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2D and 3D Transformation

2D and 3D Transformation. Pradondet Nilagupta Dept. of Computer Engineering Kasetsart University. Transformations and Matrices. Transformations are functions Matrices are functions representations Matrices represent linear transformation {2x2 Matrices}  {2D Linear Transformation}.

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2D and 3D Transformation

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  1. 2D and 3D Transformation Pradondet Nilagupta Dept. of Computer Engineering Kasetsart University 204481 Foundation of Computer Graphics

  2. Transformations and Matrices • Transformations are functions • Matrices are functions representations • Matrices represent linear transformation • {2x2 Matrices}  {2D Linear Transformation} 204481 Foundation of Computer Graphics

  3. Transformations (1/3) • What are they? • changing something to something else via rules • mathematics: mapping between values in a range set and domain set (function/relation) • geometric: translate, rotate, scale, shear,… • Why are they important to graphics? • moving objects on screen / in space • mapping from model space to screen space • specifying parent/child relationships • … 204481 Foundation of Computer Graphics

  4. ty tx hnew hold wold wnew Transformation (2/3) • Translation • Moving an object • Scale • Changing the size of an object xnew = xold + tx; ynew = yold + ty sx=wnew/wold sy=hnew/hold xnew = sxxold ynew = syyold 204481 Foundation of Computer Graphics

  5. (x,y) Transformation (3/3) • To rotate a line or polygon, we must rotate each of its vertices • Shear Original Data y Shear x Shear 204481 Foundation of Computer Graphics

  6. What is a 2D Linear Transform? Example 204481 Foundation of Computer Graphics

  7. Example Scale in x by 2 204481 Foundation of Computer Graphics

  8. Transformations: Translation (1/2) • A translation is a straight line movement of an object from one position to another. • A point (x,y) is transformed to the point (x’,y’) by adding the translation distances Tx and Ty: x’ = x + Tx y’ = y + Ty 204481 Foundation of Computer Graphics

  9. Transformations: Translation(2/2) • moving a point by a given tx and ty amount • e.g. point P is translated to point P’ • moving a line by a given tx and ty amount • translate each of the 2 endpoints 204481 Foundation of Computer Graphics

  10. Transformations: Rotation (1/4) • Objects rotated according to angle of rotation theta () • Suppose a point P(x,y) is transformed to the point P'(x',y') by an anti-clockwise rotation about the origin by an angle of  degrees, then: Given x = r cos  , y = r sin  x’ = x cos  – y sin  y’ = y sin  + y cos  204481 Foundation of Computer Graphics

  11. Transformations: Rotation (2/4) • Rotation P by  anticlockwise relative to origin (0,0) 204481 Foundation of Computer Graphics

  12. Transformations: Rotation (3/4) • Rotation about an arbitary pivot point (xR,yR) Step 1: translation of the object by (-xR,-yR) x1 = x - xR y1 = y - yR Step 2: rotation about the origin x2 = x1 cos(q) - y1sin (q) y2 = y1cos(q) - x1sin (q) Step 3: translation of the rotated object by (xR,yR) x’ = xr + x2 y’ = yr + y2 204481 Foundation of Computer Graphics

  13. Transformations: Rotation (4/4) • object can be rotated around an arbitrary point (xr,yr) known as rotation or pivot point by: x' = xr + (x - xr) cos(q) - (y - yr) sin (q) y' = yr + (x - xr) sin (q)+(y - yr) cos(q) 204481 Foundation of Computer Graphics

  14. Transformations: Scaling (1/5) • Scaling changes the size of an object • Achieved by applying scaling factors sx and sy • Scaling factors are applied to the X and Y co-ordinates of points defining an object’s 204481 Foundation of Computer Graphics

  15. Transformations: Scaling (2/5) • uniform scaling is produced when sx and sy have same value i.e. sx = sy • non-uniform scaling is produced when sx and sx are not equal - e.g. an ellipse from a circle. • i.e. sx sy x2 = sxx1 y2 = syy1 204481 Foundation of Computer Graphics

  16. Transformations: Scaling (3/5) Simple scaling - relative to (0,0) • General form: Ex: sx = 2 and sy=1 204481 Foundation of Computer Graphics

  17. Transformations: Scaling (4/5) • If the point (xf,yf) is to be the fixed point, the transformation is: x' = xf + (x - xf) Sx y' = yf + (y - yf) SyThis can be rearranged to give: x' = x Sx + (1 - Sx) xf y' = y Sy + (1 - Sy) yfwhich is a combination of a scaling about the origin and a translation. 204481 Foundation of Computer Graphics

  18. Transformations: Scaling (5/5) 204481 Foundation of Computer Graphics

  19. Transformation as Matrices Scale: x’ = sxx y’ = syy Rotation: x’ = xcos - ysin  y’ = xsin  + ycos  Translation: x’ = x + tx y’ = y + ty 204481 Foundation of Computer Graphics

  20. Transformations: Shear (1/2) Shear in x: 204481 Foundation of Computer Graphics

  21. Transformations: Shear (2/2) Shear in y: 204481 Foundation of Computer Graphics

  22. Shear in x then in y 204481 Foundation of Computer Graphics

  23. Shear in y then in x 204481 Foundation of Computer Graphics

  24. Homogeneous coordinate • As translations do not have a 2 x 2 matrix representation, we introduce homogeneous coordinates to allow a 3 x 3 matrix representation. • The Homogeneous coordinate corresponding to the point (x,y) is the triple (xh, yh, w) where: xh = wx yh = wyFor the two dimensional transformations we can set w = 1. 204481 Foundation of Computer Graphics

  25. Matrix representation 204481 Foundation of Computer Graphics

  26. Basic Transformation (1/3) • Translation 204481 Foundation of Computer Graphics

  27. Basic Transformation (2/3) • Rotation 204481 Foundation of Computer Graphics

  28. Basic Transformation (3/3) • Scaling 204481 Foundation of Computer Graphics

  29. Composite Transformation Suppose we wished to perform multiple transformations on a point: 204481 Foundation of Computer Graphics

  30. Example of Composite Transformation(1/3) • A scaling transformation at an arbitrary angle is a combination of two rotations and a scaling:R(-t) S(Sx,Sy) R(t) • A rotation about an arbitrary point (xf,yf) by and angle t anti-clockwise has matrix:T(-xf,-yf) R(t) T(xf,yf) 204481 Foundation of Computer Graphics

  31. Example of Composite Transformation(2/3) Reflection about the y-axis Reflection about the x-axis 204481 Foundation of Computer Graphics

  32. Example of Composite Transformation(3/3) Reflection about the origin Reflection about the line y=x 204481 Foundation of Computer Graphics

  33. Y Y Z X X Z 3D Transformation 204481 Foundation of Computer Graphics

  34. Basic 3D Transformations • Translation • Scale • Rotation • Shear • As in 2D, we use homogeneous coordinates (x,y,z,w), so that transformations may be composited together via matrix multiplication. 204481 Foundation of Computer Graphics

  35. 3D Translation and Scaling TP = (x + tx, y + ty, z + tz) SP = (sxx, syy, szz) 204481 Foundation of Computer Graphics

  36. 3D Rotation (1/4) Positive Rotations are defined as follows: Axis of rotation isDirection of positive rotation is x y to z y z to x z x to y 204481 Foundation of Computer Graphics

  37. 3D Rotation (2/4) • Rotation about x-axis Rx(ß)P 204481 Foundation of Computer Graphics

  38. 3D Rotation (3/4) • Rotation about y-axis Ry(ß)P 204481 Foundation of Computer Graphics

  39. 3D Rotation (4/4) • Rotation about z-axis Rz(ß)P 204481 Foundation of Computer Graphics

  40. 3D Shear • xy Shear: SHxyP 204481 Foundation of Computer Graphics

  41. P2 Z U P1 c u3 Y a ß u2 b u1 X Rotation About An Arbitary Axis (1/3) • Translate one end of the axis to the origin • Rotate about the y-axis and angle  • Rotate about the x-axis through an angle  204481 Foundation of Computer Graphics

  42. P2 Z Z U Z a c U u3 P1 c u3 µ Y Y a Y ß a ß u2 u2 u2 b b X u1 u1 X X Rotation About An Arbitary Axis (2/3) 4. When U is aligned with the z-axis, apply the original rotation, R, about the z-axis. 5. Apply the inverses of the transformations in reverse order. 204481 Foundation of Computer Graphics

  43. Rotation About An Arbitary Axis (3/3) T-1 Ry(ß) Rx(-µ) R Rx(µ) Ry(-ß) T P 204481 Foundation of Computer Graphics

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