Introduction to Computer Graphics CS 445 / 645 Lecture 11 Chapter 7: Camera Transformations

1. Introduction to Computer GraphicsCS 445 / 645Lecture 11Chapter 7: Camera Transformations Angel: Interactive ComputerGraphics

2. Announcements • Midterm is one week from today • Old tests will be available • Detailed list of chapter coverage made available • Homework on Thursday to sample coverage • Material covers up to today’s class

3. Taxonomy of Projections FVFHP Figure 6.10

4. Taxonomy of Projections

5. Parallel Projection • Center of projection is at infinity • Direction of projection (DOP) same for all points View Plane DOP Angel Figure 5.4

6. Orthographic Projections • DOP perpendicular to view plane Front Angel Figure 5.5 Top Side

7. Oblique Projections • DOP not perpendicular to view plane Cavalier (DOP  = 45o) tan(a) = 1 Cabinet (DOP  = 63.4o) tan(a) = 2 H&B

8. Oblique Parallel-Projection Oblique View Volume Transformed View Volume

9. Oblique Parallel-Projection Oblique View Volume Transformed View Volume

10. Transformation Matrix Vp HB Matrix 7-13

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

12. 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)

13. Orthographic: Screen Space Transformation (Normalization) HB Matrix: 7-7 • 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

14. 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

15. 3-Point Perspective 2-Point Perspective 1-Point Perspective Perspective Projection • How many vanishing points? Angel Figure 5.10

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

17. 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:

18. 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

19. 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?

20. A Perspective Projection Matrix • Answer:

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

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

23. Perspective vs. Parallel • Perspective projection • Size varies inversely with distance - looks realistic • Distance and angles are not (in general) preserved • Parallel lines do not (in general) remain parallel • Parallel projection • Good for exact measurements • Parallel lines remain parallel • Angles are not (in general) preserved • Less realistic looking

24. Classical Projections Angel Figure 5.3

25. 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 • GL_MODELVIEW: Object->Camera • GL_PROJECTION: Camera->Screen • glViewport(0,0,w,h): Screen->Device

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

27. 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

28. 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

29. 2 Basic Steps • Align the two coordinate frames by rotation

30. 2 Basic Steps • Translate to align origins

31. 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

32. 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?

33. 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?

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

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

36. 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

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

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

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