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Computer Graphics - PowerPoint PPT Presentation

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Computer Graphics
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  1. Computer Graphics 3D Transformations

  2. 3D Translation • Remembering 2D transformations -> 3x3 matrices, take a wild guess what happens to 3D transformations. T=(tx, ty, tz)

  3. 3D Scale S=(sx, sy, sz)

  4. 3D Rotations What does a rotation in 3D mean? Q: How do we specify a rotation? R=(rx, ry, rz, ) A: We give a vector to rotate about, and a theta that describes how much we rotate.  Q: Since 2D is sort of like a special case of 3D, what is the vector we’ve been rotating about in 2D?

  5. Rotations about the Z axis What do you think the rotation matrix is for rotations about the z axis? R=(0,0,1,) 

  6. Rotations about the X axis Let’s look at the other axis rotations R=(1,0,0,) 

  7. Rotations about the Y axis R=(0,1,0,) 

  8. Rotations around a principle axis

  9.   Rotations for an arbitrary axis u Steps: 1. Normalize vector u 2. Compute  3. Compute  4. Create rotation matrix

  10. Vector Normalization • Given a vector v, we want to create a unit vector that has a magnitude of 1 and has the same direction as v. Let’s do an example.

  11. Computing the Rotation Matrix • 1. Normalize u ( the vector we are rotating around ) • 2. Compute Rx(x) • 3. Compute Ry(y) • 4. Generate Rotation Matrix

  12. Rotations for an arbitrary axis • Unit vectors are made of direction cosines:

  13. Rotations for an arbitrary axis

  14. Rotations for an arbitrary axis

  15. Rotation Matrix

  16. Axis-Angle Rotations in OpenGL glRotatef( angle, x, y, z);

  17. Euler Angles More intuitive: represent rotations by 3 angles, one for each axis glRotatef(anglex,1,0,0); glRotatef(angley,0,1,0); glRotatef(anglez,0,0,1); Think: if we have a torus unstranformed at the origin, what will the torus look like if you have anglex=90, angley=90, and anglez=90 Transformed

  18. Gimbal Lock Rotate around these Initial orientation (x=blue, y=green, z = red) How can we gain altitude here?

  19. Gimbal Lock in OpenGL What is the problem here? glRotatef(anglex,1,0,0); glRotatef(angley,0,1,0); glRotatef(anglez,0,0,1);

  20. Avoiding Gimbal Lock: Quaternions • Quaternions represent 3D rotations in 4D using imaginary numbers – a 4-tuple Q = (w,x,y,z) • Convert Angle-axis -> Quaternion • (,x,y,z) -> ( cos( /2), xsin(/2), ysin(/2), zsin(/2) ) • Why quaternions? • No gimbal lock • Smooth interpolation between rotations (for animation)

  21. Quaternions • Quaternions have rules for multiplication, inversion, etc • See: http://www.j3d.org/matrix_faq/matrfaq_latest.html • Typical usage: • Convert from euler, matrix, or angle-axis • Do rotations • Convert to angle-axis or matrix

  22. What is going on here? frame1 frame3 frame2

  23. gluLookAt Orients and positions the “camera” gluLookat( eyex, eyey, eyez, centerx, centery, centerz, upx,upy, upz); eye – the position of the camera in world coordinates center – the camera is pointed at this point up – the direction defined to be up for the camera

  24. gluLookAt – in matrix form

  25. u  Cross Products Given two vectors, the cross product returns a vector that is perpendicular to the plane of the two vectors and with magnitude equal to the area of the parallelogram formed by the two vectors.

  26. Different Phases:Model Definition

  27. Different Phases:Transformations

  28. Different Phases:Projection

  29. Different Phases:Projection

  30. Different Phases:Rasterization

  31. Different Phases:Scan Conversion

  32. What are the steps needed?

  33. Let’s Examine the Camera • If I gave you a world, and said I want to “render” it from another viewpoint, what information do I have to give you? • Position • Which way we are looking • Which way is “up” • Aspect Ratio • Field of View • Near and Far

  34. Camera View Right View Up View Normal View Direction

  35. Camera View Up View Right What are the vectors?

  36. Graphics Pipeline So Far Object Object Coordinates Transformation Object -> World World World Coordinates Projection Xform World -> Projection Normalize Xform & Clipping Projection -> Normalized Camera Projection Coordinates Viewport Normalized Coordinates Viewport Transform Normalized -> Device Screen Device Coordinates

  37. Transformation World->Camera View Right View Up View Normal View Direction

  38. Transformation World->Camera View Right = u View Up = V View Direction = -N