CS 445 / 645: Introductory Computer Graphics

1. CS 445 / 645: Introductory Computer Graphics Geometric Transforms

2. Matrix Manipulation - Example • Drawing a car with wheels and lugnuts draw_wheel( ); for (j=0; j<5; j++) { glPushMatrix (); glRotatef(72.0*j, 0.0, 0.0, 1.0); glTranslatef (3.0, 0.0, 0.0); draw_bolt ( ); glPopMatrix ( );

3. Matrix Manipulation - Example draw_wheel( ); for (j=0; j<5; j++) { glPushMatrix (); glRotatef(72.0*j, 0.0, 0.0, 1.0); glTranslatef (3.0, 0.0, 0.0); draw_bolt ( ); glPopMatrix ( ); Global – Bottom Up Start Rot Trans

4. Matrix Manipulation - Example draw_wheel( ); for (j=0; j<5; j++) { glPushMatrix (); glRotatef(72.0*j, 0.0, 0.0, 1.0); glTranslatef (3.0, 0.0, 0.0); draw_bolt ( ); glPopMatrix ( ); Local – Top Down Start Rot Trans

5. Translation Matrices? • We can composite scale matrices just as we did rotation matrices • But how to represent translation as a matrix? • Answer: with homogeneous coordinates

6. Homogeneous Coordinates • Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector (Note that typically w = 1 in object coordinates)

7. Homogeneous Coordinates • Homogeneous coordinates seem unintuitive, but they make graphics operations much easier • Our transformation matrices are now 4x4:

8. Homogeneous Coordinates • Homogeneous coordinates seem unintuitive, but they make graphics operations much easier • Our transformation matrices are now 4x4:

9. Homogeneous Coordinates • Homogeneous coordinates seem unintuitive, but they make graphics operations mucheasier • Our transformation matrices are now 4x4:

10. Homogeneous Coordinates • Homogeneous coordinates seem unintuitive, but they make graphics operations much easier • Our transformation matrices are now 4x4:

11. Homogeneous Coordinates • How can we represent translation as a 4x4 matrix? • A: Using the rightmost column:

12. Translation Matrices • Now that we can represent translation as a matrix, we can composite it with other transformations • Ex: rotate 90° about X, then 10 units down Z:

13. Translation Matrices • Now that we can represent translation as a matrix, we can composite it with other transformations • Ex: rotate 90° about X, then 10 units down Z:

14. Translation Matrices • Now that we can represent translation as a matrix, we can composite it with other transformations • Ex: rotate 90° about X, then 10 units down Z:

15. Translation Matrices • Now that we can represent translation as a matrix, we can composite it with other transformations • Ex: rotate 90° about X, then 10 units down Z:

16. Transformation Commutativity • Is matrix multiplication, in general, commutative? Does AB = BA? • What about rotation, scaling, and translation matrices? • Does RxRy = RyRx? • Does RAS = SRA ? • Does RAT = TRA ?

17. More On Homogeneous Coords • What effect does the following matrix have? • Conceptually, the fourth coordinate wis a bit like a scale factor

18. Homogenous Coordinates • In particular, increasing w makes things smaller • We think of homogenous coordinates as defining a projective space • Increasing w “getting further away” • Will come in handy for projection matrices

19. Projection Matrix • We talked about geometric transforms, focusing on modeling transforms • Ex: translation, rotation, scale, gluLookAt() • These are encapsulated in the OpenGL modelview matrix • Can also express projection as a matrix • Next few slides: representing orthographic and perspective projection with the projection matrix

20. Orthographic Projection • Simple OrthographicTransformation • Original world units are preserved • Pixel units are preferred

21. Orthographic: Screen Space Transformation left =10 m right = 20 m top=20 m (max pixx, max pixy) (height in pixels) (0, 0) bottom=10 m (width in pixels)

22. Orthographic: Screen Space Transformation • left, right, top, bottom refer to the viewing frustum in modeling coordinates • width and height are in pixel units • This matrix scales and translates to accomplish the transition in units

23. Perspective Transformation • First discovered by Donatello, Brunelleschi, and DaVinci during Renaissance • Objects closer to viewer look larger • Parallel lines appear to converge to single point

24. Perspective Projection • In the real world, objects exhibit perspective foreshortening: distant objects appear smaller • The basic situation:

25. How tall shouldthis bunny be? Perspective Projection • When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world:

26. Viewplane X P (x, y, z) x’ = ? (0,0,0) Z Perspective Projection • The geometry of the situation is that of similar triangles. View from above: • What is x’ ? d

27. Perspective Projection • Desired result for a point [x, y, z, 1]T projected onto the view plane: • What could a matrix look like to do this?

28. A Perspective Projection Matrix • Answer:

29. A Perspective Projection Matrix • Example: • Or, in 3-D coordinates:

30. Perspective Projection • Translate model units to pixel units

31. Projection Matrices • Now that we can express perspective foreshortening as a matrix, we can composite it onto our other matrices with the usual matrix multiplication • End result: a single matrix encapsulating modeling, viewing, and projection transforms

32. A 3D Scene • Notice the presence ofthe camera, theprojection plane, and the worldcoordinate axes • Viewing transformations define how to acquire the image on the projection plane

33. Viewing Transformations • Create a camera-centered view • Camera is at origin • Camera is looking along negative z-axis • Camera’s ‘up’ is aligned with y-axis

34. 2 Basic Steps • Align the two coordinate frames by rotation

35. 2 Basic Steps • Translate to align origins

36. Creating Camera Coordinate Space • Specify a point where the camera is located in world space, the eye point • Specify a point in world space that we wish to become the center of view, the lookat point • Specify a vector in worldspace that we wish to point up in camera image, the up vector • Intuitive camera movement

37. Constructing Viewing Transformation, V • Create a vector from eye-point to lookat-point • Normalize the vector • Desired rotation matrix should map this vector to [0, 0, -1]T Why?

38. Constructing Viewing Transformation, V • Construct another important vector from the cross product of the lookat-vector and the vup-vector • This vector, when normalized, should align with [1, 0, 0]TWhy?

39. Constructing Viewing Transformation, V • One more vector to define… • This vector, when normalized, should align with [0, 1, 0]T • Now let’s composite the results

40. Compositing Vectors to Form V • We know the three world axis vectors (x, y, z) • We know the three camera axis vectors (u, v, n) • Viewing transformation, V, must convert from world to camera coordinate systems

41. Compositing Vectors to Form V • Remember • Each camera axis vector is unit length. • Each camera axis vector is perpendicular to others • Camera matrix is orthogonal and normalized • Orthonormal • Therefore, M-1 = MT

42. Compositing Vectors to Form V • Therefore, rotation component of viewing transformation is just transpose of computed vectors

43. Compositing Vectors to Form V • Translation component too • Multiply it through

44. Final Viewing Transformation, V • To transform vertices, use this matrix: • And you get this: