1. OpenGL Computer Graphics
Programming with Transformations

2. Topics Transformations in OpenGL Saving Current Transformation
Drawing 3D Scenes with OpenGL
OpenGL Functions for Modeling and Viewing

3. Transformations in OpenGL
CT: current transformation
Simplified graphics pipeline
OpenGL maintains so-called modelview matrix
Every vertex passed down the graphics pipeline is multiplied by this matrix V Q
Window-to-Viewport
Transformation S CT S V Q
Viewport
World Window
Screen
Coordinate System
Model (Master)
Coordinate System
World
Coordinate System

4. Transformations in OpenGL
OpenGL is a 3D graphics package
Transformations are 3D
How does it work in 2D?
2D drawing is done in the xy-plane, z coordinate is 0.
Translation: dz = 0
Scaling: Sz = 1
Rotation: z-roll y z x

5. Transformations in OpenGL
Fundamental Transformations
Translation: glTranslated(dx, dy, dz)
for 2D: glTranslated(dx, dy, 0)
Scaling: glScaled(sx, sy, sz)
for 2D: glScaled(sx, sy, 1.0)
Rotation: glRotated(angle, ux, uy, uz)
for 2D: glRotated(angle, 0, 0, 1)
Transformations does not set CT directly, a matrix is postmultiplied to CT
CT = CT  M

6. Transformations in OpenGL
Canvas functions
void Canvas:: initCT(void) {
glMatrixMode(GL_MODELVIEW);
glLoadIdentity(); }
void Canvas:: scale2D(double sx, double sy)
glScaled(dx, dy, 1.0);

7. Transformations in OpenGL
Canvas functions
void Canvas:: translate2D(double dx, double dy) {
glMatrixMode(GL_MODELVIEW);
glTranslated(dx, dy, 0); }
void Canvas:: rotate2D(double angle)
glRotated(angle, 0.0, 0.0, 1.0);

8. Transformations Example
Draw a house. Draw another house by rotating it through -30° and then translating it through (32, 25)
cvs.initCT();
house();
cvs.translate2D(32, 25);
cvs.rotate2D(-30.0);

9. Transformations Example

10. Transformations Example
Think of it in two different ways
Q =T(32, 25)R(-30)P
 CT = CT  T(32, 25)  R(-30)
Translate the coordinate system through (32, 25) and then rotate it through –30°
The code generated by these two ways is identical.

11. Saving Current Transformation
We can save and restore CTs using glPushMatrix() and glPopMatrix()
Manipulation of a stack of CT
After
rotate2D()
After
popCT()
Before
After
pushCT()
CT4
CT = CT3  Rot
CT3
CT3
CT3
CT3
CT2
CT2
CT2
CT2
CT1
CT1
CT1
CT1

12. Saving Current Transformation
Canvas functions
void Canvas:: pushCT(void) {
glMatrixMode(GL_MODELVIEW);
glPushMatrix(); }
void Canvas:: popCT(void)
glPopMatrix();

13. Saving CT Examples
Master coordinate system: where an object is defined
Modeling transformation: transforms an object from its master coordinate system to world coordinate system to produce an instance
Instance: a picture of an object in the scene

14. Drawing 3D Scenes with OpenGL
The concept of “camera” (eye) is used for 3D viewing
Our 2D drawing is a special case of 3D drawing
far plane y
view volume
near plane z x
eye
Viewport
window

15. Drawing 3D Scenes with OpenGL
Camera to produce parallel view of a 3D scene

16. Drawing 3D Scenes with OpenGL
Simplified OpenGL graphics pipeline VM P
clip Vp
modelview
matrix
projection
matrix
viewport
matrix

17. Drawing 3D Scenes with OpenGL
Modelview matrix = CT
Object transformation + camera transformation
Applying model matrix M then viewing matrix V

18. Drawing 3D Scenes with OpenGL
Projection matrix
Shifts and scales view volume into a standard cube (extension from –1 to 1)
Distortion can be compensated by viewport transformation later

19. Drawing 3D Scenes with OpenGL
Viewport matrix
Maps surviving portion of objects into a 3D viewport after clipping is performed
Standard cube 
block w/ x and y extending across viewport and z from 0 to 1

20. OpenGL Modeling and Viewing Functions
Modeling transformation
Translation: glTranslated(dx, dy, dz)
Scaling: glScaled(sx, sy, sz)
Rotation: glRotated(angle, ux, uy, uz)
Camera for parallel projection
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
glOrtho(left, right, bottom, top, near, far)
Example
near=2: near plane is 2 units in front of eye
far=20: far plane is 20 units in front of eye

21. OpenGL Modeling and Viewing Functions
Positioning and aiming camera
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glutLookAt(eye.x, eye.y, eye.z, // eye position
look.x, look.y, look.z, // look at point
up.x, up.y, up.z) // up vector
Up vector is often set to (0, 1, 0)
glutLookAt() builds a matrix that converts world coordinates into eye coordinates.

22. Set up a Typical Camera - Example
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
glOrtho(-3.2, 3.2, -2.4, 2.4, 1, 50)
glMatrixMode(GL_MODELVIEW);
glutLookAt(4, 4, 4,
0, 1, 0,
0, 1, 0)
(4, 4, 4)
(0, 1, 0)

23. Transformation Matrix for LookAt
Camera coordinate system
Axes: u, v, n
n = eye – look
u = up  n
v = n  u
Origin: eye (looking in the direction –n)
Transformation matrix

24. Transformation Matrix for LookAt

25. Elementary 3D Shapes Provided by OpenGL
Cube
glutWireCube(GLdouble size)
size = length of a side
Sphere
glutWireSphere(GLdouble radius, GLint nSlices, GLint nStacks)
Approximated by polygonal faces
nSlices = #polygons around z-axis
nStacks = #bands along z-axis

26. Elementary 3D Shapes Provided by OpenGL
Torus
glutWireTorus(GLdouble inRad, GLdouble outRad, GLint nSlices, GLint nStacks)
Approximated by polygonal faces
Teapots
glutWireTeapot(GLdouble size)
There are solid counterparts of the wire objects

27. Plantonic Solids Provided by OpenGL
Tetrahedron
glutWireTetrahedron()
Octahedron
glutWireOctahedron()
Dodecahedron
glutWireDodecahedron()
Icosahedron
glutWireIcosahedron()
All of them are centered at the origin

28. Plantonic Solids Provided by OpenGL

29. Cone Provided by OpenGL
glutWireCone(GLdouble baseRad, GLdouble height, GLint nSlices, GLint nStacks)
Axis coincides with the z-axis
Base rests on xy-plane and extends to z = height
baseRad: radius at z = 0

30. Tapered Cylinder Provided by OpenGL
gluCylinder(GLUquadricObj *qobj, GLdouble baseRad, GLdouble topRad, GLdouble height, GLint nSlices, GLint nStacks)
Axis coincides with the z-axis
Base rests on xy-plane and extends to z = height
baseRad: radius at z = 0
topRad: radius at z = height

31. Tapered Cylinder Provided by OpenGL
A family of shapes distinguished by the value of topRad
To draw, we have to
Deifne a new quadric object
Set drawing style
GLU_LINE: wire frame
GLU_FILL: solid rendering
Draw the object

32. Tapered Cylinder Provided by OpenGL
Example – wire frame cylinder
GLUquadricObj *qobj;
qobj = gluNewQuadric();
gluQuadricDrawStyle(qobj, GLU_LINE);
gluCylinder(qobj, baseRad, topRad, height, nSlices, nStacks);

35. #include <gl/glut.h>
//<<<<<<<<<<<<<<<<<<< axis >>>>>>>>>>>>>>
void axis(double length)
{ // draw a z-axis, with cone at end
glPushMatrix();
glBegin(GL_LINES);
glVertex3d(0, 0, 0); glVertex3d(0,0,length); // along the z-axis
glEnd();
glTranslated(0, 0,length -0.2);
glutWireCone(0.04, 0.2, 12, 9);
glPopMatrix(); }

36. //<<<<<<<<<<<<<< displayWire >>>>>>>>>>>>>>
void displayWire(void) {
glMatrixMode(GL_PROJECTION); // set the view volume shape
glLoadIdentity();
glOrtho(-2.0*64/48.0, 2.0*64/48.0, -2.0, 2.0, 0.1, 100);
glMatrixMode(GL_MODELVIEW); // position and aim the camera
gluLookAt(2.0, 2.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
// to obtain the picture shown in Figure 5.59 we have to
// change the eye location as follows
// gluLookAt(1.0, 1.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);

37. glClear(GL_COLOR_BUFFER_BIT); // clear the screen
glColor3d(0,0,0); // draw black lines
axis(0.5); // z-axis
glPushMatrix();
glRotated(90, 0, 1, 0);
axis(0.5); // x-axis
glRotated(-90, 1, 0, 0);
axis(0.5); // y-axis
glPopMatrix();
glTranslated(0.5, 0.5, 0.5); // big cube at (0.5, 0.5, 0.5)
glutWireCube(1.0);

38. glPushMatrix();
glTranslated(1.0,1.0,0); // sphere at (1,1,0)
glutWireSphere(0.25, 10, 8);
glPopMatrix();
glTranslated(1.0,0,1.0); // cone at (1,0,1)
glutWireCone(0.2, 0.5, 10, 8);
glTranslated(1,1,1);
glutWireTeapot(0.2); // teapot at (1,1,1)

39. glPushMatrix();
glTranslated(0, 1.0 ,0); // torus at (0,1,0)
glRotated(90.0, 1,0,0);
glutWireTorus(0.1, 0.3, 10,10);
glPopMatrix();
glTranslated(1.0, 0 ,0); // dodecahedron at (1,0,0)
glScaled(0.15, 0.15, 0.15);
glutWireDodecahedron();

40. glPushMatrix();
glTranslated(0, 1.0 ,1.0); // small cube at (0,1,1)
glutWireCube(0.25);
glPopMatrix();
glTranslated(0, 0 ,1.0); // cylinder at (0,0,1)
GLUquadricObj * qobj;
qobj = gluNewQuadric();
gluQuadricDrawStyle(qobj,GLU_LINE);
gluCylinder(qobj, 0.2, 0.2, 0.4, 8,8);
glFlush(); }

41. //<<<<<<<<<<<<<<<< main >>>>>>>>>>>>>>>>
void main(int argc, char **argv) {
glutInit(&argc, argv);
glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB );
glutInitWindowSize(640,480);
glutInitWindowPosition(100, 100);
glutCreateWindow("Transformation testbed - wireframes");
glutDisplayFunc(displayWire);
glClearColor(1.0f, 1.0f, 1.0f,0.0f); // background is white
glViewport(0, 0, 640, 480);
glutMainLoop(); }