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2D Geometric Transformations

2D Geometric Transformations. Contents. Definition & Motivation 2D Geometric Transformation Translation Rotation Scaling Matrix Representation Homogeneous Coordinates Matrix Composition Composite Transformations Pivot-Point Rotation General Fixed-Point Scaling

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2D Geometric Transformations

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  1. 2D Geometric Transformations cgvr.korea.ac.kr

  2. Contents • Definition & Motivation • 2D Geometric Transformation • Translation • Rotation • Scaling • Matrix Representation • Homogeneous Coordinates • Matrix Composition • Composite Transformations • Pivot-Point Rotation • General Fixed-Point Scaling • Reflection and Shearing • Transformations Between Coordinate Systems cgvr.korea.ac.kr

  3. Geometric Transformation • Definition • Translation, Rotation, Scaling • Motivation – Why do we need geometric transformations in CG? • As a viewing aid • As a modeling tool • As an image manipulation tool cgvr.korea.ac.kr

  4. Example: 2D Geometric Transformation Modeling Coordinates World Coordinates cgvr.korea.ac.kr

  5. Example: 2D Scaling Modeling Coordinates Scale(0.3, 0.3) World Coordinates cgvr.korea.ac.kr

  6. Example: 2D Rotation Modeling Coordinates Scale(0.3, 0.3) Rotate(-90) World Coordinates cgvr.korea.ac.kr

  7. Example: 2D Translation Modeling Coordinates Scale(0.3, 0.3) Rotate(-90) Translate(5, 3) World Coordinates cgvr.korea.ac.kr

  8. Example: 2D Geometric Transformation Modeling Coordinates Again? World Coordinates cgvr.korea.ac.kr

  9. Example: 2D Geometric Transformation Modeling Coordinates Scale Translate Scale Rotate Translate World Coordinates cgvr.korea.ac.kr

  10. Basic 2D Transformations • Translation • Scale • Rotation • Shear cgvr.korea.ac.kr

  11. Basic 2D Transformations • Translation • Scale • Rotation • Shear Transformations can be combined (with simple algebra) cgvr.korea.ac.kr

  12. Basic 2D Transformations • Translation • Scale • Rotation • Shear cgvr.korea.ac.kr

  13. Basic 2D Transformations • Translation • Scale • Rotation • Shear cgvr.korea.ac.kr

  14. Basic 2D Transformations • Translation • Scale • Rotation • Shear cgvr.korea.ac.kr

  15. Basic 2D Transformations • Translation • Scale • Rotation • Shear cgvr.korea.ac.kr

  16. Matrix Representation • Represent a 2D Transformation by a Matrix • Apply the Transformation to a Point Transformation Matrix Point cgvr.korea.ac.kr

  17. Matrix Representation • Transformations can be combined by matrix multiplication Transformation Matrix Matrices are a convenient and efficient way to represent a sequence of transformations cgvr.korea.ac.kr

  18. 2×2 Matrices • What types of transformations can be represented with a 2×2 matrix? 2D Identity 2D Scaling cgvr.korea.ac.kr

  19. 2×2 Matrices • What types of transformations can be represented with a 2×2 matrix? 2D Rotation 2D Shearing cgvr.korea.ac.kr

  20. 2×2 Matrices • What types of transformations can be represented with a 2×2 matrix? 2D Mirror over Y axis 2D Mirror over (0,0) cgvr.korea.ac.kr

  21. 2×2 Matrices • What types of transformations can be represented with a 2×2 matrix? 2D Translation NO!! Only linear 2D transformations can be Represented with 2x2 matrix cgvr.korea.ac.kr

  22. 2D Translation • 2D translation can be represented by a 3×3 matrix • Point represented with homogeneous coordinates cgvr.korea.ac.kr

  23. Basic 2D Transformations • Basic 2D transformations as 3x3 Matrices Translate Scale Rotate Shear cgvr.korea.ac.kr

  24. Homogeneous Coordinates • Add a 3rd coordinate to every 2D point • (x, y, w) represents a point at location (x/w, y/w) • (x, y, 0) represents a point at infinity • (0, 0, 0) Is not allowed y 2 (2, 1, 1) or (4, 2, 2) or (6, 3, 3) 1 x 1 2 Convenient Coordinate System to Represent Many Useful Transformations cgvr.korea.ac.kr

  25. Matrix Composition • Transformations can be combined by matrix multiplication • Efficiency with premultiplication • Matrix multiplication is associative cgvr.korea.ac.kr

  26. Matrix Composition • Rotate by  around arbitrary point (a,b) • Scale by sx, sy around arbitrary point (a,b) (a,b) (a,b) cgvr.korea.ac.kr

  27. (xr,yr) (xr,yr) (xr,yr) (xr,yr) Pivot-Point Rotation Translate Rotate Translate cgvr.korea.ac.kr

  28. (xf,yf) (xf,yf) (xf,yf) (xf,yf) General Fixed-Point Scaling Translate Scale Translate cgvr.korea.ac.kr

  29. Reflection • Reflection with respect to the axis • x• y• xy y 1 y y 1 1’ 1’ 2 3 2 3 3’ 2 3’ 2 x x x 3 2’ 3’ 1 2 1’ cgvr.korea.ac.kr

  30. y y y y 1 2 3 x x x x 2’ 3’ 1’ Reflection • Reflection with respect to a Line • Clockwise rotation of 45  Reflection about the x axis  Counterclockwise rotation of 45 y=x cgvr.korea.ac.kr

  31. y y y y x x x x Shear • Converted to a parallelogram x’ = x + shx · y, y’ = y • Transformed to a shifted parallelogram (Y = Yref) x’ = x + shx ·(y-yref), y’ = y (1,1) (0,1) (2,1) (3,1) (0,0) (1,0) (0,0) (1,0) (Shx=2) (2,1) (1,1) (1,1) (0,1) (1/2,0) (3/2,0) (0,0) (1,0) (0,-1) (Shx=1/2, yref=-1) cgvr.korea.ac.kr

  32. y x Shear • Transformed to a shifted parallelogram (X = Xref) x’ = x, y’ = shy ·(x-xref) + y (1,2) (0,3/2) y (1,1) (0,1) (0,1/2) (1,1) x (0,0) (1,0) (-1,0) (Shy=1/2, xref=-1) cgvr.korea.ac.kr

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