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Affine Transformations

Affine Transformations. Jim Van Verth NVIDIA Corporation ( jim@essentialmath.com ). Topics. What’s an affine transformation? How to generate various forms Things to watch out for. What is it?. A mapping between affine spaces Preserves lines (& planes) Preserves parallel lines

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Affine Transformations

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  1. Affine Transformations Jim Van Verth NVIDIA Corporation (jim@essentialmath.com)

  2. Topics • What’s an affine transformation? • How to generate various forms • Things to watch out for

  3. What is it? • A mapping between affine spaces • Preserves lines (& planes) • Preserves parallel lines • But not angles or distances • Can represent as • (note we’re using column vectors)

  4. Affine Space • Collection of points and vectors • Can be represented using frame • Within frame, vector and point j = (0,1) O i = (1,0)

  5. Affine Transformation • Key idea: map from space to space by using frames • Note how axes change (A) • Note how origin changes (y)

  6. Example: Translation • Axes don’t change • Origin moves by y y

  7. Example: Rotation • Axes change • Origin doesn’t move • But what is A?

  8. Example: Rotation • Follow the axes: • New axes go in columns of matrix (-sin q, cos q) (cos q, sin q) q q

  9. Example: Scale • Axes change • Origin doesn’t move • Again, what is A?

  10. Example: Scale • Follow the axes: • New axes go in columns of matrix (0, 0.5) (1.5, 0)

  11. Example: Reflection • Axes change • Origin doesn’t move • But what, oh what, is A?

  12. Example: Reflection • Follow the axes: • New axes go in columns of matrix (0, 1) (-1, 0)

  13. Example: Shear • Axes change • Origin doesn’t move • Hey, hey, what is A?

  14. Example: Shear • Follow the axes: • New axes go in columns of matrix (0.5, 1) (1, 0)

  15. Transform Types • Rigid-body transformation • Translation • Rotation • Deformable transformation • Scale • Reflection • Shear

  16. Combining Transforms • Simple function composition

  17. Combining Transforms • Order is important! • Scale, then rotate

  18. Combining Transforms • Order is important! • Rotate, then scale

  19. Local and World Frames • Objects built in local frame • Want to place in world frame • Local-to-world transformation • Same basic idea: determine where axes and origin end up in world frame

  20. World to Local Frame • Similar, but in reverse • Want transformation relative to local frame • Often easier to just take the inverse • Example: world-to-view transformation

  21. Inverse • Reverses the effect of a transformation • Easy to do with formula: • For matrix, just use matrix inverse

  22. Matrix Form • Necessary for many APIs • Is easy • Concatenate by matrix multiply • Can be faster on vector architectures • Takes more storage, though

  23. Object-centered Transform • Often get this case • Already have local-to-world transform • Want rotate/scale/whatever around local origin, not world origin • How?

  24. Object-Oriented Transform • One way • Translate to origin • Rotate there • Translate back

  25. Object-Oriented Transform • Using formula: • Translate to origin • Rotate there • Translate back • This works with arbitrary center in world frame!

  26. Alternate Format • Problem: want to • Translate object in space and change rotation and scale arbitrarily • Handle rotation/scale separately • Can’t do easily with matrix format or Ax + y form • Involves SVD, Polar decomposition • Messy, but we can do better using…

  27. Rigid Body Transforms • Any sequence of translation and rotation transformations • Not scale, reflection or shear • Object shape is not affected (preserves angles and lengths) • Usually include uniform scale despite this

  28. Alternate Format • Scale – one uniform scale factor s • Rotation – matrix R • Translation – single vector t

  29. Alternate Format • Want to concatenate transforms T1, T2 in this form, or • Do this by

  30. Alternate Format • Advantages • Clear what each part does • Easier to change individual elements • Disadvantages • Eventually have to convert to 4x4 matrix anyway for renderer/video card • 4x4 faster on vector architecture

  31. Alternate Format • Matrix conversion

  32. Inverting Rigid Body Xforms • We can easily invert our rigid body transforms symbolically:

  33. Inverting Rigid Body Xforms (2) • In fact, the result itself may be written as a rigid body transform:

  34. References • Rogers, F. David and J. Alan Adams, Mathematical Elements for Computer Graphics, 2nd Ed, McGraw-Hill, 1990. • Watt, Alan, 3D Computer Graphics, Addison-Wesley, Wokingham, England, 1993.

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