1. Computer Graphics (Spring 2003)
COMS 4160, Lecture 9: Transformations 2
Ravi Ramamoorthi

2. Motivation Modeling Objects in convenient coordinate system
Transform into world space, eye space
Viewing: project world space into 2D rectangle
Virtual cameras
Types of projections (perspective, orthographic)
Window system

3. Rotations Very brief overview (not in detail)
For little more detail, some useful websites:

4. Rotations 2D (simple) 3D (complicated)
Matrix representation, commutative
3D (complicated)
Non-commutative
Counterclockwise
Matrix reps for axis transforms

5. Euler Angles Rotate about Z, then new X, then new Z [Goldstein pp 107]
Product of axis transforms
Efficient, 3 parameters
Bad for numerics, splining

6. Axis-Angle Single axis, angle about it
3 params (axis is normalized)
Somewhat intuitive
Many of same numerical problems as Euler-Angles
Formula (axis-angle to rotation matrix)
Dual Matrix of v

7. Viewing 3D Pipeline (pp 230 FvDFH) Projection 3D -> 2D
Clip: near and far planes (viewing volume)

8. The Whole Viewing Pipeline
Eye coordinates
Model coordinates
Perspective
transformation
Model
transformation
Screen coordinates
World coordinates
Viewport
transformation
Camera
transformation
Window coordinates
Raster
transformation
Device coordinates
Slide courtesy Greg Humphreys

9. Projections To lower dimensional space Taxonomy [Fig 6.13, pp 237]
Preserve straight lines
Trivial example: Drop one coordinate
We are concerned with 3D -> 2D
Taxonomy [Fig 6.13, pp 237]
Perspective [realistic images, no proportions preserved]
Parallel [preserves proportions]
Orthographic (common in graphics)

10. Perspective Center of projection, rays, projection plane
Distance from center of projection to projection plane is finite. If infinite, projection rays parallel
Foreshortening: Key feature
Size inverse proportional to distance
Cameras, visual system are similar

11. Perspective or Parallel?
If the distance between the center of projection and the projection plane is finite, the projection is called a perspective projection. Otherwise, it’s a parallel projection. A A A’ A’ B B B’ B’
C (at infinity) C
Parallel
Perspective

12. Perspective Distortions
Lines remain lines, everything else is distorted…
Vanishing points for parallel lines
Shape distortions
Parallel Projections (parallel lines are parallel)
Not realistic but preserve size (useful in mechanical drawings)

13. Parallel (Orthographic)
Projection vector (direction of rays)
Perpendicular to projection plane : Orthographic
Otherwise, oblique.
Far off objects in perspective are approx. orthographic
Uniform foreshortening
Parallelism preserved, angles are not
More info in book
We now concentrate on perspective…

14. Perspective mathematically 1
Intuition: divide by z
Eye as a point (or pinhole camera)
Easy in eye space (homogeneous coordinates)
Simple form: equation 6.4 page 255
Full mathematics next week

15. Overhead View of Our Screen
Looks like we’ve got some nice similar triangles here!
Next few slides courtesy Greg Humphreys

16. The Perspective Matrix
This “division by z” can be accomplished by a 4x4 matrix
We’re in homogeneous coordinates, so if we multiply some point by P, we get
Recall that to get the actual 3D point, you divide by w, giving , which was the point on our screen. Now just drop the third coordinate!

17. Viewing in OpenGL
OpenGL has multiple matrix stacks - transformation functions right-multiply the top of the stack
Two most important stacks: GL_MODELVIEW and GL_PROJECTION
Points get multiplied by the modelview matrix first, and then the projection matrix
This is only really necessary for convenience -- you could ignore all of this and put your entire viewing matrix into one or the other. However, if one is changing and the other one isn’t, you’ll save time

18. OpenGL Example void SetUpViewing() {
// The viewport isn’t a matrix, it’s just state...
glViewport( 0, 0, window_width, window_height );
// Set up camera transformation first
glMatrixMode( GL_PROJECTION );
glLoadIdentity();
gluPerspective( 60, 1, 1, 1000 ); // fov, aspect, near, far
gluLookAt( 3, 3, 2, // eye point
0, 0, 0, // look at point
0, 0, 1 ); // up vector
// Set up the model transformation
glMatrixMode( GL_MODELVIEW );
glRotatef( theta, 0, 0, 1 ); // rotate the model
glScalef( zoom, zoom, zoom ); // scale the model }