1. Introduction to Computer Graphics: Viewing Transformations Rama C
Introduction to Computer Graphics: Viewing Transformations Rama C. Hoetzlein, 2011 Cornell University Lecture Notes

2. Topics Covered
1. Vector & Matrix Algebra 2. Affine Transformations 3. Homogeneous Coordinates 4. Perspective Projection 5. Projection Pipeline

16. Complete Perspective Matrix
Lets us specify all characteristics of the projective volume.
glMatrixMode ( GL_PROJECTION ) glFrustum ( l, r, b, t, near, far);
gluPerspective ( fov, asp, near, far); gluOrtho2D ( l, r, b, t );
OpenGL Red Book

17. Viewing Pipeline
What does the world look like through the camera?

18. scene with camera
camera image
Goal: Create a projected image of the scene from viewpoint of camera.

20. Model Space Local objects coordinates centered on the origin X+ Z+
drawCylinder () p

21. World Space { Objects in their place in the world
How do we look through this camera?
camera
glTranslatef ( .. ) { M
glRotatef ( .. )
drawCylinder ()
p = M p
world

22. Camera Transforms Step 1. Translate everything to camera origin

23. Camera Transforms Step 2. Rotate around Y-axis by camera angle

24. Camera Transforms Step 3. Rotate around X-axis by camera tilt

25. { View Space Objects oriented with the camera along the Z-axis
glMatrixMode (
MODELVIEW ) V {
gluLookAt (..)
glTranslatef ( .. ) M {
glRotatef ( .. )
drawCylinder () V {
p = Rcz Rcy Tcam M p
p = V M p
view
view

26. Normalized Projection Space
Objects multiplied by perspective matrix, to fit in unit cube.
glMatrixMode (
PROJECTION ) P {
gluPerspective(..)
glMatrixMode (
MODELVIEW ) V {
gluLookAt (..)
glTranslatef ( .. ) M {
glRotatef ( .. )
drawCylinder ()
p = P V M p
eye

27. Clipping Objects clipped to unit cube for rasterization.
glMatrixMode (
PROJECTION ) P {
gluPerspective(..)
glMatrixMode (
MODELVIEW ) V {
gluLookAt (..)
Look here!
glTranslatef ( .. ) M {
glRotatef ( .. )
drawCylinder ()
p = clip ( P V M p )
clip

28. Screen Space
Objects as they appear (flattened) on the screen. Divide by w to get 2D coordinates.
glMatrixMode (
PROJECTION ) P {
gluPerspective(..)
glMatrixMode (
MODELVIEW ) V {
gluLookAt (..)
glTranslatef ( .. ) M {
glRotatef ( .. )
drawCylinder ()
p = divw ( clip ( P V M p ) )
screen

29. Summary…

30. VIEWING PIPELINE Model Space World Space View Space M V P Screen Space
Clipping
persp. div
clip
Normalized Dev. Coordinates
Depth Sorting
p = divw ( clip ( P V M p ) )
screen
Rasterization

31. VIEWING PIPELINE – in OpenGL
OpenGL has two stored matricies, one for the projection, and one for the combined model-view.
glMatrixMode ( GL_PROJECTION ); gluPerspective ( .. );
glMatrixMode ( GL_MODELVIEW );
gluLookAt ( fx, fy, fz, tx, ty, tz );
glRotatef ( 10, 0, 0, 1 );
drawCylinder ( .. );
tell OpenGL to setup projection creates a perspective matrix (P)
tell OpenGL to setup model-view creates a view matrix (V) assigns rotation as model matrix (M) requests polygons to be drawn
The clipping and divide by w happen automatically in hardware.

32. MULTIPLE OBJECTS Whats wrong with this?
The scene has one view matrix (V) for the camera, but.. each object has its own MODELVIEW matrix, VM.
glMatrixMode ( GL_PROJECTION ); gluPerspective ( .. );
glMatrixMode ( GL_MODELVIEW );
gluLookAt ( fx, fy, fz, tx, ty, tz );
glRotatef ( 10, 0, 0, 1 );
drawCylinder ( .. );
glRotatef ( 20, 1, 0, 0 ); P V M1 M2
Whats wrong with this?

33. MULTIPLE OBJECTS
Right! (I hope).. Each object needs to set up the proper VM matrix.
glMatrixMode ( GL_PROJECTION ); gluPerspective ( .. );
glMatrixMode ( GL_MODELVIEW );
gluLookAt ( fx, fy, fz, tx, ty, tz );
glGetMatrixfv ( view_matrix );
glLoadMatrixf ( view_matrix );
glRotatef ( 10, 0, 0, 1 );
drawCylinder ( .. );
glRotatef ( 20, 1, 0, 0 ); P V M1
V M2
obj1 = P V M1 obj2 = P V M2
save V