin(270)+4, r*cos (270)) yy1 = 0.5* cos(deg_to_rad * theta); yy2 = 0.5* cos(deg_to_rad * (theta+90)); yy3 = 0.5* cos(deg_to_rad * (theta+180)); yy4 = 0.5* cos(deg_to_rad * (theta+270)); *Rotation at (4,0) using variables Orbital rotation using transformation

13 Procedure to rotate the square (orbital rotation) 1. Rotate the square at (4,0) with increment of 1 0 (theta) Translate (-4, 0, 0) Rotate (theta, Z axis) Translate ( 4,0,0) 2. When square rotates full 360 0, the square will be rotated by 1 0 at (0,0) Rotate (theta2, Z axis) OpenGL 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);

14 Source Code: transformation.cpp glpushmatrix(); 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);

15 Modeling and animation of 2D robot arm) 1. Model the robot arm with all the arm horizontally Transformation required: translation 2. Rotate all the arm using the arm1 rotation angle Transformation required: rotation 3. Rotate the arm2 using the rotation angle arm2. Transformation required: translation before after, rotation

16 Initial modeling L1: (0,0) to (5,0) glbegin( GL_LINES ); glvertex2f( 0, 0); glvertex2f( 5, 0); glpushmatrix(); gltranslatef(5,0,0); Translate the line by 5,0,0 glbegin( GL_LINES ); glcolor3f(0.0,1.0,0.0); glvertex2f( 0, 0); glvertex2f( 2, 0); L2: (0,0) to (2,0)

17 Rotation by 45 o glpushmatrix(); glrotatef(45,0,0, 1.0); glbegin( GL_LINES ); glvertex2f( 0, 0); glvertex2f( 5, 0); Rotate L1 and L2 by 45 o (Z axis) glpushmatrix(); gltranslatef(5,0,0); glbegin( GL_LINES ); glcolor3f(0.0,1.0,0.0); glvertex2f( 0, 0); glvertex2f( 2, 0);

18 Rotate L2 by -15 o Step 1. Translate by -5,0,0 2. Rotate by -15 at Z axis 3. Translate by 5,0,0 glpushmatrix(); glrotatef(45,0,0, 1.0); glbegin( GL_LINES ); glvertex2f( 0, 0); glvertex2f( 5, 0); 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);