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Two dimensional geometric transformations

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Two dimensional geometric transformations

  1. 1. Two-Dimensional Geometric Transformations
  2. 2. Two-Dimensional Geometric Transformations • Basic Transformations • Translation • Rotation • Scaling • Composite Transformations • Other transformations • Reflection • Shear
  3. 3. Translation •Translation transformation •Translation vector or shift vector T = (tx, ty) •Rigid-body transformation •Moves objects without deformation y y P’ T T p x x
  4. 4. Rotation Rotation transformation (anticlockwise) x’=rcos(θ+Φ)= rcosθcosΦ - rsin θsinΦ y’=rsin(θ+Φ)= rcosθsinΦ + rsin θcosΦ y x=rcosΦ y=rsin Φ P’ (x’,y’) x’=x cosθ - ysinθ y’=xsinθ+ycosθ θ P(x,y) r Φ x P’= R· P
  5. 5. Rotation General Pivot point rotation x’=xr+(x- xr)cosθ - (y- yr)sinθ y P’ (x’,y’) y’=yr+(x- xr)sinθ+(y- yr)cosθ θ P(x,y) r Φ (xr,yr) x
  6. 6. Scaling Scaling transformation  Scaling factors, sx and sy y y x x
  7. 7. Scaling  Fixed point
  8. 8. Matrix Representations and Homogeneous Coordinates •Homogeneous Coordinates •Matrix representations • Translation
  9. 9. Matrix Representations  Matrix representations  Scaling  Rotation
  10. 10. Composite Transformations Translations
  11. 11. Composite Transformations Scaling
  12. 12. Composite Transformations Rotations
  13. 13. General Pivot-Point Rotation •Rotations about any selected pivot point (xr,yr) •Translate-rotate-translate
  14. 14. General Pivot-Point Rotation
  15. 15. General Fixed-Point Scaling Scaling with respect to a selected fixed position (x f,yf)
  16. 16. General Fixed-Point Scaling Translate-scale-translate
  17. 17. Concatenation Properties Matrix multiplication is associative. A·B ·C = (A·B )·C = A·(B ·C) Transformation products may not be commutative  Be careful about the order in which the composite matrix is evaluated.  Except for some special cases: Two successive rotations Two successive translations Two successive scalings rotation and uniform scaling
  18. 18. Concatenation Properties Reversing the order  A sequence of transformations is performed may affect the transformed position of an object.
  19. 19. Reflection A transformation produces a mirror image of an object. Axis of reflection A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object 180 0 about the reflection axis. Rotation path Axis in xy plane: in a plane perpendicular to the xy plane. Axis perpendicular to xy plane: in the xy plane.
  20. 20. Reflection Reflection about the x axis
  21. 21. Reflection Reflection about the y axis
  22. 22. Reflection Reflection relative to the coordinate origin
  23. 23. Reflection Reflection of an object relative to an axis perpendicular to the xy plane through P rfl
  24. 24. Reflection Reflection about the line y = x
  25. 25. Shear The x-direction shear relative to x axis If shx = 2:
  26. 26. Shear The x-direction shear relative to y = yref If shx = ½ yref = -1: 1 1/2 3/2
  27. 27. Shear The y-direction shear relative to x = xref 3/2 If shy = ½ xref = -1: 1/2 1

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