1. Graphics Graphics Lab @ Korea University cgvr.korea.ac.kr 2D Geometric Transformations 고려대학교 컴퓨터 그래픽스 연구실

2. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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

3. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Geometric Transformation Definition 물체의 좌표를 바꾸는 것 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

4. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Example: 2D Geometric Transformation Modeling Coordinates World Coordinates

5. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Example: 2D Scaling Modeling Coordinates World Coordinates Scale(0.3, 0.3)

6. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Example: 2D Rotation Modeling Coordinates Scale(0.3, 0.3) Rotate(-90) World Coordinates

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

8. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Example: 2D Geometric Transformation Modeling Coordinates World Coordinates Again?

9. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Example: 2D Geometric Transformation Modeling Coordinates World Coordinates Scale Translate Scale Rotate Translate

10. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Basic 2D Transformations Translation Scale Rotation Shear

11. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Basic 2D Transformations Translation Scale Rotation Shear Transformations can be combined (with simple algebra)

12. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Basic 2D Transformations Translation Scale Rotation Shear

13. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Basic 2D Transformations Translation Scale Rotation Shear

14. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Basic 2D Transformations Translation Scale Rotation Shear

15. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Basic 2D Transformations Translation Scale Rotation Shear

16. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Matrix Representation Represent a 2D Transformation by a Matrix Apply the Transformation to a Point Transformation Matrix Point

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

18. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 2×2 Matrices What types of transformations can be represented with a 2×2 matrix? 2D Identity 2D Scaling

19. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 2×2 Matrices What types of transformations can be represented with a 2×2 matrix? 2D Rotation 2D Shearing

20. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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)

21. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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

22. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 2D Translation 2D translation can be represented by a 3×3 matrix Point represented with homogeneous coordinates

23. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Basic 2D Transformations Basic 2D transformations as 3x3 Matrices Translate Shear Scale Rotate

24. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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 12 1 2 x y (2, 1, 1) or (4, 2, 2) or (6, 3, 3) Convenient Coordinate System to Represent Many Useful Transformations

25. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Linear Transformations Linear transformations are combinations of … Scale Rotation Shear, and Mirror Properties of linear transformations Satisfies: Origin maps to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition

26. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Affine Transformations Affine transformations are combinations of Linear transformations, and Translations Properties of affine transformations Origin does not map to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition

27. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Projective Transformations Projective transformations… Affine transformations, and Projective warps Properties of projective transformations Origin does not map to origin Lines map to lines Parallel lines do not necessarily remain parallel Ratios are not preserved Closed under composition

28. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Matrix Composition Transformations can be combined by matrix multiplication Efficiency with premultiplication Matrix multiplication is associative

29. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Matrix Composition Rotate by  around arbitrary point (a,b) Scale by sx, sy around arbitrary point (a,b) (a,b)

30. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Pivot-Point Rotation TranslateRotateTranslate (x r,y r )

31. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr General Fixed-Point Scaling TranslateScaleTranslate (x f,y f )

32. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Reflection Reflection with respect to the axis x 축에 대한 반사 y 축에 대한 반사 xy 축 ( 원점 ) 에 대한 y 축에 대한 반사 x 축에 대한 반사원점에 대한 반사 x y1 32 1’ 3’2’ x y 1 32 1’ 3’2 x y 3 1’ 3’2 1 2

33. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Reflection with respect to a Line Clockwise rotation of 45  Reflection about the x axis  Counterclockwise rotation of 45 Reflection y=x 에 대한 반사 x y1 32 1’ 3’2’ x y x y x y

34. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Shear Converted to a parallelogram x’ = x + sh x · y, y’ = y Transformed to a shifted parallelogram (Y = Yref) x’ = x + sh x · ( y-y ref ), y’ = y x 축으로 밀림 (Sh x =2) 선분에 대한 밀림 (Sh x =1/2, y ref =-1) x y x y x y x y (0,0)(1,0) (1,1) (0,1) ( 0,0 )( 1,0 ) ( 1,1 ) (0,1) (0,0)(1,0) (3,1)(2,1) ( 1/2,0 ) ( 3/2,0 ) ( 2,1 ) ( 1,1 ) (0,-1)

35. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Shear Transformed to a shifted parallelogram (X = Xref) x’ = x, y’ = sh y · ( x-x ref ) + y 선분에 대한 밀림 (Sh y =1/2, x ref =-1) x y x y (-1,0) ( 0,0 )( 1,0 ) ( 1,1 ) ( 0,1 ) ( 0,1/2 )( 1,1 ) ( 1,2 ) ( 0,3/2 )